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Micro-local analysis of contact Anosov flows and band
structure of the Ruelle spectrum

Frédéric Faure, Masato Tsujii

► To cite this version:

Frédéric Faure, Masato Tsujii. Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum. Communications of the American Mathematical Society, 2024, 4 (15), pp.641-745. 10.1090/cams/40 . hal-04875349

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# MICRO-LOCAL ANALYSIS OF CONTACT ANOSOV FLOWS AND BAND STRUCTURE OF THE RUELLE SPECTRUM

FRÉDÉRIC FAURE AND MASATO TSUJII

**ABSTRACT.** We develop a geometric micro-local analysis of contact Anosov flow, such as the geodesic flow on negatively curved manifold. This micro-local analysis is based on the wavepacket transform discussed by Faure and Tsujii [Ann. H. Lebesgue 6 (2023), pp. 331–426]. The main result is that the transfer operator is well approximated (in the high frequency limit) by the quantization of the Hamiltonian flow naturally defined from the contact Anosov flow and extended to some vector bundle over the symplectic set. This has a few important consequences: the discrete eigenvalues of the generator of transfer operators, called Ruelle spectrum, are structured into vertical bands. If the right-most band is isolated from the others, most of the Ruelle spectrum in it concentrates along a line parallel to the imaginary axis and, further, the density satisfies a Weyl law as the imaginary part tends to infinity. Some of these results were announced by Faure and Tsujii [C. R. Math. Acad. Sci. Paris 351 (2013), pp. 385–391].

## CONTENTS

<table><tr><td>1. Introduction</td><td>642</td></tr><tr><td>2. Micro-local analysis of a non-vanishing vector field X</td><td>661</td></tr><tr><td>3. Micro-local analysis of a contact vector field X on (M, A) near Σ = ℝA \ {0}</td><td>673</td></tr><tr><td>4. Micro-local analysis of a contact Anosov vector field X near Σ = ℝA \ {0}</td><td>683</td></tr><tr><td>5. Proof of Theorem 1.7 (band spectrum)</td><td>702</td></tr><tr><td>6. Proof of Theorem 1.5 (emergence of quantum dynamics)</td><td>706</td></tr><tr><td>7. Proof of Theorem 1.11 (Weyl law)</td><td>707</td></tr><tr><td>8. Proof of Theorem 1.13 (accumulation of eigenvalues on narrower bands)</td><td>716</td></tr><tr><td>9. Proof of Theorem 1.18 (horocycle operators)</td><td>717</td></tr><tr><td>Appendix A. General notations used in this paper</td><td>719</td></tr><tr><td>Appendix B. More information about flows</td><td>719</td></tr><tr><td>Appendix C. Bargmann transform and metaplectic operators</td><td>721</td></tr><tr><td>Appendix D. Linear contracting maps</td><td>738</td></tr><tr><td>Acknowledgments</td><td>742</td></tr><tr><td>References</td><td>742</td></tr></table>

Received by the editors May 25, 2022, and, in revised form, September 14, 2023, March 6, 2024, and August 24, 2024.

2020 Mathematics Subject Classification. Primary 37D20, 37D35, 37C30, 81Q20, 81Q50.

**Key words and phrases.** Transfer operator, Ruelle resonances, decay of correlations, semi-classical analysis.

The first author was partially supported by French ANR Adyct, Grant Number ANR-20-CE40-0017 during this work. The second author was partially supported by JSPS KAKENHI Grant Number 15H03627 and 22340035 during this work.
# 1. INTRODUCTION

## 1.1. Broad mathematical context.
In this paper we consider a **deterministic chaotic dynamical system** defined by a contact Anosov vector field $X$ on a closed manifold $M$ that generates a flow $\phi^t : M \to M$. A typical example is the **geodesic flow for a negatively curved Riemannian manifold**. These flows possess the Anosov property also known as uniform hyperbolicity: each trajectory is very unstable with sensitivity to initial conditions, and most of them look complex and random (i.e. chaotic) even though this system is deterministic. Long time prediction of the dynamics therefore seems difficult. For such chaotic dynamical systems a fruitful approach initiated in the 1970s has been to consider not individual orbits but a smooth collection of them, i.e. the evolution of a cloud of points under the flow. It appears that this cloud of points equidistributes over long time, i.e., converges weakly to some measure called equilibrium and thus providing a predictive tool for these systems. We can moreover refine this description by considering corrections to this evolution towards the equilibrium, giving an asymptotic series that describes the long time behavior of the cloud of points. Technically the clouds of points will be modeled by a smooth function $u \in C^\infty(M; \mathbb{C})$ (or more generally by a smooth section of some arbitrary vector bundle) and we are interested in describing the evolution $u \circ \phi^t$ for long times $t \gg 1$ in the weak sense, i.e., “correlation functions” $t \in \mathbb{R} \mapsto \langle v|u \circ \phi^t \rangle \in \mathbb{C}$ for any smooth test function $v$.

Important progress has been made by Pollicott, Ruelle and others who were able to describe these correlation functions using an asymptotic expansion over some discrete set of “Pollicott-Ruelle resonances”, which are the poles of the resolvent $z \in \mathbb{C} \to (z-X)^{-1}$ and more recently, that appear to be the discrete eigenvalues of the vector field $X: \mathcal{H} \to \mathcal{H}$ in a suitable Hilbert space $\mathcal{H}$ (an anisotropic Sobolev space) [37–39], [2].

More recently researchers have been able to provide more precise information on the location of these eigenvalues and associated eigenvectors for some specific models. This paper presents the special case where the vector field $X$ is an Anosov contact vector field. We show that the Pollicott-Ruelle eigenvalues are organized into vertical bands in the spectral plane $\mathbb{C}$ and provide consequences for the description of correlation functions $t \mapsto \langle v|u \circ \phi^t \rangle$ defined above. The main result presented here is that these corrections to the equilibrium are small fluctuations that behave as if governed by a quantum evolution equation, cf. Theorem 1.5. We can say roughly that a **quantum dynamics emerges from the correlation functions of a deterministic chaotic dynamical system**. Technically we use **micro-local analysis**, which appears to be a very fruitful approach in this domain [15, 16], [11].

*Related papers.* We present now previous works specifically related to the topics of band structure of the Ruelle spectrum for contact Anosov dynamics and the emergence of an effective quantum dynamics as expressed in Eq. (1.13), using micro-local analysis. This paper is a continuation along the following series of papers.

* In [13], the simplest toy model has been considered. This dynamical system is a contact $U(1)$-extension of the Arnold's cat map $M = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ on $\mathbb{T}^2$ (or any hyperbolic $M \in SL_2(\mathbb{Z})$). Band structure of the Ruelle spectrum and
emergence of the quantum cat map dynamics has been shown. This model is
also called “prequantum cat map” since its construction follows the prequan-
tization procedure of Kostant, Souriau, Kirillov that is equivalent to a contact
U (1)-extension of the Hamiltonian dynamics.

Remark 1.1. The symplectic form $\Omega$ on $\mathbb{T}^2$ is not exact so we need to consider a non-trivial line bundle $L \to \mathbb{T}^2$ with curvature $\Omega$, called prequantum line bundle. On the opposite, in the model of contact Anosov flows considered in this paper the symplectic form $\Omega$ on $\Sigma \subset T^*M$ is exact since $\Omega = d\theta$ with $\theta$ being the Liouville one-form (2.12). Hence the prequantum line bundle $L \to \Sigma$ is trivial and we have decided to ignore it, although its presence manifests all along the paper, for example in the definition of the twist operator (2.42). For a more correct geometrical description, we have to twist by $L$ in the tensor product $\mathcal{F} \otimes L \to \Sigma$ in (4.27).

* In [18], the previous model is generalized. We consider a $U(1)$-extension of an arbitrary symplectic Anosov map $\phi : M \to M$ (also called prequantum Anosov map as explained before). Band structure of the Ruelle spectrum and emergence of the quantum dynamics has been shown. The models in [13, 18] can be considered as toy models (or simplified models) for contact Anosov flows considered in this paper because they are analogous (but not equivalent) to a time one contact Anosov flow map. These toy models are indeed at the core of this paper, we will come back to them in Appendices C and D.

* The paper [17] announced the band spectrum for contact Anosov flows, (less precise) Weyl law and the method presented in this paper.

* In the paper [19], one has considered the interesting choice of the bundle $F = |\det E_s|^{1/2}$, giving the trivial bundle (1.14). This model is treated in relation with the “semi-classical zeta function” that generalizes the Selberg zeta function to non-constant curvature. Technically, the case of the Hölder continuous bundle $F = |\det E_s|^{1/2}$ is more tricky than the case of a smooth bundle $F$ and needs to consider an extension of the dynamics to a Grassmannian bundle.

* Mihajlo Cekic and Colin Guillarmou in [26] prove the first band for contact Anosov flows in dimension 3 using horocycle operators.

* For the special case of hyperbolic manifolds, endowed with an algebraic structure, i.e. high symmetry described by Lie groups, band spectrum of Anosov dynamics has been studied in [10], [27], [30], [4].

* In different situations, band structure and Weyl law for the spectrum of resonances has been studied in [42], [41] for convex obstacles and by S. Dyatlov in [9] for regular normally hyperbolic trapped sets.

1.2. Model considered in this paper.

**Contact Anosov flows:** In this paper we consider a smooth contact Anosov vector field $X$ on a closed contact manifold $(M, \mathcal{A})$ with contact one form $\mathcal{A}$. A particular and important example is with $(\mathcal{N}, g_\mathcal{N})$ a closed Riemannian manifold with negative curvature, the geodesic vector field $X$ is a contact Anosov vector field on the unit cotangent bundle $M = T_1^*\mathcal{N}$, where the contact one form $\mathcal{A} = pdq$ is the Liouville one-form.

The vector field $X$ is considered as a derivation $X : C^\infty(M) \to C^\infty(M)$ and generates the pullback operators by the flow $\phi^t : M \to M$, i.e. $e^{tX}u = u \circ \phi^t$ with $u \in C^\infty(M)$
and $t \in \mathbb{R}$. More generally, if $F$ is a vector bundle over $M$, we will consider a derivation $X_F$ acting on smooth sections $C^\infty(M;F)$ of $F$ over<fnref n="1" /> $X$, generator of the pull back operator

$$ (1.1) \qquad e^{tX_F} : C^\infty(M;F) \to C^\infty(M;F). $$

As an example, we may consider $F = \Lambda^*(M)$, the bundle of differential forms, and $X_F$ being the Lie derivative.

**Question of long time behavior:** Anosov flows are a typical model of chaotic dynamics with sensitivity to initial conditions. A typical question in dynamical systems theory that is addressed in this paper is to describe the long time action of the flow generated by $X$ on smooth sections, i.e. for any given smooth sections $u, v \in C^\infty(M;F)$, describe the correlation $\langle v|e^{tX_F}u \rangle_{L^2}$ for $t \to \infty$. Here $L^2 = L^2_{dm}(M;F)$ with the invariant contact volume $dm$ (3.2) on $M$ and an arbitrary Hermitian metric on $F$. In other words we are looking for a “good description” of the pull back operator $e^{tX_F}$ for $t \to \infty$.

In the case of a trivial bundle $F = \mathbb{C}$, it is known from Sinaǐ [40] that contact Anosov flows are **mixing**, i.e. for any smooth functions $u, v \in C^\infty(M;\mathbb{C})$, we have $\langle v|e^{tX}u \rangle_{L^2} \xrightarrow{t \to +\infty} \langle v|1 \rangle_{L^2} \langle \frac{1}{\mathrm{Vol}(M)}|u \rangle_{L^2}$. This mixing property dominates the long time behavior and says that the operator $e^{tX}$ converges (in the weak sense) to the rank one operator $\Pi_0 = 1 \langle \frac{1}{\mathrm{Vol}(M)}| \rangle_{L^2}$.

**Problem 1.2 (Aim).** In this paper we will be interested in finding a description of the operator $e^{tX_F}$ in the limit of large $t$, up to an error that decays as $e^{-\Lambda t}$ with an arbitrary large rate $\Lambda > 0$.

**Discrete Ruelle spectrum and anisotropic Sobolev space:** Although the space $L^2(M;F)$ may be natural to consider in order to describe the operator $e^{tX_F}$, this is not what we will do. We will instead consider a family of generalized Sobolev spaces $H_W(M;F)$, defined from a weight function $W$ on $T^*M$. By choosing a good weight $W$, we can get that the essential spectrum of the generator $X_F$ is in $\{z \in \mathbb{C}, \mathrm{Re}(z) < -\Lambda\}$ with arbitrary large $\Lambda > 0$ [20, thm 2.11], revealing some **intrinsic discrete spectrum** called **Ruelle resonances** in $\{z \in \mathbb{C}, \mathrm{Re}(z) > -\Lambda\}$. See Figure 1.2. This fact has been essentially first obtained by Butterley and Liverani in [6]. The function space $H_W(M;F)$ is called anisotropic Sobolev space [25], [3], [15, 16] because it is a Hilbert space of distributional sections that contains smooth sections $C^\infty(M;F)$ and its order depends on the stable/unstable directions of the dynamics.

This discrete spectrum governs the long time behavior of $e^{tX_F}$ but only little<fnref n="2" /> is known about it for a general Anosov flow (or Axiom A flow). In the special case of a contact Anosov vector field $X$, much more can be said about the Ruelle spectrum of $X_F$ and about the effective action of the operator $e^{tX_F}$. This is the subject of this paper.

<footnote n="1">¹See Definition (2.4).</footnote>

<footnote n="2">²Ruelle spectrum can be calculated explicitly in only exceptional cases. Appendix D.3 presents the Ruelle spectrum of a very simple toy model on $\mathbb{R}$. See [17, Prop. 4.1] for the Ruelle spectrum of the geodesic flow on a compact hyperbolic surface.</footnote>
1.3. Main result: Emergence of an effective quantum dynamics.

1.3.1. Known results beyond the exponential mixing description. In order to motivate the analysis and few results presented in this paper, let us discuss again about the question of long time behavior of the dynamics $e^{tX}$ in the case of a contact Anosov vector field $X$ (i.e. trivial bundle $F = \mathbb{C}$ for this discussion) and known results so far. Recall the result of **mixing** given above, that is the weak convergence $e^{tX} \xrightarrow{t \to +\infty} \Pi_0 = 1 \langle \frac{1}{\mathrm{Vol}(M)} | \cdot | \rangle_{L^2}$. Beyond this, there is **exponential mixing** obtained by C. Liverani [33], that is $\exists \epsilon > 0, \exists C > 0, \forall t \ge 0, \|e^{tX} - \Pi_0\|_{H_W(M)} \le Ce^{-t\epsilon}$. Even better, there is an explicit description of the **exponential small corrections** [46] in terms of a finite number of Ruelle eigenvalues of $X$ in $\mathcal{H}_W(M)$, $z_j = a_j + i\omega_j \in \mathbb{C}$ with real part $\gamma_0^+ + \epsilon < \dots < a_{j+1} \le a_j \le \dots < a_1 < a_0 = 0$ and associated finite rank spectral projectors $\Pi_j$, that can be expressed as follows:

$$ (1.2) \qquad \exists \epsilon > 0, C_{\epsilon} > 0, \forall t \ge 0, \left\| e^{tX} - R_t \right\|_{H_W(M)} \le C_{\epsilon} e^{t(\gamma_0^+ + \epsilon)}, $$

with a finite rank operator<fnref n="3" />

$$ (1.3) \qquad R_t = \sum_{z_j, \mathrm{Re}(z_j) > \gamma_0^+ + \epsilon} e^{tX} \Pi_j, $$

with the explicit threshold

$$ (1.4) \qquad \gamma_0^+ := \lim_{t \to +\infty} \log \left\| e^{tX_{\gamma_0}} \right\|_{L^\infty}^{1/t} < 0, $$

where $X_{\gamma_0}$ is the Lie derivative of the vector field acting on section of the line bundle $\mathcal{F}_0(E_s) := |\det E_s|^{-1/2} \to M$, $E_s \subset TM$ is the vector bundle over $M$ of the stable directions of the Anosov flow, $\det E_s = E_s^{\lambda_d}$ is the determinant bundle from $E_s$, with $d = \dim E_s$, $|\det E_s|^{1/2}$ is the half densities bundle and $|\det E_s|^{-1/2} = (|\det E_s|^{1/2})^*$ is its dual.

*Remark 1.3.*

* We have $\gamma_0^+ < 0$ because $E_s$ is the stable bundle and therefore under the pull-back operator $e^{tX}$, the induced action of the line bundle $|\det E_s|$ is expanding hence on $\mathcal{F}_0(E_s) = |\det E_s|^{-1/2}$ it is contracting. We have the estimate $\gamma_0^+ < -\frac{d}{2}\lambda_{\min}$ with the minimal Lyapunov exponent $\lambda_{\min} > 0$ and $d = \dim E_s = \frac{1}{2}(\dim M - 1)$.

* Except for the obvious eigenvalue $z_0 = 0$ of $X$ (whose eigenspace is constant functions), the set of other eigenvalues $\{z_j\}_j$ that enter in the sum (1.3) is quite unknown in general and may be empty.

* Considering an arbitrary vector bundle $F \to M$ changes the operator $R_t$ and the estimate $\gamma_0^+$, see below.

<footnote n="3">³In case of a simple eigenvalue $z_j$ this gives $e^{tX} \Pi_j = e^{z_j t} \Pi_j$. Notice also that $z_0 = 0$ is simple, hence $e^{tX} \Pi_0 = \Pi_0$ corresponding to the “mixing main term” so called “equilibrium”.</footnote>
1.3.2. Looking for a better description of the dynamics up to arbitrary small rate $e^{-\Lambda t}$.
In this paper our aim (Problem 1.2) is to give a description of the dynamics $e^{tX}$ similar to (1.2) but with an error term $Ce^{-\Lambda t}$ with arbitrary large rate $\Lambda > 0$. So we have to overcome the threshold $\gamma_0^+$. For this we do some micro-local analysis of the pullback operator $e^{tX}$ which means that we describe precisely its action on functions (or sections) with high frequencies (i.e. roughly speaking considering in any local chart only Fourier components with large Fourier variable $|\xi| \gg 1$, $\xi \in \mathbb{R}^{\dim M}$).

With the **micro-local approach**, it is a general fact that we can describe the operator $e^{tX}$ in terms of the action of the flow induced on the cotangent bundle $T^*M$ (in other terms $e^{tX}$ is considered as a Fourier Integral Operator (F.I.O.). In the special case of contact Anosov flow we show that only a neighborhood of the symplectic subset $\Sigma = \mathbb{R}^*A \subset T^*M$ has to be considered, where $A$ is the contact one-form, because (in some sense explained in this paper) “outside of $\Sigma$, the norm of $e^{tX}$” decays faster than $e^{-\Lambda t}$ with arbitrary large $\Lambda > 0$. As a result, we obtain an effective description of $e^{tX}$ in terms of a Hamiltonian flow on the symplectic subset $\Sigma$ but considering also its action on the symplectic normal bundle $N \to \Sigma$, needed to get the neighborhood of $\Sigma$. This micro-local description of the operator $e^{tX}$ is in terms of **quantization** of the “classical Hamiltonian action” $e^{tX_f}$ giving an operator $\text{Op}_\Sigma(e^{tX_f})$.

The **main drawback** of the micro-local approach is its inability to describe the “low frequency subspace of functions” (that corresponds effectively to a finite dimensional subspace since $M$ is compact) and for this reason, our approximation below in Theorem 1.5 will contain an unknown finite rank operator $R_{\sigma,t}$, similar to the operator $R_t$ in (1.2). Optimistically we can consider that a finite rank operator is negligible compared to the infinite rank quantum operator $\text{Op}_\Sigma(e^{tX_f})$ that will enter in the description.

Let us now explain more precisely this micro-local approach, the effective classical dynamics that appears in a first result presented in Theorem 1.5.

1.3.3. *Effective classical dynamics.*

**Bundle** $\mathcal{F}_k(E_s) \to M$ **and its dynamics:** We now go to our first aim (Problem 1.2) and consider again a general vector bundle $F \to M$. For this, for every $k \in \mathbb{N}$, we introduce the Hölder continuous finite rank vector bundle over $M$:

$$ (1.5) \qquad \mathcal{F}_k(E_s) := |\det E_s|^{-1/2} \otimes \text{Pol}_k(E_s) \otimes F \to M, $$

where $\text{Pol}_k(E_s)$ is the bundle of homogeneous polynomials of degree $k$ on $E_s$. The bundle $E_s \to M$ is Hölder continuous but smooth along the flow and $E_s$ directions. Let $C^\beta(M; \mathcal{F}_k(E_s))$ be the space of $\beta$-Hölder continuous sections. Then we consider $X_{\mathcal{F}_k} : C^\beta(M; \mathcal{F}_k(E_s)) \to C^\beta(M; \mathcal{F}_k(E_s))$ being the derivation of sections of $\mathcal{F}_k(E_s)$ over $X(C^1$ along the flow direction). $X_{\mathcal{F}_k}$ is naturally induced from the initial derivation $X_F$ and is the generator of the group of operators $e^{tX_{\mathcal{F}_k}} : C^\beta(M; \mathcal{F}_k(E_s)) \to C^\beta(M; \mathcal{F}_k(E_s))$, $t \in \mathbb{R}$. We introduce the related quantities that generalize (1.4)

$$ (1.6) \qquad \gamma_k^{\pm} := \lim_{t \to \pm\infty} \log \left\| e^{tX_{\mathcal{F}_k}} \right\|_{L^\infty(M; \mathcal{F}_k(E_s))}^{1/t}. $$

*Remark 1.4.* We can give obvious (but rough) estimates for $\gamma_k^\pm$ as follows. If $0 < \lambda_{\min} \le \lambda_{\max}$ denote the minimal and maximal Lyapunov exponents of the Anosov flow $\phi^t$, we have

$$ (1.7) \qquad -\left(\frac{d}{2} + k\right) \lambda_{\max} + c_F \le \gamma_k^- \le \gamma_k^+ \le -\left(\frac{d}{2} + k\right) \lambda_{\min} + C_F, $$
where $\dim M = 2d+1$ and $c_F \le C_F$ depend on $X_F$. For example, if $X_F = X$ is the vector field itself with $F = \mathbb{C}$, we have $c_F = C_F = 0$ and from (1.7), the pinching condition $\frac{\lambda_{\max}}{\lambda_{\min}} < 1 + \frac{1}{k}$ implies $\gamma_{k+1}^+ < \gamma_k^-$ that we call the **spectral gap assumption** later.

**A notation** $S_\beta(E_s)$: For every point $m \in M$, $S(E_s(m))$ is the space of Schwartz functions on the vector space $E_s(m)$. However the bundle $E_s \to M$ is only $\beta$-Hölder, i.e. the map $m \in M \to E_s(m)$ is Hölder continuous. For simplicity we will denote

$$ (1.8) \qquad S_{\beta}(E_s) := C^{\beta}(M; S(E_s)) $$

the space of $\beta$-Hölder continuous sections valued in $S(E_s(m))$ at each point $m \in M$. Similarly $S'_\beta(E_s) := C^\beta(M; S'(E_s))$ is the space of sections valued in Schwartz distributions $S'(E_s(m))$, or more precisely distributional extensions of $S(E_s(m))$, the dual of Schwartz forms $S(E_s(m); \Lambda^d(E_s(m)))$, so that polynomials are included in $S'(E_s(m))$. **Taylor projectors** $T_k$: The homogeneous polynomials $\text{Pol}_k(E_s)$ are obtained as the image of the “Taylor projector” (see Section C.10 for a definition) $T_k: S_\beta(E_s) \to \text{Pol}_k(E_s) \subset S'_\beta(E_s)$ that we extend to a bundle map $T_k \in L(\mathcal{F}(E_s), \mathcal{F}'(E_s))$,

$$ (1.9) \quad T_k: \mathcal{F}(E_s) := |\det E_s|^{-1/2} \otimes S_\beta(E_s) \otimes F \to \mathcal{F}'(E_s) := |\det E_s|^{-1/2} \otimes S'_\beta(E_s) \otimes F. $$

For any given $K \in \mathbb{N}$, we denote $T_{[0,K]} := \bigoplus_{k=0}^K T_k$.

**Bundle** $\mathcal{F}_k(N_s) \to \Sigma$: We consider the symplectic submanifold

$$ \Sigma := \mathbb{R}^*\mathcal{A} = \{\omega\mathcal{A}(m), \omega \in \mathbb{R}^*, m \in M\} $$

being the **symplectization** of the contact manifold $(M, \mathcal{A})$. This manifold $\Sigma$, as a submanifold of $T^*M$, is also the trapped set (or non-wandering set) of the induced dynamics $\tilde{\phi}^t = (d\phi^t)^*$ on $T^*M$ and therefore invariant. We consider the symplectic normal bundle $N = (T\Sigma)^\perp$, its stable component $N_s \subset N$ and construct the bundle$^4$ $\mathcal{F}_k(N_s) := |\det N_s|^{-1/2} \otimes \text{Pol}_k(N_s) \otimes F \to \Sigma$. The operator $e^{tX_{\mathcal{F}_k}}$ defined above extends to

$$ (1.10) \qquad e^{tX_{\mathcal{F}_k}} : C^\beta(\Sigma; \mathcal{F}_k(N_s)) \to C^\beta(\Sigma; \mathcal{F}_k(N_s)) $$

over the Hamiltonian flow $\tilde{\phi}^t = (d\phi^t)^*$: $\Sigma \to \Sigma$. The family of operators $(e^{tX_{\mathcal{F}_k}})_{t \in \mathbb{R}}$ in (1.10) will be called the **classical dynamics**. For any $t \in \mathbb{R}$, we also denote the flow $e^{tX_{\mathcal{F}}}: C^\beta(\Sigma; \mathcal{F}(N_s)) \to C^\beta(\Sigma; \mathcal{F}(N_s))$ with similar definition $\mathcal{F}(N_s) := |\det N_s|^{-1/2} \otimes S_\beta(N_s) \otimes F$ and that $e^{tX_{\mathcal{F}}}\tilde{T}_{[0,K]} = \bigoplus_{k=0}^K e^{tX_{\mathcal{F}_k}}$ with Taylor projectors $\tilde{T}_k: S_\beta(N_s) \to \text{Pol}_k(N_s)$ and $\tilde{T}_{[0,K]} := \bigoplus_{k=0}^K \tilde{T}_k$.

1.3.4. General remarks about quantization. Before continuing the presentation, we first make a pause to say few important words about quantization. These remarks will be important to understand the content of Theorem 1.5.

* In the most common situation, on a given manifold $M$, quantization denoted $\text{Op}(.)$ maps a function $a \in S(T^*M; \mathbb{C})$ called “symbol” to a bounded operator $\text{Op}(a) : L^2(M) \to L^2(M)$ called Pseudo-Differential Operator (P.D.O.). There are many different definitions for $\text{Op}(.)$, and it is important to know that **they are all equivalent at leading order in the limit of high frequencies.** At

<footnote>$^4$From the precise definition of $N_s$ in Lemma 4.9, it will appear that this bundle $\mathcal{F}_k(N_s) \to \Sigma$ is isomorphic to the pull back of $\mathcal{F}_k(E_s) \to M$ under the projection $\pi: \Sigma \to M$.</footnote>
leading order the map Op(.) satisfies some very interesting and useful uni-
versal properties as the “boundness theorem”, “composition theorem” that is
homomorphism of algebra, “trace formula”, etc. (under some additional hy-
pothesis called “symbol classes” [43. (1.4) p.3]) and our analysis will rely on
these properties. For these reasons the precise definition of Op(.) is not very
crucial for the moment.

* The simplest example of Op(.) called ordinary quantization is given in local charts $x \in \mathbb{R}^n$ on $M$ with dual coordinates $\xi \in \mathbb{R}^n$ on $T_x^*M$, by [43, (1.3) p.2]

$$
(\mathrm{Op}(a) u)(x) = \int a(x, \xi) (\mathcal{F}u)(\xi) e^{i(\xi|x)} d\xi,
$$

where $(\mathcal{F}u)(\xi)$ is Fourier transform of $u \in S(\mathbb{R}^n)$. Weyl quantization is dif-
ferent and has the advantage that a real valued symbol $a$ gives a self-adjoint
operator $\mathrm{Op}(a)$ [43, (14.1) p.67].

* In geometric quantization people introduce an operator $\mathcal{T}:C^\infty(M) \to S(T^*M)$, called Bargman transform or FBI transform, wavelet transform, wave-packet transform (depending on the context, and additional structures for its construction) and they define Toeplitz quantization by

$$
\mathrm{Op}(a) := \mathcal{T}^{\dagger} a \mathcal{T},
$$

where $a$ on the right hand side stands for the multiplication operator by $a$ and
$\mathcal{T}^\dagger : S(T^*M) \to C^\infty(M)$ is its $L^2$ adjoint satisfying $\mathrm{Op}(1) = \mathcal{T}^\dagger \mathcal{T} = \mathrm{Id}$. We
will mainly use this quantization in this paper.

* It is also possible to define the quantization of a symplectomorphism $\tilde{\phi} : T^*M \to T^*M$ giving an operator $\mathrm{Op}(\tilde{\phi}) : L^2(M) \to L^2(M)$ called Fourier Integral Operator (F.I.O.) (e.g. [48]). In geometric quantization the definition is

$$
(1.11) \qquad \mathrm{Op}(\tilde{\phi}) := \mathcal{T}^{\dagger} Y_{\tilde{\phi}}^{1/2} \tilde{\phi}^{-\cdot \circ} \mathcal{T},
$$

where $\tilde{\phi}^{-\circ} : S(T^*M) \to S(T^*M)$ is the push-forward operator and $Y_{\tilde{\phi}}^{1/2} \in C^\infty(T^*M; \mathbb{R}^+)$ is the “metaplectic correction” (C.17) to get an almost unitary operator.

* It is possible to consider a vector bundle $F \to M$ and define the quantization of a symbol $a \in S(T^*M; \mathrm{End}(F))$ giving $\mathrm{Op}(a) : L^2(M;F) \to L^2(M;F)$.

1.3.5. Quantization in our setting. In this paper in order to study the operator $e^{tX_F} : C^\infty(M;F) \to C^\infty(M;F)$ in (1.1), we will need all these previous and different variations around quantization and moreover another variation: instead of considering the whole cotangent bundle $T^*M$ as the symplectic manifold, we will restrict to a vicinity of the symplectic submanifold $\Sigma := \mathbb{R}^*A \subset T^*M$. To have a good description of a vicinity of $\Sigma$, we will use its normal bundle $N = (T\Sigma)^\perp \to \Sigma$ and the exponential map $\exp_N : N \to T^*M$ that is a local diffeomorphism from a vicinity of the zero section of $N$ to a vicinity of $\Sigma$ in $T^*M$. See Figure 1.1.
<figure><img src="image_3.png" /><figcaption>FIGURE 1.1. The dynamics induced on $T^*M$ scatters on the trapped set $\Sigma \subset T^*M$ defined in (3.7). $\Sigma$ is a line bundle over $M$, a symplectic submanifold of $T^*M$ and at every point $\rho = \omega A(m) \in \Sigma$, where $\omega$ called frequency is the coordinate along the line, the symplectic-normal bundle $N(\rho) = (T_\rho \Sigma)^\perp$ (a symplectic linear subspace of $T_\rho T^*M$) splits into unstable/stable subspaces, $N(\rho) = N_u(\rho) \oplus N_s(\rho)$, see (3.11). The **main geometrical object** considered in this paper is this fibration $N_s \to \Sigma \to M$. Beware that for a geodesic flow on $(N, g)$, this fibration sequence continues with $M = (T^*N)_1 \to N$.</figcaption></figure>

Then we will use a wave-packet transform given by some operators $\mathcal{J}_{N_s} : C^\infty(M;F) \to C^\beta(\Sigma; \mathcal{F}(N_s))$ and a converse $\mathcal{J}_N^\Delta : C^\beta(\Sigma; \mathcal{F}(N_s)) \to C^\infty(M;F)$ defined by

$$ \mathcal{J}_{N_s} \stackrel{(4.34)}{=} B_{N_s, N_u}^\dagger \chi_\Sigma^\mu (\widetilde{\exp_N})^\circ \mathcal{J} $$

and

$$ \mathcal{J}_{N_s}^\Delta \stackrel{(4.35)}{=} \mathcal{J}^\dagger (\widetilde{\exp_N}^{-1})^\circ \chi_\Sigma^\mu B_{N_s, N_u}, $$

where $\mathcal{T} : C^\infty(M;F) \to S(T^*M;F)$ is a wave-packet transform, $(\widetilde{\exp_N})^\circ : S(T^*M;F) \to S(\Sigma; S(N))$ is the pull back of $\exp_N$ with some additional phase in (3.35), $\chi_\Sigma^\mu$ is some cutoff function to a vicinity of the zero section of $N$ at distance $\omega^{\mu/2}$ in (3.34) and $B_{N_s,N_u} : C^\beta(\Sigma; S_\beta(N_s)) \to C^\beta(\Sigma; S(N))$ in (4.30) is a bundle-wise Bargman transform. With this, we define the quantization of the classical dynamics (1.10) with the additional symbol $\tilde{T}_{[0,K]}$ by

$$ (1.12) \quad \mathrm{Op}_\Sigma (e^{tX_s} \tilde{T}_{[0,K]}) \stackrel{(4.55)}{=} \mathcal{J}_{N_s}^\Delta Y_t^{1/2} e^{tX_s} \tilde{T}_{[0,K]} \mathcal{J}_{N_s} : C^\infty(M;F) \to C^\infty(M;F) $$

called **quantum evolution operators**. In the analysis, the semi-classical parameter is the frequency $\omega$ along the flow direction (equivalently the parameter along $\Sigma$) and the **semi-classical limit** is $|\omega| \to \infty$ (usually denoted $2\pi\hbar = 1/|\omega| \to 0$).
<figure><img src="image_4.png" /><figcaption>FIGURE 1.2. The dots represent the intrinsic Ruelle discrete spectrum of the derivation $X_F$ in a Sobolev space $H_W(M; F)$. From [20, thm 2.11], the essential spectrum is in a (brown) vertical band that can be moved arbitrarily far on the left by changing the weight $W$, and reveals this intrinsic discrete spectrum. The right most eigenvalues in the first band $B_0$ dominate the emerging behavior of $e^{tX_F}$ for $t \gg 1$. In figure (b) for the special case of the bundle $F = |\det E_s|^{1/2}$, the first band coincides with the imaginary axis.</figcaption></figure>
1.3.6. *Emergence of an effective quantum dynamics.* To express the result, we also need an operator $\text{Op}(\chi_{\Sigma,\sigma}) : C^{\infty}(M;F) \to C^{\infty}(M;F)$ defined in (1.22) that restricts functions micro-locally to their vicinity of $\Sigma$ at distance $\sigma > 0$. Theorem 1.5 shows how the pull-back operator $e^{tX_F}$ in (1.1) is well approximated by the quantum evolution operator $\text{Op}_{\Sigma}(e^{tX_F} T_{[0,K]})$ in (1.12).

**Theorem 1.5 (Emergence of quantum dynamics).** For any $K \in \mathbb{N}$ and $\varepsilon > 0$, we can choose an anisotropic Sobolev space $\mathcal{H}_W(M;F)$, such that $\exists C > 0, \forall t > 0, \exists \sigma_t > 0, \forall \sigma > \sigma_t$, there exists a finite rank operator $R_{\sigma,t}$ and

$$ (1.13) \quad \left\| e^{tX_F} - \text{Op}(\chi_{\Sigma,\sigma}) \text{Op}_{\Sigma}(e^{tX_F} \tilde{T}_{[0,K]}) \text{Op}(\chi_{\Sigma,\sigma}) - R_{\sigma,t} \right\|_{\mathcal{H}_W(M;F)} \le C e^{t(\gamma_{K+1}^+ + \varepsilon)}. $$

The proof of Theorem 1.5 is given in Section 6.

*Remark 1.6.* From (1.7) we have $\gamma_k^+ \to -\infty$, hence the term $e^{t(\gamma_{K+1}^+ + \varepsilon)}$ in (1.13) decays very fast for $t \to +\infty$ when $K$ is large, Theorem 1.5 is a kind of answer to our aim 1.2. One interpretation of Theorem 1.5 is that for large time $t \gg 1$, an **effective quantum dynamics emerges from the contact Anosov dynamics.** This quantum dynamics is the quantization of the Hamiltonian dynamics (1.10) that takes place on $\Sigma$.

To show (1.13) we will use micro-local analysis directly on $T^*M$ following the approach proposed in [20], using a metric $g$ on $T^*M$ compatible with the symplectic form, that measures the size of wave-packets (sometimes called coherent states) in accordance with the uncertainty principle.

1.3.7. *Few comments about the result (1.13).*

* As explained above, in the case of a trivial bundle $F = \mathcal{C}$, contact Anosov flows are **mixing** $e^{tX} \xrightarrow{\text{weak}} \Pi_0 = 1\langle\frac{1}{\text{Vol}(M)}|\cdot|_{L^2}\rangle$. Although this mixing property dominates the long time behavior, the limit rank one operator $1\langle\frac{1}{\text{Vol}(M)}|\cdot|_{L^2}\rangle$ belongs to the finite rank operator $R_{\sigma,t}$ in (1.13) that we do not describe in this paper. We instead consider terms that are exponentially small compared to it, but that belong to an infinite dimensional effective space. In our analysis the rank of the operator $R_{\sigma,t}$ may increase with $t$ because we use micro-local analysis (i.e. high frequency analysis) where the cutoff $\omega$ we need in frequency increases with $t$. The operator $R_{\sigma,t}$ represents the remainder in low frequencies that we are not able to describe with this approach. See also footnote 6 for more comments.

* In (1.5), the vector bundle $F \to M$ is arbitrary but notice that the **special choice** of the bundle $F = |\det E_s|^{1/2}$ is particularly interesting because it gives

$$ (1.14) \quad \mathcal{F}_0 = |\det E_s|^{-1/2} \otimes \text{Pol}_0(E_s) \otimes |\det E_s|^{1/2} = \mathcal{C}, $$

i.e. the trivial bundle over $\Sigma$. The paper [19] is devoted to that case.<fnref n="5" /> Eq. (1.6) gives $\gamma_0^\pm = 0$ and $\gamma_1^+ < 0$. For that case and taking $K = 0$, Eq. (1.13) shows that for $t \gg 1$, the operator $e^{tX_F}$ is well described by the dominant term $\text{Op}_{\Sigma}(e^{tX_F} T_0)$ that is the “quantization” of the dynamics $X$ itself. From Theorem 1.7, for $\omega =$

<footnote n="5">⁵$F = |\det E_s|^{1/2}$ is not a smooth bundle but only Hölder continuous and technically we consider a smooth extension to a Grassmannian bundle in [19].</footnote>
$\text{Im}(z) \to \infty$, the eigenvalues of $X_F$ accumulate on the imaginary axis $\text{Re}(z) = \gamma_0^\pm = 0$ with density given by the Weyl law, separated by a uniform spectral gap $\gamma_1^+ < 0$. See Figure 1.2(b).

* In the special case of the **geodesic flow** on a surface $N$ of constant and negative curvature, i.e. $N = \Gamma \setminus SL_2(\mathbb{R})/SO_2(\mathbb{R})$, giving $M = \Gamma \setminus SL_2(\mathbb{R})$, with $\Gamma$ being a co-compact subgroup of $SL_2(\mathbb{R})$ (that contains $-\text{Id}$), there is no remainder operator $R_t$ in (1.13) and the dominant quantum operator $\text{Op}_\Sigma(e^{tX_f} \tilde{T}_0)$ has the same spectrum as the wave operators $\exp(\pm it\sqrt{\Delta - \frac{1}{4}})$ on $S(N)$ [10, 17] see Figure 4.1(a) in [17, Fig 4.1(a) p.389]. This operator is indeed considered in physics as the Schrödinger evolution operator, giving the quantum description of a free particle. For the case of geodesic flow on a non-constant negatively curved Riemannian manifold $(N, g)$, let us observe that the symplectic phase space $\Sigma$ is (symplectically) isomorphic to a double cover of the cotangent bundle $T^*N \setminus \{0\}$ and that the principal symbol of the generator of the leading quantum operator $\text{Op}_\Sigma(e^{tX_f} \tilde{T}_0)$ in (1.13) is the frequency $\omega$, equal to the principal symbol of $\sqrt{\Delta}$ on $T^*N$ under this isomorphism, where $\Delta = d^\dagger d$ is the Laplace Beltrami operator on $S(N)$. However we do not expect that the spectra of both operators coincide in general. We will investigate this question in a future work. See also [4] for related results and discussions.

* In semi-classical analysis, the Egorov Theorem [43, p.26] or the WKB approximation [5, p. 11] shows that **classical Hamiltonian dynamics emerges in the high frequency limit (and finite time) of quantum dynamics**. This is how in physics, geometrical optics is derived from electromagnetic waves, Newtonian (and Hamilton) mechanics is derived from quantum waves mechanics of Schrödinger, etc. Inspired from these physical phenomena this is how quantization has been defined in mathematics, for example standard quantization [43, p.2] that defines a pseudo-differential operator on $S(\mathbb{R}^n)$ from a Hamiltonian function (symbol) on $\mathbb{R}^{2n}$, or geometric quantization in a more geometric setting [47]. In this paper we have exhibited the converse and maybe **unexpected phenomena: how quantum mechanics emerges from the classical mechanics** when this later is chaotic. From the mathematical point of view, an interesting consequence is that it furnishes a **natural quantization** of a given classical dynamics among all possible quantizations. This natural quantization has indeed the preferable properties (that characterize it) that the semi-classical Van-Vleck formula and semi-classical Trace formula are asymptotically exact, i.e. they have error terms that decay exponentially fast with $t \to \infty$ at large but fixed $\omega$. This provides a kind of generalization of Selberg theory to non-constant curvature. For more discussions, see [18, Section 1.5,1.6,1.7] and [19]. From a physical point of view one may wonder if quantum phenomena observed in experimental data may emerge from an underlying deterministic but chaotic dynamics, see e.g. the last paragraph in [35]. In Section 3.6 we give Theorem 3.21 that concerns a general contact flow (i.e. without Anosov assumption) so this may concern a general geodesic flow and show the emergence of quantum dynamics near the invariant set $\Sigma$.
1.4. **Sketch of proof.** The proof of the emerging quantum dynamics in (1.13) can be summarized by the following mechanisms that will be explained in details in this paper and illustrated in Figure 1.1.

(1) In the limit of high frequencies, evolution of functions (or sections) by the pull-back operator $e^{tX_F}$ is well described on the cotangent bundle $T^*M$ with the induced flow $\tilde{\phi}^t := (d\phi^t)^*$, $t \in \mathbb{R}$. This is because $e^{tX_F}$ is a Fourier integral operator (i.e. has micro-local property (2.39)). We introduce a specific metric $g$ on $T^*M$, compatible with the symplectic form, and then we define an $L^2$-isometric “wave-packet transform” $\mathcal{T} : C^\infty(M; F) \to S(T^*M; F)$, that allows to use **micro-local analysis** on $T^*M$ for the pull back operator $e^{tX_F}$. The unit boxes for the metric $g$ correspond to the effective size of wave-packets and reflect the uncertainty principle.

(2) For a contact Anosov flow, the dynamics induced on the cotangent bundle $T^*M$ is a “scattering dynamics” on the trapped set $\Sigma = \mathbb{R}^*A \subset T^*M$ and $\Sigma$ is symplectic and normally hyperbolic. See Figure 1.1. In terms of dynamics, the subset $\Sigma$ is the non-wandering set for the flow $\tilde{\phi}^t$ on $T^*M$ and the orbits on the outside of $\Sigma$ go to infinity either as $t \to +\infty$ or $t \to -\infty$. As a consequence, for large time $|t| \gg 1$, the outer part of the trapped set $\Sigma$ has a negligible contribution, because information escapes to infinity (i.e. the Sobolev norm measured by the specific weight $W$ decays). This will be given in Theorem 4.6. So only the dynamics on $\Sigma$ plays a role for our purpose. But due to the uncertainty principle in $T^*M$, we still have to consider a neighborhood of $\Sigma$ in $T^*M$. We will hence consider a neighborhood of $\Sigma$ of a given size $\langle\omega\rangle^{\mu/2}$ in (3.34) (measured by the metric $g$), with some $0 < \mu < 1$. It is important to remark that for large frequencies $\omega$ this neighborhood of size $\langle\omega\rangle^{\mu/2}$ projected down on $M$ is scaled by $\langle\omega\rangle^{-1/2}$ (from the definition of the metric $g$ on $T^*M$ in (2.25), see Remark 2.7) and get size $\asymp \omega^{-(1-\mu)/2} \to 0$ that goes to zero as $\omega$ goes to infinity. This will allow us to use **the linearization of the dynamics $\tilde{\phi}^t$** as a local approximation.

(3) In the neighborhood of size $\langle\omega\rangle^{\mu/2}$ of the trapped set $\Sigma$ (i.e. the non-wandering set) that matters, there is a micro-local decoupling between the directions tangent to $\Sigma$ and those (symplectically) normal to $\Sigma$, represented by a normal vector bundle denoted $N = (T\Sigma)^\perp \to \Sigma$. The dynamics on the normal direction $N$ is hyperbolic and responsible for the emergence of polynomial functions along the stable direction $N_s$ (that projects to $E_s$). By considering the Taylor operators $\tilde{T}_k$ as symbols, we introduce approximate projectors $\text{Op}_\Sigma(\tilde{T}_k)$ on $C^\infty(M; F)$ that micro-locally (i.e. seen in $S(T^*M; F)$ after the wave-packet transform $\mathcal{T}$) restrict functions to the symplectic trapped set $\Sigma$ and that are polynomial valued along $N_s$ with degree $k$. This projector plays a similar role as the Bergman projector (or Szegö projector) in geometric quantization. What remains for large time is an effective Hilbert space of “quantum waves” that live on the trapped set $\Sigma$, valued in the vector bundle $\mathcal{F}_k$ defined in (1.5).

The very simple toy model that is useful to have in mind is given in Section D.3. It explains in particular the emergence of polynomials $\text{Pol}_k(E_s)$ as in (4.27).
**1.5. Consequences and other results.** There are many consequences of the effective description (1.13) of the dynamics. In this paper we describe a few of them that are illustrated in Figure 1.2. We will follow general ideas from classical-quantum correspondence principles for quantization in the case where the classical symbol is an operator valued function on a symplectic manifold $\Sigma$ (this situation is also present in physics for elastic waves, electromagnetic waves, Dirac equation etc.), and we will derive the following results for the operator $X_F$.

**1.5.1. Discrete spectrum in vertical bands.** The generator of the classical dynamics (1.10) is $X_{\mathcal{F}_k}$ and its spectrum (some essential spectrum in fact) in $L^2(\Sigma; \mathcal{F}_k)$ is contained in the vertical band $B_k := [\gamma_k^-, \gamma_k^+] \times i\mathbb{R}$ with $\gamma_k^\pm$ defined in (1.6). We always have $\gamma_{k+1}^- < \gamma_k^-$ and $\gamma_{k+1}^+ < \gamma_k^+$ but this does not guarantee that the bands are separated by gaps, i.e. that $\gamma_{k+1}^+ < \gamma_k^-$, $B_{k+1} \cap B_k = \emptyset$.

In the semi-classical limit, i.e. for $\omega = \text{Im}(z) \to \pm\infty$ on the spectral plane $z \in \mathbb{C}$, Theorem 1.7 shows that the spectrum of $X_F$ is discrete and asymptotically contained in these vertical bands $\cup_{k \in \mathbb{N}} B_k$ and the resolvent is uniformly bounded in the gaps between the bands (if they exist). This is illustrated in Figure 1.2.

**Theorem 1.7 (Band structure of the Ruelle spectrum).** For any $\epsilon > 0$, $C > 0$, there exists $C_\epsilon > 0$, $\omega_\epsilon > 0$ such that the Ruelle eigenvalues spectrum $(X_F)$ are contained in the following spectral domain that consists of a union of a **"low frequency horizontal band"** and **"vertical bands"**:

$$ (1.15) \qquad (\text{spect}(X_F) \cap \{\text{Re}(z) > -C\}) \subset \{|\text{Im}(z)| \le \omega_\epsilon\} \cup \left( \bigcup_{k \in \mathbb{N}} \{\text{Re}(z) \in [\gamma_k^-, \epsilon, \gamma_k^+ + \epsilon]\} \right), $$

and the resolvent operator is uniformly bounded in the "gaps":

$$ (1.16) \qquad \|(z - X_F)^{-1}\|_{\mathcal{H}_W} \le C_\epsilon, \quad \forall z \in \mathbb{C} \text{ s.t. } |\text{Im}(z)| > \omega_\epsilon, \text{Re}(z) \in [\gamma_{k+1}^+ + \epsilon, \gamma_k^- - \epsilon]. $$

The proof is given in Section 5.

Notice that if the gap does not exist, i.e. if the bands overlap ($\gamma_{k+1}^+ \ge \gamma_k^-$ for every $k \in \mathbb{N}$) then the content of Theorem 1.7 is empty except for the right most limit $\gamma_0^+$.

From a general theorem in semi-group theory [12, p.276, Thm1.18 p.307, prop 1.15 p.305], Theorem 1.7 that concerns the generator $X_F$ is equivalent to the following result that concerns the semi-group $(e^{tX_F})_{t \ge 0}$ and illustrated in Figure 1.3.
<figure><img src="image_5.png" /><figcaption>FIGURE 1.3. Band spectrum of $e^{tX_F}$ in $\mathcal{H}_W(M;F)$ for $t > 0$, from Theorem 1.8. For $k \in \mathbb{N}$, and $\gamma_k^{\pm}$ defined in (1.6), the operator $e^{tX_F}$ has discrete spectrum in the annulus $e^{t\gamma_{k+1}^{\pm}} < |z| < e^{t\gamma_k^{\pm}}$ (if $\gamma_{k+1}^{\pm} < \gamma_k^{\pm}$) called “spectral gap”. It has also discrete spectrum on $|z| > e^{t\gamma_0^{\pm}}$ and possibly essential spectrum elsewhere (the colored annuli). This picture is somehow the exponential of Figure 1.2.</figcaption></figure>

**Theorem 1.8 (Ring spectrum).** Let $k \in \mathbb{N}$. For any $t > 0$, the operator $e^{tX_F} : \mathcal{H}_W(M;F) \to \mathcal{H}_W(M;F)$ has discrete spectrum on the ring $\{z \in \mathbb{C}, e^{t\gamma_{k+1}^{\pm}} < |z| < e^{t\gamma_k^{\pm}}\}$ (if non-empty, i.e. if $\gamma_{k+1}^{\pm} < \gamma_k^{\pm}$) and on $|z| > e^{t\gamma_0^{\pm}}$.

*Remark 1.9.* An immediate consequence of Theorem 1.8, if $\gamma_{k+1}^{\pm} < \gamma_k^{\pm}$ for some $k \in \mathbb{N}$, is the possibility to perform a contour integral of the resolvent of $e^{tX_F}$ at fixed value of $t > 0$, on the circle $C_{t,\epsilon} := \{z \in \mathbb{C} \text{ s.t. } |z| = e^{t(\gamma_{k+1}^{\pm}+\epsilon)}\}$ with $\epsilon > 0$ such that $\gamma_{k+1}^{\pm} + \epsilon < \gamma_k^{\pm}$ and such that there is no eigenvalue on this circle. Using Cauchy integral and holomorphic functional calculus, we get a spectral projector

$$ \Pi_{[0,k]} = \mathrm{Id} - \frac{1}{2\pi i} \oint_{C_{t,\epsilon}} (z - e^{tX_F})^{-1} dz $$

on the first $k$ external bands and one can approximate $e^{tX_F}$ for $t > 0$ by the spectral restriction $e^{tX_F} \Pi_{[0,k]}$ with the following estimate:

$$ (1.17) \qquad \left\| e^{tX_F} - e^{tX_F} \Pi_{[0,k]} \right\|_{\mathcal{H}_W(M;F)} \le C_{\epsilon} e^{t(\gamma_{k+1}^{\pm}+\epsilon)}, $$
where $e^{tX_F} \Pi_{[0,k]}$ is compared to a quantum evolution operator in this paper. This operator can be used to describe with great accuracy the **decay of correlations** $\langle v | e^{tX_F} u \rangle_{L^2(M;F)}$ as $t \to \infty$, that is the question$^6$ raised in the beginning of this paper.

**Remark 1.10.** The discrete spectra of $X_F$ (and eigenspace) are said “intrinsic” because they do not depend on the space $\mathcal{H}_W(M;F)$, see [16, Thm 1.5]. However more refined spectral properties as the norm of the resolvent $\|(z - X_F)^{-1}\|_{\mathcal{H}_W(M;F)}$ depend on the choice of the space $\mathcal{H}_W(M;F)$ and play an important role in this paper. For example the uniform boundness of the norm of the resolvent in the gaps given in Theorem 1.7 (implying Theorem 1.8) is obtained with the weight function $W$ and Sobolev space $\mathcal{H}_W(M;F)$ from [20] and defined in (4.15) because we have decay outside a parabolic neighborhood of the trapped set $\Sigma$, saturating the uncertainty principle. This property of uniform boundness is not true with the weight function and Sobolev space defined in [16] where the decay takes place only outside a (much bigger) conical neighborhood of the trapped set $\Sigma$.

1.5.2. *Weyl law.* Recall that $\dim M = 2d + 1$, with $d = \dim E_{s,u} = \dim N_{u,s}$. For some given $k \in \mathbb{N}$, suppose $B_k \cap (\bigcup_{k' \neq k} B_{k'}) = \emptyset$, i.e. that the band $B_k$ is isolated. Theorem 1.11 shows that the density of discrete eigenvalues of $X_F$ in band $B_k$ in the limit $\omega \to \infty$ converges to rank $(\mathcal{F}_k) \text{Vol}(M) \frac{\omega^d}{(2\pi)^{d+1}}$ with

$$ (1.18) \qquad \text{rank} (\mathcal{F}_k) = \binom{k+d-1}{d-1} \text{rank}(F). $$

It is analogous to the usual Weyl law.

**Theorem 1.11** (Weyl law for isolated bands). For any $k \in \mathbb{N}$ and $\epsilon > 0$, such that $\gamma_{k+1}^+ < \gamma_k^- - \epsilon$ and $\gamma_k^+ + \epsilon < \gamma_{k-1}^-$ (this last condition is only for $k \ge 1$)

$$ (1.19) \qquad \lim_{\delta \to +\infty} \limsup_{\omega \to \pm\infty} \left| \frac{1}{|\omega|^d} \mathfrak{H}\{\text{spect}(X_F) \cap \text{Box}_{\omega,\delta}\} - \frac{\text{rank}(\mathcal{F}_k) \text{Vol}(M)}{(2\pi)^{d+1}} \right| = 0 $$

with the spectral box

$$ \text{Box}_{\omega, \delta} := \{z \in ([\gamma_k^- - \epsilon, \gamma_k^+ + \epsilon] \times i[\omega, \omega + \delta])\}. $$

The proof is obtained in Section 7.

**Remark 1.12.** As the proof shows, the result holds true more generally for a group of isolated bands indexed by $k \in [k_1, k_2]$ with $k_1 \le k_2$, i.e. assuming $\gamma_{k_2+1}^+ < \gamma_{k_2}^-$ and $\gamma_{k_1}^+ < \gamma_{k_1-1}^-$. For the case $k_1 = 0$ we only have to assume $\gamma_{k_2+1}^+ < \gamma_{k_2}^-$. Then in (1.19), the interval $[\gamma_k^- - \epsilon, \gamma_k^+ + \epsilon]$ has to be replaced by $[\gamma_{k_2}^- - \epsilon, \gamma_{k_1}^+ + \epsilon]$ and rank $(\mathcal{F}_k)$ has to be replaced by rank $(\bigoplus_{k=k_1}^{k_2} \mathcal{F}_k)$.

<footnote>$^6$Remark that, thanks to the spectral gap assumption $\gamma_{k+1}^+ < \gamma_k^-$ and thanks to the abstract definition of the spectral projector $\Pi_{[0,k]}$ we obtain Eq. (1.17) without being embarrassed by the finite rank operator $R_{\sigma,l}$ described in Section 1.3.7, whose rank may grow with $t$.</footnote>
1.5.3. **Ergodic concentration of the spectrum.** We define in (8.1) the **maximal and minimal exponents** $\tilde{\gamma}_k^- \le \tilde{\gamma}_k^+$ of the bundle maps $e^{tX_F} \kappa$ with respect to the contact volume on $\Sigma$, satisfying $[\tilde{\gamma}_k^-, \tilde{\gamma}_k^+] \subset [\gamma_k^-, \gamma_k^+]$. Theorem 1.13 in addition to Theorem 1.11 shows that most of the eigenvalues in band $B_k$ belong to the narrower band Re $(z) \in [\tilde{\gamma}_k^-, \tilde{\gamma}_k^+]$.

**Theorem 1.13** (Ergodic concentration of the spectrum). Let $k \in \mathbb{N}$. Assume that $\gamma_{k+1}^+ < \gamma_k^-$ and $\gamma_k^+ < \gamma_{k-1}^-$ (this last condition is only for $k \ge 1$). For any $\epsilon > 0$ small enough, we have

$$ (1.20) \qquad \lim_{\delta \to +\infty} \limsup_{\omega \to \pm\infty} \frac{1}{|\omega|^d} \#\{\text{spect}(X_F) \cap \text{Strips}_{\omega, \delta}\} = 0 $$

with

$$ \text{Strips}_{\omega, \delta} := \{z \in ([\gamma_k^- - \epsilon, \gamma_k^+ + \epsilon] \setminus [\tilde{\gamma}_k^- - \epsilon, \tilde{\gamma}_k^+ + \epsilon]) \times [\omega, \omega + \delta]\}. $$

The proof is given in Section 8. For example, in the special case of band $k = 0$ and rank $(F) = 1$ we have rank $(\mathcal{F}_0) = 1$, and from ergodicity of $X$, most of eigenvalues accumulate on the vertical line

$$ \tilde{\gamma}_0^- = \tilde{\gamma}_0^+ = \frac{1}{\text{Vol}(M)} \int_M Ddm $$

being the space average of the “damping function” $D \in C(M; \mathbb{R})$ defined by $D(m) := V(m) + \frac{1}{2}\text{div}X_{/E_s}(m)$ with the potential function $V$ given in (2.5), see [18].

1.5.4. **Emerging quantum dynamics on vector bundles.** The band structure of the Ruelle spectrum and Weyl law described above reflect in fact a deeper geometric phenomenon that can be explained in terms of geometric quantization. Let us explain this.

**Wave-packet transform $\mathcal{J}$:** In paper [20, def. 4.23], see also (2.33), we introduce a wave-packet transform (here $S(T^*M; F)$ is the space of Schwartz sections of the bundle $F \to T^*M$)

$$ (1.21) \qquad \mathcal{J} : C^\infty(M; F) \to S(T^*M; F), $$

satisfying $\mathcal{J}^\dagger \mathcal{J} = \text{Id}_{L^2(M; F)}$ and that is the basic tool used for micro-local analysis. For example, using the characteristic function $\chi_{\Sigma, \sigma} = 1_{\{\|\rho_u + \rho_s\|_{g\rho} \le \sigma\}} : T^*M \to [0, 1]$

defined in (3.30), for the neighborhood of the set $\Sigma = \mathbb{R}^*A \subset T^*M$ at a distance $\sigma > 0$ measured by a specific metric $g$ (given in Lemma 2.8), we define the operator

$$ (1.22) \qquad \text{Op}(\chi_{\Sigma, \sigma}) = \mathcal{J}^\dagger \chi_{\Sigma, \sigma} \mathcal{J} : C^\infty(M; F) \to C^\infty(M; F) $$

that restricts functions to their micro-local components near $\Sigma$.

Theorem 4.6 will show that the norm decays exponentially fast outside the trapped set $\Sigma$ with an exponential rate, i.e. we have $\|e^{tX_F} (\text{Id} - \text{Op}(\chi_{\Sigma, \sigma}))\|_{\mathcal{H}_W(M)} \le Ce^{-t\Lambda}$ for $t \ge 0$, with a rate $\Lambda > 0$ arbitrarily large if $\sigma$ is large enough (depending on $t$). This implies that for our study, we can consider only the component of the dynamics $e^{tX} \text{Op}(\chi_{\Sigma, \sigma})$ near the trapped set.
**Quantization:** Recall the vector bundle $\mathcal{F}(E_s) \to M$ defined in (1.9) and the lifted bundle $\mathcal{F}(N_s) \to \Sigma$. For a map $a \in C^\beta(M; \text{End}(\mathcal{F}(E_s)))$ (or even $a \in C^\beta(M; L(\mathcal{F}(E_s), \mathcal{F}'(E_s)))$ and time $t \in \mathbb{R}$, we will define a quantum operator in Definition 4.27

$$ \mathrm{Op}_{\Sigma}(e^{tX_{\mathcal{F}}}\tilde{a}) : C^{\infty}(M; F) \to C^{\infty}(M; F), $$

where $\tilde{a} \in C^\beta(\Sigma; L(\mathcal{F}(N_s)))$ is the lifted map.

For two operators $A, B$, we define in (3.32) an equivalence relation $A \approx B$ if their difference in any $\sigma$-neighborhood of the trapped set $\Sigma$ and for large enough frequencies becomes “negligible”. A consequence is: if $A \approx B$ then for any $\sigma > 0$,

$$ (1.23) \qquad \left\| \mathrm{Op}(\chi_{\Sigma, \sigma})(A - B) \mathrm{Op}(\chi_{\Sigma, \sigma}) (\mathrm{Id} - \mathrm{Op}(\chi_{\omega})) \right\|_{\mathcal{H}_W(M)} \xrightarrow{\omega \to \infty} 0, $$

where $\mathrm{Op}(\chi_{\Sigma, \sigma})$ is given in (1.22) and for $\omega > 0$, $\mathrm{Op}(\chi_{\omega}) : S'(M; F) \to S(M; F)$ is a smoothing operator hence compact that removes frequencies larger than $\omega$, defined as follows: let $g_M$ be an arbitrary smooth metric on $M$. Let $\chi_{\omega} : T^*M \to \mathbb{R}^+$ given by

$$ (1.24) \qquad \chi_{\omega}(\cdot) = 1_{\{\|\cdot\|_{g_M} \le \omega\}} $$

and let

$$ (1.25) \qquad \mathrm{Op}(\chi_{\omega}) := \mathcal{T}^{\dagger} \chi_{\omega} \mathcal{T}. $$

In Theorem 4.29 we will obtain

**Theorem 1.14** (Approximation of the dynamics by quantum operator). *For any $t \in \mathbb{R}$,*

$$ (1.26) \qquad e^{tX_{\mathcal{F}}} \approx \mathrm{Op}_{\Sigma}(e^{tX_{\mathcal{F}}}). $$

In Theorem 4.37 we will obtain$^7$

**Theorem 1.15** (Composition formula). *For any symbols*

$$ a, b \in C^{\beta}(M; L(\mathcal{F}_{[0,K]}(E_s))), $$

any $t, t' \in \mathbb{R}$, we have

$$ (1.27) \qquad \mathrm{Op}_{\Sigma}(e^{tX_{\mathcal{F}}\tilde{a}}) \mathrm{Op}_{\Sigma}(e^{t'X_{\mathcal{F}}\tilde{b}}) \approx \mathrm{Op}_{\Sigma}(e^{tX_{\mathcal{F}}} \tilde{a} e^{t'X_{\mathcal{F}}} \tilde{b}). $$

In particular, since for any $k, k' \in \mathbb{N}$ we have for the Taylor projectors $[T_k, T_{k'}] = \delta_{k=k'} T_k$ and $[T_k, e^{tX_{\mathcal{F}}}] = 0$, we will get in Corollary 4.41 that

$$ (1.28) \qquad \mathrm{Op}_{\Sigma}(e^{tX_{\mathcal{F}}\tilde{T}_k}) \mathrm{Op}_{\Sigma}(e^{t'X_{\mathcal{F}}\tilde{T}_{k'}}) \approx \delta_{k=k'} \mathrm{Op}_{\Sigma}(e^{(t+t')X_{\mathcal{F}}\tilde{T}_k}). $$

We will get a general continuity Theorem 4.31,

**Theorem 1.16** (Continuity theorem for F.I.O.). *There exists $C > 0$, such that for any symbol $a \in C^\beta(M; L(\mathcal{F}_{[0,K]}(E_s)))$ and $t \in \mathbb{R}$,*

$$ (1.29) \qquad \left\| \mathrm{Op}_{\Sigma}(e^{tX_{\mathcal{F}}\tilde{a}}) \right\|_{\mathcal{H}_W(M)} \le C \left\| e^{tX_{\mathcal{F}}\tilde{a}} \right\|_{\mathcal{H}_W(\mathcal{F})}. $$

As a corollary we will obtain in Corollary 4.35 some estimates that reflect Eq. (1.6) at the quantum level, such as

<footnote>$^7$Recall from Section 1.3 that a symbol $a \in C^\beta(M; \text{End}(\mathcal{F}_{[0,K]}(E_s)))$ can be extended to $a \circ T_{[0,K]} \in C^\beta(M; L(\mathcal{F}, \mathcal{F}'))$ using Taylor projectors $T_{[0,K]} : \mathcal{F}(E_s) \to \mathcal{F}_{[0,K]}(E_s) \subset \mathcal{F}'(E_s)$.</footnote>
**Corollary 1.17.** For any $k \in \mathbb{N}$, $\forall \epsilon > 0$, $\exists C_{k,\epsilon} > 0, \forall t \ge 0$,

$$ (1.30) \qquad \left\| \mathrm{Op}_{\Sigma} (e^{tX_{\Sigma}} \tilde{T}_k) \right\|_{\mathcal{H}_W(M)} \le C_{k,\epsilon} e^{t(\gamma_k^+ + \epsilon)}, $$

$$ (1.31) \qquad \left\| \mathrm{Op}_{\Sigma} (e^{-tX_{\Sigma}} \tilde{T}_k) \right\|_{\mathcal{H}_W(M)} \le C_{k,\epsilon} e^{-t(\gamma_k^- - \epsilon)}. $$

The band spectrum of the Ruelle spectrum in Theorem 1.7 and Theorem 1.8 is a direct manifestation of these boundness estimates together with the algebraic structure of (1.28).

In Theorem 4.43 we will get some trace formula that will be used to prove Weyl law in Section 7.

**1.5.5. Horocycle operators.** In the special case of the geodesic flow on $M = \Gamma \backslash \mathrm{SL}_2(\mathbb{R})$, the Lie algebra of the $\mathrm{sl}_2(\mathbb{R})$ provides left invariant vector fields $u \in C^{\infty}(M; E_u)$, $s \in C^{\infty}(M; E_s)$ that satisfies

$$ (1.32) \qquad [s, u] = X $$

and called respectively unstable/stable horocycle vector fields (and more generally for the algebra of so $(n, 1)$), see [10, 21, 27] where it is shown that these operators $s, u$ map respectively band $B_k$ to band $B_{k-1}, B_{k+1}$.

In the general case of a contact Anosov vector field $X$, we propose below some “**aproximate horocycle operators**” $\mathrm{Op}_{\Sigma}(\iota_s)$, $\mathrm{Op}_{\Sigma}(u\odot)$ that satisfy some commutator relation similar to (1.32) but in the semi-classical limit $\omega \gg 1$. They also respectively map band $B_k$ to $B_{k\mp 1}$ (at first order).

If $u \in C^\beta(M; E_u)$, $s \in C^\beta(M; E_s)$ are respectively Hölder continuous sections of the unstable/stable bundles of the Anosov dynamics, we naturally associate to them the following “operator-valued symbols” (see Section 9 for a more detailed definition)

$$ (1.33) \qquad \iota_s : C^\beta(\Sigma; \mathcal{F}_k(N_s)) \to C^\beta(\Sigma; \mathcal{F}_{k-1}(N_s)) $$

by point-wise tensor-contraction with the polynomial in (1.5) and

$$ (1.34) \qquad u \odot : C^\beta(\Sigma; \mathcal{F}_k(N_s)) \to C^\beta(\Sigma; \mathcal{F}_{k+1}(N_s)) $$

by point-wise symmetric tensor product. We will see in Lemma 9.2 that they satisfy the point-wise Weyl algebra (also called symplectic Clifford algebra)

$$ (1.35) \qquad [\iota_s, u \odot] \tilde{T}_k = \omega(\cdot)((dA)(s, u)) \tilde{T}_k, $$

where $\omega(\cdot) : \omega\mathcal{A}(m) \in \Sigma \to \omega \in \mathbb{R}$ is the frequency map, $d\mathcal{A}$ is the symplectic form on $E_u \oplus E_s$ and $\tilde{T}_k$ the projector on $\mathcal{F}_k$.

**Theorem 1.18 (Quantized Weyl algebra).** Let $K \in \mathbb{N}$ and $\tilde{T}_{[0,K]}$ the projector on $\mathcal{F}_{[0,K]}$. For $k < K$ we have

$$ (1.36) \qquad [\mathrm{Op}_{\Sigma}(\iota_s \tilde{T}_{[0,K]}), \mathrm{Op}_{\Sigma}((u\odot) \tilde{T}_{[0,K]})] \mathrm{Op}_{\Sigma}(\tilde{T}_k) \approx \mathrm{Op}_{\Sigma}(\omega(\cdot)(d\mathcal{A})(s, u) \tilde{T}_k) $$
and for any $t \in \mathbb{R}$,

$$ (1.37) \qquad e^{tX_F} \mathrm{Op}_{\Sigma}(\iota_s \tilde{T}_k) \approx \mathrm{Op}_{\Sigma}(\iota_{d\phi^t s} \tilde{T}_k) e^{tX_F}, $$

$$ (1.38) \qquad e^{tX_F} \mathrm{Op}_{\Sigma}((u\odot)\tilde{T}_k) \approx \mathrm{Op}_{\Sigma}(((d\phi^t u)\odot)\tilde{T}_k) e^{tX_F}. $$

The proof is obtained in Section 9.
*Remark 1.19.* To compare (1.36) with (1.32), notice that $\omega(\cdot)$ is the principal symbol of $X$, see [20, prop.4.42].

## 1.6. Organization of the paper.

**First part, setup of a symbolic calculus for contact Anosov vector field.** The first part of paper is organized by starting with a general model and then adding more and more assumptions to arrive at the model we are interested in and get more specific results.

In Section 2 we present the micro-local analysis of a general non-vanishing vector field $X$ on a manifold $M$. The strategy is to lift the analysis of the pullback operator $e^{tX}$ on the cotangent bundle $T^*M$ using the wave-packet transform $\mathcal{T}: C^\infty(M) \to S(T^*M)$. In the high frequency limit (semi-classical limit), the linearization of the flow $\tilde{\phi}^t$ induced on $T^*M$ gives a good description of the pullback operator $e^{tX}$ in terms of the differential map $d\tilde{\phi}^t$ on $TT^*M$. This is obtained in Theorem 2.16.

In Section 3 we assume furthermore that $X$ is a Reeb vector field, i.e. defined from smooth contact one-form $\mathcal{A}$ on $M$. The purpose of this section is to obtain a precise description of the dynamics $e^{tX}$ not on all $T^*M$ but only in a vicinity of the symplecticization subset $\Sigma := (\mathbb{R} \setminus \{0\})\mathcal{A} \subset T^*M$. This subset $\Sigma$ is an invariant set for the dynamics $\tilde{\phi}^t$ induced on $T^*M$. Our description of the operator $e^{tX}$ in Theorem 3.19 uses the differential $d\tilde{\phi}^t$ on the vector bundle $TT^*M$ restricted to the base $\Sigma \subset T^*M$ and denoted $T_\Sigma T^*M$. We get an alternative description in Theorem 3.15 in terms of the normal bundle $N := (T\Sigma)^\perp \to \Sigma$. Both descriptions will be used later.

In Section 4 we assume furthermore that $X$ is a contact Anosov vector field on $M$, the main subject of this paper. First, from this assumption, we have that the outside of the set $\Sigma$ is negligible for the description of $e^{tX}$ for large time. This is Theorem 4.6 and obtained using escape functions, anisotropic Sobolev norm, results from previous papers. Secondly, the normal bundle over $\Sigma$ splits as $N = N_s \oplus N_u$ with stable and unstable invariant sub-bundles. It is there that we introduce the vector bundles $\mathcal{F}_k(N_s) = |\det N_s|^{-1/2} \otimes \text{Pol}_k(N_s) \otimes F \to \Sigma$ for every $k \in \mathbb{N}$. In Theorem 4.16 we obtain a description of the operator $e^{tX}$ in terms of the dynamics induced on these bundles $\mathcal{F}_k$. We show that this description is a “geometric quantization framework” with some adapted symbolic calculus. We introduce symbols in Definition 4.24 and quantization in Definition 4.27 giving P.D.O. and F.I.O. The rest of the section is to obtain classical and useful results in symbolic calculus, as continuity Theorem 4.31, composition Theorem 4.37, Egorov theorem 4.41 and Trace formula 4.43.

**Second part, some consequences.** In the second part of the paper we use the setup of the first part to deduce some specific spectral properties of a contact Anosov vector field $X$. In Section 5, we show how to use these properties to deduce the band structure of the Ruelle spectrum of the vector field $X$. In Section 6, we deduce Theorem 1.5 about emergence of quantum dynamics. Section 7 gives the proof of the Weyl law and Section 8 explains the concentration in narrower bands. Section 9 gives definitions and proofs of Theorem 1.18 for approximate horocycle operators.

**Appendices:** Appendix A gives some convention of notations used in this paper. Appendix B contains additional comments about flows. Appendices C and D are quite important and sustain the main argument used in this paper: for the analysis of the differential $d\tilde{\phi}^t$ on $T_\Sigma T^*M$, we use Bargmann transform, i.e. wave-packet transform
on a Euclidean vector space with metaplectic operators Op(d$\tilde{\phi}^t$) obtained by quantization of the bundle map of linear symplectic maps d$\tilde{\phi}^t : TT^*M \to TT^*M$. All the details of this are given in Appendix C. We also use results for linear expanding maps given in Appendix D.

## 2. MICRO-LOCAL ANALYSIS OF A NON-VANISHING VECTOR FIELD X

In this section we consider a general smooth and non-vanishing vector field $X$ (not necessarily Anosov) on a smooth closed manifold $M$ (not necessarily contact). We review some results given in [20]. As in (1.21), we define the wave packet transform that is a linear operator $\mathcal{T} : C^\infty(M) \to S(T^*M)$ and an isometry from $S^2 L^2(M; dm)$ to $L^2(T^*M; \frac{d\rho}{(2\pi)^{\dim M}})$, i.e. satisfies $\mathcal{T}^\dagger \mathcal{T} = \text{Id}$ where $\mathcal{T}^\dagger$ is the $L^2$-adjoint operator. Then we recall the Theorem of propagation of singularities 2.13, saying that the Schwartz kernel of the lifted pull back operator $\mathcal{T} e^{tX} \mathcal{T}^\dagger$ on $S(T^*M)$, with $t \in \mathbb{R}$, decays very fast outside the graph of the flow $\tilde{\phi}^t := (d\phi^t)^* : T^*M \to T^*M$ acting on the cotangent space $T^*M$. Moreover by linearization, we give in Theorem 2.16 an approximate description of this Schwartz kernel in the neighborhood of the graph of $\tilde{\phi}^t$.

### 2.1. Vector field X and pull back operator $e^{tX}$.
Let $M$ be a $C^\infty$ closed connected manifold. Let $X \in C^\infty(M; TM)$ be a $C^\infty$ vector field on $M$, considered as a first order differential operator acting on smooth functions $C^\infty(M)$, i.e. in local coordinates $y = (y_1, y_2, \dots, y_{\dim M})$, $X = \sum_{j=1}^{\dim M} X_j(y) \frac{\partial}{\partial y_j}$. For $t \in \mathbb{R}$, let

$$ (2.1) \qquad \phi^t : M \to M $$

be the $C^\infty$ flow defined by $\frac{d(u \circ \phi^t)}{dt} = Xu, \forall u \in C^\infty(M)$, i.e.

$$ (2.2) \qquad e^{tX}u := u \circ \phi^t = (\phi^t)^\circ u, $$

where $(\phi^t)^\circ = e^{tX}$ denotes the **pull back operator** acting on functions (following notations in Appendix A). See Appendix B.1 for additional remarks about the transfer operator.

### 2.2. More general pull back operator $e^{tX_F}$ with generator $X_F$. 
It will be interesting to consider the following more general situation. Let $\pi : F \to M$ be a smooth complex vector bundle of finite rank over $M$. Let $X_F$ be a **derivation over $X$**, i.e. $X_F$ is a first order differential operator acting on sections of $F$

$$ (2.3) \qquad X_F : C^\infty(M; F) \to C^\infty(M; F) $$

satisfying the Leibniz rule

$$ (2.4) \qquad X_F(fu) = X(f)u + fX_F(u), \quad \forall f \in C^\infty(M; \mathbb{C}), \forall u \in C^\infty(M; F). $$

**Example 2.1.** If $F$ is the bundle of differential forms $F = \Lambda^*(TM)$ (or a more general tensor field). The Lie derivative $X_F$ (obtained from the differential $d\phi^t$) satisfies (2.4), so $X_F$ is a derivation over $X$.

<footnote n="8">⁸Here $dm$ is an arbitrary smooth measure on $M$ and $d\rho$ the canonical Liouville measure on $T^*M$.</footnote>
**Example 2.2.** A simple but useful example is the trivial rank one bundle $F = M \times \mathbb{C}$. Let $V \in C^\infty(M; \mathbb{C})$ called the **potential** function. The operator

$$ (2.5) \qquad X_F = X + V $$

satisfies (2.4), where $V$ is seen as the multiplication operator by the function $V$. By integration we get, with $u \in C^\infty(M)$,

$$ (2.6) \qquad e^{tX_F} u = \underbrace{e^{\int_0^t V \circ \phi^s ds}}_{\text{amplitude transport}} \underbrace{u \circ \phi^t}_{\text{transport}} . $$

More remarks about general pull back operators $e^{tX_F}$ are given in Section B.2.

**2.3. Lifted flow** $\tilde{\phi}^t : T^*M \to T^*M$ and its **generator** $\tilde{X}$. In this section we define and describe the flow $\tilde{\phi}^t = (d\tilde{\phi}^t)^*$ : $T^*M \to T^*M$ induced from the flow $\phi^t$ in (2.1). It will appear in Theorem 2.13, that for the analysis of $e^{tX}$, this flow $\tilde{\phi}^t$ on $T^*M$ plays a major role.

**2.3.1. Frequency function** $\omega$. We define the **frequency function** $\omega \in C^\infty(T^*M; \mathbb{R})$ by

$$ (2.7) \qquad \forall \rho \in T^*M, \quad \omega(\rho) := \rho(X), $$

that measures the oscillations of $\rho$ along the flow direction $X$.

**Remark 2.3.** In terms of micro-local analysis, the function $i\omega$ is the **principal symbol** of the vector field $X$, see [16, footnote page 332.].

**2.3.2. The lifted flow** $\tilde{\phi}^t : T^*M \to T^*M$. For $m \in M$, we denote $d\tilde{\phi}^t : T_mM \to T_{\tilde{\phi}^t(m)}M$, the differential of the flow map $\phi^t : M \to M$. By duality we get a map

$$ (2.8) \qquad \tilde{\phi}^t := (d\tilde{\phi}^t)^* : T_m^*M \to T_{\tilde{\phi}^{-t}(m)}^*M, $$

defined by

$$ (2.9) \qquad \langle v | (d\tilde{\phi}^t)^* (\rho) \rangle_{T_m M \times T_m^* M} = \langle d\tilde{\phi}^t v | \rho \rangle_{T_{\tilde{\phi}^t(m)} M \times T_{\tilde{\phi}^t(m)}^* M}, \quad \forall \rho \in T_{\tilde{\phi}^t(m)}^* M, v \in T_m M, m \in M. $$

Notice that $\tilde{\phi}^t$ is a lift of the inverse map $\phi^{-t}$ map, i.e.

$$ (2.10) \qquad \pi(\tilde{\phi}^t(\rho)) = \phi^{-t}(\pi(\rho)), $$

with the bundle projection map $\pi : T^*M \to M$.

**2.3.3. The canonical Liouville one form** $\theta$ and the **symplectic two form** $\Omega$. Let $\theta$ be the canonical Liouville one form on the vector bundle $\pi : T^*M \to M$ [34, p.90], [7, p.7], [8, p.56]: at any point $\rho \in T^*M$, $\theta_\rho$ is defined by

$$ (2.11) \qquad \forall V \in T_\rho T^*M, \quad \theta_\rho(V) := \rho(d\pi(V)). $$

Using the local coordinates $(y_j)_{j=1,...,\dim M}$ on $M$ and dual coordinates $(\eta_j)_j$ on $T^*M$, this gives

$$ (2.12) \qquad \theta = \sum_{j=1}^{\dim M} \eta_j dy_j. $$

Let $\Omega$ be the canonical symplectic form

$$ (2.13) \qquad \Omega := d\theta = \sum_{j=1}^{\dim M} d\eta_j \wedge dy_j, $$
that gives a bundle isomorphism $\check{\Omega} : TT^*M \to T^*T^*M$ defined by

$$ \forall V \in TT^*M, \quad \check{\Omega}(V) := \Omega(V, \cdot) \in T^*T^*M. $$

Using notation in Appendix A for the pullback, the Definition (2.11) of $\theta$ on $T^*M$ is equivalent to the fact that for any smooth one form $\alpha \in C^\infty(M; T^*M)$, considered as a map $\alpha : M \to T^*M$, considering its differential $d\alpha : TM \to TT^*M$, we have

$$ (2.14) \qquad (d\alpha)^\circ \theta = \alpha, $$

and $\text{Ker}\theta = \text{Ker}(d\pi)$ away from the zero section. Taking the differential we get

$$ (2.15) \qquad (d\alpha)^\circ \Omega = d\alpha. $$

We define the **Euler (or Liouville) vector field** on $T^*M$

$$ (2.16) \qquad \mathcal{E} := \check{\Omega}^{-1}(\theta). $$

In coordinates, $\mathcal{E} = \sum_{j=1}^{\text{dim}M} \eta_j \frac{\partial}{\partial \eta_j}$. Using Cartan formula we get for the Lie derivatives

$$ \mathcal{L}_{\mathcal{E}}\theta = \theta, \quad \mathcal{L}_{\mathcal{E}}\Omega = \Omega. $$

**2.3.4. The vector field $\tilde{X}$ on $T^*M$.** Let us write $\tilde{X}$ for the vector field on $T^*M$ that generates the flow $\tilde{\phi}^t$ (2.8). Notice from (2.10) that

$$ (2.17) \qquad (d\pi)(\tilde{X}) = -X. $$

**Lemma 2.4** ([23, Thm 2.124 p.112, in coordinates], [36, prop. 1.21 p.15]). *The Lie derivative of $\theta$ by the vector field $\tilde{X}$ vanishes:*

$$ (2.18) \qquad \mathcal{L}_{\tilde{X}}\theta = 0, $$

and $\tilde{X}$ is determined by

$$ (2.19) \qquad \tilde{X} = \check{\Omega}^{-1}(d\omega), $$

i.e. $\tilde{X}$ is the Hamiltonian vector field on $(T^*M, \Omega)$, with Hamiltonian function $\omega$, Eq. (2.7).

*Proof.* [23, Thm 2.124 p.112, for a proof using coordinates], [36, Prop. 1.21 p.15] Let $\rho \in T^*M$ and $V \in T_\rho(T^*M)$. For any $t \in \mathbb{R}$ we have

$$ \begin{align*} \left\langle \left(d\tilde{\phi}^t\right)^* \theta_{\tilde{\phi}^t(\rho)} \right\rangle (V) &= \theta_{\tilde{\phi}^t(\rho)} \left\langle \left(d\tilde{\phi}^t\right) V \right\rangle_{(2.11)} = \tilde{\phi}^t(\rho) \left\langle d\pi \left(d\tilde{\phi}^t(V)\right) \right\rangle \\ &= \left\langle \left(d\phi^t\right)^* (\rho) \left| d\phi^{-t}(d\pi(V)) \right. \right\rangle_{(2.9)} = \rho(d\pi(V)) = \theta_\rho(V) \\ \Leftrightarrow \left(d\tilde{\phi}^t\right)^* \theta_{\tilde{\phi}^t(\rho)} &= \theta_\rho \Leftrightarrow e^{t\mathcal{L}_{\tilde{X}}\theta} = \theta \Leftrightarrow \mathcal{L}_{\tilde{X}}\theta = 0. \end{align*} $$

This gives (2.18). Notice that

$$ (2.20) \qquad \omega(\rho) = \rho(X) = -\rho(d\pi(\tilde{X})) = -\theta_{\rho}(\tilde{X}) = -\iota_{\tilde{X}}\theta_{\rho}. $$

Hence, using Cartan formula $\mathcal{L}_{\tilde{X}}\theta = d(\iota_{\tilde{X}}\theta) + \iota_{\tilde{X}}d\theta$ we get

$$ d\omega = -d(\iota_{\tilde{X}}\theta) = -\mathcal{L}_{\tilde{X}}\theta + \iota_{\tilde{X}}d\theta = \Omega(\tilde{X}, \cdot), $$

that gives (2.19). □
**Remark 2.5.** As a consequence of (2.19) we have $\tilde{X}(\omega) = \Omega(\tilde{X}, \tilde{X}) = 0$, i.e. for every $\omega \in \mathbb{R}$, the frequency level

$$ (2.21) \qquad \Sigma_{\omega} := \omega^{-1}(\omega) = \{\rho \in T^*M, \ X(\rho) = \omega\} \subset T^*M $$

that is an affine sub-bundle of $T^*M$ is preserved by $\tilde{\phi}^t$.

**Remark 2.6.** For $\omega \neq 0$, the one form $-\frac{1}{\omega}\theta$ is a contact one form on the frequency level $\Sigma_{\omega}$ and $\tilde{X}$ is its Reeb vector field, because $(-\frac{1}{\omega}\theta)(\tilde{X})_{(2.20)} = 1$ and $d(-\frac{1}{\omega}\theta)(\tilde{X}, \cdot) = -\frac{1}{\omega}\Omega(\tilde{X}, \cdot) = -\frac{1}{\omega}d\omega_{(2.21)} = 0$.

**2.4. Metric g on $T^*M$.** Let us suppose that we have an atlas $M = \cup_{j=1}^J U_j$ where $U_j$ are open subsets on $M$ such that for $j = 1, \dots, J$,

$$ (2.22) \qquad \kappa_j : m \in U_j \to y = (x, z) \in V_j \subset \mathbb{R}_x^{\dim M-1} \times \mathbb{R}_z $$

are flow box coordinates, i.e. $(d\kappa_j)(X) = \frac{\partial}{\partial z}$. On each chart $U_j$, let us denote $\eta = (\xi, \omega) \in \mathbb{R}^{\dim M-1} \times \mathbb{R}$ the dual coordinates associated to $(x, z)$. Hence $(y, \eta) = (x, z, \xi, \omega)$ are coordinates on $T^*(\mathbb{R}_x^{\dim M-1} \times \mathbb{R}_z)$.

Let us recall the definition and properties of the metric $g$ on $T^*M$ introduced in [20, section 4.1.2] with parameters that we take here $\alpha^{\perp} = \frac{1}{2}$ and $\alpha^{\parallel} = 0$. For $s \in \mathbb{R}$, we set

$$ (2.23) \qquad \langle s \rangle := (1 + s^2)^{1/2} \underset{|s| \gg 1}{\sim} |s|. $$

For $\eta \in \mathbb{R}^{\dim M}$, let

$$ (2.24) \qquad \delta^{\perp}(\eta) := \langle |\eta| \rangle^{-1/2}. $$

In coordinates, we define the metric $g$ on $T^*\mathbb{R}^{\dim M} = \mathbb{R}_{\varrho}^{2\dim M}$ at a point $\varrho = (y, \eta) \in \mathbb{R}^{\dim M} \times \mathbb{R}^{\dim M}$ by

$$ (2.25) \qquad g(\varrho) := \left(\frac{dx}{\delta^{\perp}(\eta)}\right)^2 + \left(\frac{d\xi}{\delta^{\perp}(\eta)^{-1}}\right)^2 + dz^2 + d\omega^2. $$

This gives a metric $g_j := (d\kappa_j)^* g$ on $T^*U_j$. It is shown in [20, Lemma 4.6] that on each intersection $U_j \cap U_{j'}$, the metrics $g_j$ and $g_{j'}$ are relatively bounded by each other uniformly in $\eta$. Each metric $g_j$ is compatible with the canonical symplectic form $\Omega$ on $T^*M$ given in (2.13).

**Remark 2.7.** An important property of the metric $g$ that will be essential in our analysis is that the size of a unit ball projects on the coordinates $x$ as a set of size of order $\delta^{\perp}(\eta) = \langle |\eta| \rangle^{-1/2}$ that goes to zero as $|\eta| \to +\infty$. See [20, Figure 2.3].

We recall the definition that $\Omega$, $g$ are compatible in Definition C.13.

**Lemma 2.8.** There exists a global smooth metric $g$ on $T^*M$ that is also compatible with $\Omega$ and relatively bounded with respect to every metric $g_j$ on any chart $T^*U_j$.
*Proof.* Let $G$ be a global metric on $T^*M$ defined by

$$ (2.26) \qquad G := \sum_{j=1}^{J} |\det d\kappa_j| (\chi_j^2 \circ \kappa_j \circ \pi) \cdot g_j, $$

where $\pi: T^*M \to M$ denotes the bundle projection and $\chi_j \in C_0^\infty(\kappa_j(U_j); \mathbb{R}^+)$ form a quadratic partition of unity on $M$, i.e. $\sum_j \chi_j^2 \circ \kappa_j |\det d\kappa_j| = 1$, see [20, Lemma 4.2]. Recall that on each intersection $U_j \cap U_{j'}$, the metrics $g_j$ and $g_{j'}$ are uniformly equivalent (i.e. relatively bounded) that we write $g_j \asymp g_{j'}$. Consequently, from finiteness of the sum (2.26), on any chart $T^*U_j$, we have

$$ (2.27) \qquad G \asymp g_j. $$

We define

$$ (2.28) \qquad K := \check{\Omega}^{-1} \check{G} : TT^*M \to TT^*M, $$

where $\check{G}, \check{\Omega}: TT^*M \to T^*T^*M$ are defined from $G, \Omega$ as in (A.1). We consider the polar decomposition of $K$ written $K = J|K|$ where $|K| = \sqrt{K^\dagger g K}$ is positive definite Hermitian (bundle maps over $T^*M$) and $J$ is an almost complex structure because, using that $\check{G}^* = \check{G}$ and $\check{\Omega}^* = -\check{\Omega}$ we get $K^\dagger g = \check{G}^{-1} K^* \check{G} = \check{G}^{-1} \check{G}^* \check{\Omega}^{-1} \check{G} = -\check{\Omega}^{-1} \check{G} = -K$, hence $J^2 = \frac{K^2}{(K^\dagger g K)} = -\text{Id}$. Since $g_j$ is compatible with $\Omega$ then $\check{g}_j = \check{\Omega} J_j$

with $J_j^2 = -\text{Id}$. Thus $\check{\Omega} K = \check{G}_{(2.28)} \asymp \check{g}_{(2.27)}$ $g_j = \check{\Omega} J_j$ gives $K \asymp J_j$, then

$$ (2.29) \qquad |K| \asymp \text{Id}. $$

We define the metric $g$ by

$$ (2.30) \qquad \check{g} := \check{\Omega} J : TT^*M \to T^*T^*M. $$

By construction, the global metric $g$ is smooth, compatible with $\Omega$ and relatively bounded with respect to every $g_j$ on any chart $T^*U_j$ because $\check{g} = \check{\Omega} J_{(2.29)} \asymp \check{\Omega} J|K| = \check{\Omega} K_{(2.28)} = \check{G} \asymp \check{g}_j$. $\square$

We will denote $\text{dist}_g(\rho', \rho)$ the distance between two points $\rho, \rho' \in T^*M$ according to the metric $g$. It is shown in [20, Lemma 4.5] that the metric $g$ is **geodesically complete**.

We will also use a property of slow variation of the metric $g$ given in [20, Lemma 4.12] and recalled later in (2.59). Finally we have from [20, Lemma 4.6] the “Lipschitz property of $g$”, namely, for any $t \in \mathbb{R}$, there exists a constant $C_t > 0$ such that for any $\rho, \rho' \in T^*M$,

$$ (2.31) \qquad \text{dist}_g(\tilde{\phi}^t(\rho), \tilde{\phi}^t(\rho')) \le C_t \text{dist}_g(\rho, \rho'), $$

where $\tilde{\phi}^t: T^*M \to T^*M$ is defined in (2.8).

2.5. **Wave-packet transform $\mathcal{T}$**. As in [20, def. 4.23], for every coordinate chart $U_j \subset M$ with $j \in \{1, \dots, J\}$ and $\rho \in T^*U_j$ we define a wave packet<fnref n="9" /> $\Phi_{j,\rho} \in C^\infty(M; \mathbb{C})$ and

$$ (2.32) \qquad \Phi_{j,\rho}(y') \sim a_{\rho} \chi(y' - y) \exp\left(i\eta \cdot (y' - y) - \|y' - y\|_{g_{\rho}}^2\right) $$

<footnote n="9">⁹We refer to [20, def. 4.23] for the precise expression of a wave packet $\Phi_{j,\rho}$. Here it is enough to say that on a chart $U_j$, for a given $\rho = (y, \eta) \in T^*\mathbb{R}^{n+1}$ and in local coordinates $y' \in \mathbb{R}^{n+1}$, then for large $|\eta| \gg 1$, the function $\Phi_{j,\rho}$ is equivalent to a **Gaussian wave packet** in “vertical Gauge”:</footnote>
the “wave packet transform”

$$
(2.33) \qquad \mathcal{T} : \begin{cases} C^\infty (M; \mathbb{C}) & \to S (T^*M; \mathbb{C}^J) \\ u & \to (\langle \Phi_{j,\rho} | u \rangle_{L^2(M)})_{\rho \in T^*M, j \in \{1, \dots, J\}} \end{cases},
$$

that satisfies the “resolution of identity on $C^\infty(M)$” [20, prop. 4.24]

$$
(2.34) \qquad \mathrm{Id}_{/C^{\infty}(M)} = \mathcal{T}^{\dagger} \mathcal{T},
$$

where

$$
\mathcal{T}^{\dagger} : L^2 \left( T^*M \otimes \mathbb{C}^J ; \frac{d\rho}{(2\pi)^{\dim M}} \right) \to L^2 (M ; dm)
$$

is the $L^2$-adjoint of $\mathcal{T}$ where $d\rho \equiv dyd\eta$ is the canonical measure on $T^*M$ and $dm$ an
arbitrary smooth measure on $M$. We will introduce the **wave packet projector** onto
the image of $\mathcal{T}$, see [20, prop. 4.27]:

$$
(2.35) \qquad \mathcal{P} := \mathcal{T} \mathcal{T}^{\dagger}.
$$

Remark 2.9. Below for simplicity, we will ignore $\otimes \mathbb{C}^J$ in the notation, i.e. ignore chart components, though keeping it in mind. The operator $\mathcal{T}^\dagger : S'(T^*M) \to C^\infty(M)$ is smoothing.

2.6. Description of the operator $e^{tX}$ with the bundle $TT^*M$.

Remark 2.10. From now on we will often use the knowledge about Bargmann trans-
form that is summarized in Appendix C and we ask the readers to check the notation
and their knowledge. We will often refer to this Appendix C.

We first introduce a definition that will be used to express that some operator $R$ is
“under control” or “negligible” in our analysis. It will mean that the Schwartz kernel
of $\mathcal{R}\mathcal{R}\mathcal{T}^\dagger$ decays very fast outside the graph of $\tilde{\phi}^t$ defined in (2.8) and moreover that on
the graph, the Schwartz kernel is bounded by the polynomial growth $\langle|\rho|\rangle^m$.

**Definition 2.11.** Let $t \in \mathbb{R}$ and $m \in \mathbb{R}$. We define $\Psi_{\tilde{\phi}^t}^m$ as the set of operators $R : C^\infty(M) \to S'(M)$ such that for any $N > 0$, there exists a constant $C_{N,t} > 0$ such that for any $\rho, \rho' \in T^*M$

$$
(2.36) \quad \left| \langle \delta_{\rho'} | \mathcal{R} \mathcal{T}^{\dagger} \delta_{\rho} \rangle_{L^2(T^*M)} \right| \le C_{N,t} \left\langle \operatorname{dist}_g (\rho', \tilde{\phi}^t(\rho)) \right\rangle^{-N} \langle |\rho'| \rangle^m .
$$

A useful consequence of (2.36) is Lemma 2.12 that uses the truncation operator
$\mathrm{Op}(\chi_\omega)$ defined in (1.25).

**Lemma 2.12.** If $R \in \Psi_{\tilde{\phi}^t}^m$ with some $m < 0$, then $\exists C_t > 0$,

$$
(2.37) \qquad \| R (\mathrm{Id} - \mathrm{Op}(\chi_\omega)) \|_{L^2(M)} \le C_t \omega^m \xrightarrow{\omega \to +\infty} 0.
$$

<footnote>with $y' = (x', z') \in \mathbb{R}^{n+1}$ and where $\chi \in C_0^\infty(\mathbb{R}^{n+1})$ is some cutoff function with $\chi \equiv 1$ near the origin and
$a_\varphi > 0$ is such that $\|\Phi_{j,\varphi}\|_{L^2(\mathbb{R}^{n+1})} = 1$ and $\|y' - y\|_{g_\varphi}^2 = |z' - z|^2 + \left|\frac{(x'-x)}{\delta^\perp(\eta)}\right|^2$ is obtained from the metric
(2.25).</footnote>
**Proof.** We follow the same notations and techniques as in [20, section 4.2]. We have that

$$ \begin{aligned} \|R(\mathrm{Id} - \mathrm{Op}(\chi_{\omega}))\|_{L^2(M)} &= \|\mathcal{J}R(\mathrm{Id} - \mathrm{Op}(\chi_{\omega}))\mathcal{J}^{\dagger}\|_{L^2(T^*M)} \\ &= \|\mathcal{J}R\mathcal{J}^{\dagger}(\mathcal{P} - \mathcal{P}\chi_{\omega}\mathcal{P})\|_{L^2(T^*M)}. \end{aligned} $$

From [20, Prop. 4.27], we have $\forall N \ge 0, \exists C_N > 0, \forall \rho, \rho' \in T^*M,$

$$ (2.38) \qquad |\langle \delta_{\rho'} | \mathcal{P} \delta_{\rho} \rangle_{L^2(T^*M)}| \le C_N \langle \mathrm{dist}_g(\rho', \rho) \rangle^{-N}, $$

i.e. the Schwartz kernel of $\mathcal{P}$ decays fast outside the diagonal and is uniformly bounded on the diagonal. Using definition of $\chi_{\omega}$ in (1.24) we have (the exponent 1/2 comes from the metric $g$ in (2.25)), $\forall N \ge 0, \exists C_N > 0, \forall \rho, \rho' \in T^*M,$

$$ |\langle \delta_{\rho'} | \mathcal{P} \chi_{\omega} \mathcal{P} \delta_{\rho} \rangle_{L^2(T^*M)}| \le C_N \langle \mathrm{dist}_g(\rho', \rho) \rangle^{-N} \langle \max(0, \omega^{-1/2} (\omega - \|\rho\|_{gM})) \rangle^{-N}. $$

Together with (2.36) we deduce that $\forall N \ge 0, \exists C_{N,t} > 0, \forall \rho, \rho' \in T^*M,$

$$ \begin{aligned} |\langle \delta_{\rho'} | \mathcal{J} R (\mathrm{Id} - \mathrm{Op}(\chi_{\omega})) \mathcal{J}^{\dagger} \delta_{\rho} \rangle_{L^2(T^*M)}| &\le C_{N,t} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} \langle |\rho| \rangle^m \\ &\qquad \langle \max(0, \omega^{-1/2} (\omega - \|\rho\|_{gM})) \rangle^{-N}. \end{aligned} $$

Using Shur Lemma [20, Lemma 4.38], that estimates the $L^2$ norm operator from the Schwartz kernel, we deduce that if $m < 0$, then $\exists C_t > 0,$

$$ \|R(\mathrm{Id} - \mathrm{Op}(\chi_{\omega}))\|_{L^2(M)} \le C_t \omega^m \underset{\omega \to +\infty}{\to} 0. $$

□

**2.6.1. Propagation of singularities.** First estimate for the wave front set. Theorem 2.13 is similar to the description of evolution of the **wave-front set** in micro-local analysis as in [43, Prop. 9.5, page 29.] and often called “propagation of singularities”. It gives some first description of the pull back operator $e^{tX}$ that we will need to improve later. Recall the notation $\Psi_{\tilde{\phi}^t}^0$ from Definition 2.11.

**Theorem 2.13** (Propagation of singularities [20, Thm 4.51]). For each $t \in \mathbb{R}$,

$$ (2.39) \qquad e^{tX} \in \Psi_{\tilde{\phi}^t}^0. $$

**Remark 2.14.** If $\rho'$ and $\tilde{\phi}^t(\rho)$ are in bounded distance from each other, Theorem 2.13 says nothing very informative for the value of the Schwartz kernel $\langle \delta_{\rho'} | \mathcal{J} e^{tX} \mathcal{J}^{\dagger} \delta_{\rho} \rangle_{L^2(T^*M)}$. Theorem 2.16 will complete this lack of information by giving an approximate expression for the Schwartz kernel of $\mathcal{J} e^{tX} \mathcal{J}^{\dagger}$ in the neighborhood of the graph of $\tilde{\phi}^t$, and this will be useful later to get Theorem 3.15. We first need to introduce some operators.

**2.6.2. Definition of some operators.** Below we view the (usual) space of Schwartz functions $S(TT^*M)$ as the set of functions $u : \rho \in T^*M \to u_{\rho} \in S(T_{\rho}T^*M)$, i.e. we have a natural identification, with induced topology,

$$ (2.40) \qquad S(TT^*M) \equiv S(\rho \in T^*M; S(T_{\rho}T^*M)). $$
**The operator** $\widetilde{\exp}^\circ$. The Riemannian manifold $(T^*M, g)$ is geodesically complete [20, Lemma 4.5]. Let us write

$$ (2.41) \qquad \exp : TT^*M \to T^*M $$

for the exponential map associated to the metric $g$ on $T^*M$ of Lemma 2.8. From this map we have the pull back operator $\exp^\circ : S(T^*M) \to S(TT^*M)$ but we consider instead the **twisted pull back operator** defined as follows

$$ (2.42) \qquad \widetilde{\exp}^\circ := e^{i\varphi} \exp^\circ : C^\infty(T^*M) \to C^\infty(TT^*M), $$

with a phase function $\varphi : TT^*M \to \mathbb{R}$ that corresponds to a change of trivialization to pass from the vertical gauge given by the wave-packet transform (2.32) to the local radial gauge as explained in Appendix C.2. Explicitly, this phase function $\varphi$ is given by, for $\rho \in T^*M, v \in T_\rho T^*M$,

$$ (2.43) \qquad \varphi(v) := -\theta(v) - \frac{1}{2} D^2 f(v) $$

with the Liouville one form $\theta(v) = \rho(d\pi(v))$ given in (2.11) and in the second part the function $f := \theta \circ (d\exp)_\rho - \theta_\rho : T_\rho T^*M \to \mathbb{R}$ made with the differential $(d\exp)_\rho : T_\rho T^*M \to TT^*M$ (with the canonical identification of vector spaces $TT_\rho T^*M \equiv T_\rho T^*M$). One has $f(0) = 0$ and $Df(0) = 0$, so the quadratic form $D^2f : T_\rho T^*M \to \mathbb{R}$ called Hessian is well defined.

**Remark 2.15.** Geometrically the phase $\varphi$ reflects the existence of a (trivial) complex line bundle $L$ over $T^*M$ with connection given by the canonical Liouville one form with respect to a global trivialization. This line bundle is usually called the prequantum line bundle. As mentioned before in Remark 1.1, we can ignore this line bundle since it is trivial so we can use a global section, giving phases like in (2.42). To understand better the phase $\varphi$ in (2.42) it may be useful to express it in local coordinates. Let $y \in \mathbb{R}^{\dim M}$ be local coordinates on $M$ as in (2.22) and $\eta \in \mathbb{R}^{\dim M}$ be dual coordinates on $T^*M$. For a (local) flat metric $g$ on $T^*M$ one has $\rho' = \exp_\rho(v) = (y_\rho + y_\nu, \eta_\rho + \eta_\nu)$ hence $v' = (d\exp)_\rho(v) = (y_\nu, \eta_\nu)$. The Liouville one-form is $\theta_\rho(v') = \eta_\rho y_\nu$ and $\theta_{\rho'}(v') = \eta_{\rho'} y_\nu = (\eta_\rho + \eta_\nu)y_\nu$ giving the function $f(v) = \eta_\nu y_\nu$ which is quadratic, i.e. $D^2f = f$ (due to absence of non-linearities in this simple setting of flat metric). The first term in (2.43) is $e^{-i\theta_\rho(v)} = e^{-i\eta_\rho y_\nu}$ and is there to remove high oscillations coming from wave-packet transform at point $\rho \in T^*M$. The second term in (2.43) is $e^{-\frac{i}{2}D^2f(v)} = e^{-\frac{i}{2}\eta_\nu y_\nu}$ and is used to pass from vertical gauge to radial gauge in accordance with formula (C.6). In summary, for Euclidean metric the phase function is

$$ (2.44) \qquad \varphi_{\rho}(y_{\nu}, \eta_{\nu}) = -\eta_{\rho} y_{\nu} - \frac{1}{2} \eta_{\nu} y_{\nu}. $$

**Truncation operator** $\chi^\lambda$. Let $0 < \lambda < 1/2$ that is fixed$^{10}$ in this paper. The operator $\chi^\lambda$ truncates functions in a ball of radius$^{11}$ $\langle\|\rho\|_{g_M}^{\lambda/2}\rangle$ centered on the zero section of $TT^*M$:

$$ (2.45) \qquad \chi^\lambda := \mathbb{1}\{\|\cdot\|_{g_M} \le \langle\|\rho\|_{g_M}\rangle^{\lambda/2}\} : S(TT^*M) \to S'(TT^*M), $$

<footnote>$^{10}$From the result (2.49) and estimate (2.37), we see that $\lambda$ close to zero gives better results, but in this paper it is enough to take $\lambda = 1/4$.</footnote>

<footnote>$^{11}$Here and after, $|\rho| = \|\rho\|_{g_M}$ is measured with respect to an arbitrary metric $g_M$ on $M$.</footnote>
i.e. for $u \in S(TT^*M)$, $\rho \in T^*M$, $v \in T_\rho T^*M$,

$$ (2.46) \qquad (\chi^\lambda u)(v) = \begin{cases} u(v) & \text{if } \|v\|_{g_\rho} \le \langle \|\rho\|_{g_M} \rangle^{\lambda/2} \\ 0 & \text{otherwise.} \end{cases}. $$

**The restriction operator $r_0$.** Let

$$ r_0 : S(TT^*M) \to S(T^*M) $$

be the map that restricts a function $u \in S(TT^*M)$ (written $u(\rho, v)$, $\rho \in T^*M$, $v \in T_\rho T^*M$), to its value at the zero section $(r_0 u)(\rho) := u(\rho, 0)$.

**The metaplectic operator $\tilde{\text{Op}}(d\tilde{\phi}^t)$.** Recall the map $\tilde{\phi}^t : T^*M \to T^*M$ defined in (2.8). Its differential $d\tilde{\phi}^t : TT^*M \to TT^*M$ gives a push-forward operator that we denote $(d\tilde{\phi}^t)^{-\circ} : S(TT^*M) \to S(TT^*M)$. As in (C.17) we define the “metaplectic correction”

$$ v(d\tilde{\phi}^t) := \left( \det \left( \frac{1}{2} \left( \mathrm{Id} + \left( (d\tilde{\phi}^t)^{-1} \right)^{\dagger} (d\tilde{\phi}^t)^{-1} \right) \right) \right)^{1/2}, $$

that is a positive function on $T^*M$. We also denote $\mathcal{P} : S(TT^*M) \to S(TT^*M)$ the Bergman projector in radial gauge as defined in (C.26), that is here a fiber-wise operator. As in Definition C.24, we define the fiber-wise metaplectic operator over the map $\tilde{\phi}^t$:

$$ (2.47) \qquad \tilde{\text{Op}}(d\tilde{\phi}^t) = (\mathcal{P}(d\tilde{\phi}^t))^{1/2} \mathcal{P}(d\tilde{\phi}^t)^{-\circ} \mathcal{P}. $$

### 2.6.3. Linear approximation of the kernel.
In Theorem 2.16 we use and compose the operators defined in Section 2.6.2 with the operator $\mathcal{T}$ in (2.33), and obtain a good approximation of the pull back operator $e^{t\mathcal{X}}$ that improves the information given in Theorem 2.13. The rough idea of this approximation is based on two properties: first, for $\rho \in T^*M$, with $\|\rho\|_{g_M} \gg 1$, the metric is slowly varying hence the metric $g$ can be approximated by a Euclidean metric in a neighborhood of $\rho$. Second, the unit balls of the metric $g$, transversely to the flow, have very small size projected on the base $M$ (see Remark 2.7), hence the dynamics can be approximated by its linearization, i.e. by the action of the differential $d\tilde{\phi}^t$.

**Theorem 2.16** (“More precise expression for the propagation of singularities”). For any $0 < \lambda < \frac{1}{2}$, for any $t \in \mathbb{R}$, we have

$$ (2.48) \qquad e^{t\mathcal{X}} = (\mathcal{T}^\dagger r_0) \tilde{\text{Op}}(d\tilde{\phi}^t) (\chi^\lambda \exp^{\mathcal{P}} \mathcal{T}) + R_t $$

with some operator $R_t$ that satisfies

$$ (2.49) \qquad R_t \in \Psi^{-\frac{1}{2}+\lambda} \cdot $$

**Remark 2.17.** Since $-\frac{1}{2} + \lambda < 0$, we can apply Lemma 2.12 that guarantees that the remainder operator $R_t$ restricted to high frequencies is arbitrary small in operator norm.

**Remark 2.18.** The proof below shows that we also have the simpler expression

$$ (2.50) \qquad e^{t\mathcal{X}} = \mathcal{T}^\dagger (Y(d\tilde{\phi}^t))^{1/2} (\tilde{\phi}^t)^{-\circ} \mathcal{T} + R_t $$

with the push-forward operator $(d\tilde{\phi}^t)^{-\circ} : S(T^*M) \to S(T^*M)$ and the multiplication operator by the metaplectic correction $(Y(d\tilde{\phi}^t))^{1/2} : S(T^*M) \to S(T^*M)$. We already
have commented this expression in (1.11). One reason for us to later use (2.48) instead of (2.50) is that (2.48) does not contain $(Y(d\tilde{\phi}^t))^{1/2}$, but contains instead the operator $\tilde{\text{Op}}(d\tilde{\phi}^t)$ that is fiberwise unitary in $L^2$, see Appendix C.7 (in fact $\tilde{\text{Op}}(d\tilde{\phi}^t)$ contains $(Y(d\tilde{\phi}^t))^{1/2}$). On the other hand, one advantage of (2.50) compared to (2.48) is that it deals directly with function in $S(T^*M)$ instead of the more elaborate space $S(TT^*M)$. Later we will meet a similar situation, see Remark 3.20.

*Proof.* Let $0 < \lambda < \frac{1}{2}$ and $t \in \mathbb{R}$. We consider the operator $R_t$ defined in (2.48), lifted to the cotangent space:

$$ (2.51) \qquad \tilde{R}_t := \mathcal{J} R_t \mathcal{J}^\dagger \\ = \mathcal{J} e^{tX} \mathcal{J}^\dagger - (\mathcal{J} \mathcal{J}^\dagger) r_0 \tilde{\text{Op}}(d\tilde{\phi}^t) (\chi^\lambda \widetilde{\exp^\circ})(\mathcal{J} \mathcal{J}^\dagger). $$

According to Definition 2.11, to get (2.49), we have to show the following estimate for the Schwartz kernel of $\tilde{R}_t$, that for any $N > 0$, there exists a constant $C_{N,t} > 0$ such that for any $\rho, \rho' \in T^*M$

$$ (2.52) \qquad |\langle \delta_{\rho'} | \tilde{R}_t \delta_{\rho} | \rangle| \le C_{N,t} \langle \text{dist}_g (\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} |\langle |\rho| \rangle|^{-\frac{1}{2}+\lambda}. $$

We will split the computation in two parts: (A) far from the graph of $\tilde{\phi}^t$ and (B) near the graph.

**Step (A), far from the graph of $\tilde{\phi}^t$.** Let us show that $\forall \lambda' > \lambda, \exists C_{t,\lambda'}, \forall \rho, \rho' \in T^*M$,

$$ (2.53) \qquad \text{dist}_g (\rho', \tilde{\phi}^t (\rho)) > C_{t,\lambda'} |\langle |\rho| \rangle|^{\lambda'/2} \Rightarrow \langle \delta_{\rho'} | r_0 \tilde{\text{Op}}(d\tilde{\phi}^t) (\chi^\lambda \widetilde{\exp^\circ}) \delta_\rho \rangle = 0. $$

From definition of the exponential map we have $\forall \rho'' \in T^*M, \upsilon'' \in T_{\rho''}'' T^*M$,

$$ (2.54) \qquad \text{dist}_g (\rho'', \exp((\rho'', \upsilon'')) = ||\upsilon''||_{g_{\rho''}}. $$

From definitions of $\chi^\lambda$ and $\widetilde{\exp^\circ}$, we have that $\langle \delta_{\rho''}, \upsilon'' | (\chi^\lambda \widetilde{\exp^\circ}) \delta_\rho \rangle \neq 0 \text{ iff } \exp((\rho'', \upsilon'')) = \rho$ and

$$ (2.55) \qquad \text{dist}_g (\rho'', \rho) = \frac{||\upsilon''||_{g_{\rho''}}}{(2.54)} \le \frac{|||\rho''||}{(2.45)}^{\lambda/2}. $$

The operator $\tilde{\text{Op}}(d\tilde{\phi}^t)$ is a bundle map over the map $\tilde{\phi}^t$ hence $\langle \delta_{\rho'}, \upsilon' | \tilde{\text{Op}}(d\tilde{\phi}^t) \delta_{\rho''}, \upsilon'' \rangle$ vanishes if $\rho' \neq \tilde{\phi}^t(\rho'')$. Hence, setting $\rho'' = \tilde{\phi}^{-t}(\rho')$, we have that

$$ (2.56) \qquad \langle \delta_{\rho'} | r_0 \tilde{\text{Op}}(d\tilde{\phi}^t) (\chi^\lambda \widetilde{\exp^\circ}) \delta_\rho \rangle \neq 0 \Rightarrow \text{dist}_g (\rho', \tilde{\phi}^t(\rho)) \underset{(2.31)}{\asymp} \text{dist}_g (\tilde{\phi}^{-t}(\rho'), \rho) \underset{(2.55)}{\le} |||\rho''||^{\lambda/2}. $$

From [20, Lemma 4.14] we have that $\forall 0 < \lambda < 1, \forall \epsilon > 0, \exists C_\epsilon > 0, \forall \rho, \rho'$,

$$ \text{dist}_g (\rho'', \rho) \le |||\rho''||^{\lambda/2} \Rightarrow |||\rho''|| \le C_\epsilon |||\rho||^{1+\epsilon}, $$

hence the right hand side of (2.56) becomes $\text{dist}_g(\rho', \tilde{\phi}^t(\rho)) \le C_{t,\lambda'} |||\rho||^{\lambda/2}$ for any $\lambda' > \lambda$ and we get (2.53).

We assume that $\text{dist}_g(\rho', \tilde{\phi}^t(\rho)) > C_{t,\lambda'} |||\rho||^{\lambda/2}$, i.e. points $(\rho, \rho') \in T^*M \times T^*M$ are “far” from the graph of $\tilde{\phi}^t$. We have that $e^{tX} \in \Psi_{\tilde{\phi}^t}^0$ and $(\mathcal{J}^\dagger r_0) \tilde{\text{Op}}(d\tilde{\phi}^t) (\chi^\lambda \widetilde{\exp^\circ} \mathcal{J}) \in$
$\Psi_{\tilde{\phi}t}^0$. This implies that $R_t \in \Psi_{\tilde{\phi}t}^0$. For any $N > 0$, take $M > \frac{1}{\lambda'}$ and $N' = N + M$. We have

$$ \begin{aligned} & |\langle \delta_{\rho'} | \tilde{R}_t \delta_{\rho} \rangle| \\ & \leq \sum_{(2.39)} C_{N',t}^{\prime} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N'} \leq C_{N',t}^{\prime\prime} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-M} \\ & \leq \sum_{\text{hyp.}} C_{N',t}^{\prime\prime} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} (C_{t,\lambda'} \langle |\rho| \rangle^{\lambda'/2})^{-M} \\ & \leq C_{N',t,\lambda'}^{\prime\prime\prime} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} \langle |\rho| \rangle^{-M\lambda'/2} \\ & \leq C_{N,t} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} \langle |\rho| \rangle^{-1/2} \end{aligned} $$

that gives (2.52).

**Step (B), near the graph of $\tilde{\phi}^t$.** We assume now that

$$ (2.57) \qquad \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \leq C_{t,\lambda'} \langle |\rho| \rangle^{\lambda'/2} $$

with $\lambda < \lambda' < 1$, i.e. points $(\rho, \rho') \in T^*M \times T^*M$ are “near” the graph of $\tilde{\phi}^t$. Let us explain the strategy that we will pursue. Using the slow variation of the metric $g$ given in [20, Lemma 4.12], we will (1) approximate the metric $g_{\rho'}$ by the Euclidean metric at point $\tilde{\phi}^t(\rho)$. From (2.25), the projected points $\pi(\rho')$, $\pi(\tilde{\phi}^t(\rho))$ on $M$ are at distance from each other that is $O(|\rho|^{\lambda'/2-1/2})$ transversely to the flow direction and this distance goes to zero if $|\rho| \to \infty$. So we will (2) approximate the map $\tilde{\phi}^t$ by its differential, i.e. neglect non-linear terms. Finally (3), even for a linear map on a Euclidean space, we need to show that the effect of the cutoff $\chi^\lambda$ is negligible.

Having these three approximations in mind we write

$$ |\langle \delta_{\rho'} | \tilde{R}_t \delta_{\rho} \rangle| \leq R_1 + R_2 + R_3, \quad (2.51) $$

with

$$ \begin{aligned} R_1 &= |\langle \delta_{\rho'} | (\mathcal{T}e^{tX}\mathcal{T}^\dagger - \mathcal{B}_{\tilde{\phi}^t(\rho)}e^{tX}\mathcal{B}_{\rho}^\dagger) \delta_{\rho} \rangle|, \\ (2.58) \qquad R_2 &= |\langle \delta_{\rho'} | (\mathcal{B}_{\tilde{\phi}^t(\rho)}e^{tX}\mathcal{B}_{\rho}^\dagger - e^{-i\varphi_{\tilde{\phi}^t(\rho)}}\tilde{\mathrm{O}}\mathrm{p}_{\rho}(d\tilde{\phi}_{\rho}^t)e^{i\varphi_{\rho}}) \delta_{\rho} \rangle|, \\ R_3 &= |\langle \delta_{\rho'} | (e^{-i\varphi_{\tilde{\phi}^t(\rho)}}\tilde{\mathrm{O}}\mathrm{p}_{\rho}(d\tilde{\phi}_{\rho}^t)e^{i\varphi_{\rho}} - (\mathcal{T}\mathcal{T}^\dagger)r_0\tilde{\mathrm{O}}\mathrm{p}(d\tilde{\phi}^t)(\chi^\lambda \widetilde{\exp}^\circ)(\mathcal{T}\mathcal{T}^\dagger)) \delta_{\rho} \rangle|, \end{aligned} $$

where $\mathcal{B}_\rho = \mathcal{B}_{(V),\rho} : S(\mathbb{R}^{\mathrm{dim}M}) \to S(\mathbb{R}^{2\mathrm{dim}M})$ is the Bargman transform in vertical gauge (C.8) using local charts on $M$ and defined from the Euclidean metric $g_\rho$. We have the metaplectic operator $\tilde{\mathrm{O}}\mathrm{p}_\rho(d\tilde{\phi}_\rho^t) = (Y(d\tilde{\phi}_\rho^t))^{1/2} \mathcal{P}_{\tilde{\phi}^t(\rho)}(d\tilde{\phi}_\rho^t)^{-\circ} \mathcal{P}_\rho$ with the Bergman projector in radial gauge $\mathcal{P}_\rho = \mathcal{B}_{(R)\rho}\mathcal{B}^\dagger_{(R),\rho}$. We have the phase function $\varphi_\rho$ defined in (2.43) and expressed in (2.44) for the Euclidean metric, that is to pass from vertical gauge to radial gauge.

(1) In the term $R_1$, the operator $\mathcal{T}$ is constructed from the metric $g$ whereas the operator $\mathcal{B}_g$ is constructed from the local Euclidean metric $g_\rho$ in $\mathcal{B}_\rho$. We use [20, Lemma 4.12] that the metric $g$ varies slowly on $T^*M$: for any $0 \le \gamma < 1$,
there exist $N > 0$ and $C > 0$ such that for any $\varrho, \varrho' \in \mathbb{R}^{2(n+1)}$ and $v \in \mathbb{R}^{2(n+1)}$,

$$ (2.59) \quad \max \left\{ \frac{\|v\|_{g_{\varrho'}}}{\|v\|_{g_{\varrho}}}, \frac{\|v\|_{g_{\varrho}}}{\|v\|_{g_{\varrho'}}} \right\} \le 1 + C \langle |\varrho| \rangle^{-(1-\gamma)/2} \langle |\varrho| \rangle^{-\gamma/2} \|\varrho' - \varrho\|_{g_{\varrho}} \rangle^N. $$

If we take $\gamma = \lambda + \epsilon$ with $\epsilon > 0$, we get that for $\|\varrho' - \varrho\|_{g_{\varrho}} < \langle |\varrho| \rangle^{\frac{1}{2}(\lambda+\epsilon)}$ then

$$ \left| \frac{\|v\|_{g_{\varrho'}}}{\|v\|_{g_{\varrho}}} - 1 \right| \le C \langle |\varrho| \rangle^{-\frac{1}{2}(1-\lambda-\epsilon)}, $$

$$ \text{hence } |R_1| \le C_{N,t} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} \langle |\rho| \rangle^{-\frac{1}{2}+\lambda}. $$

(2) For the term $R_2$, we start from

$$ \mathcal{B}_{\tilde{\phi}^t(\rho)} e^{tX} \mathcal{B}_{\rho}^{\dagger} \stackrel{(2.2)}{=} \mathcal{B}_{\tilde{\phi}^t(\rho)} (\phi^t)^{\circ} \mathcal{B}_{\rho}^{\dagger} $$

and in local chart in a small neighborhood of $\pi(\rho)$, $\pi(\tilde{\phi}^t(\rho)) \in M$ of size

$$ (2.60) \quad O_t \left( \langle |\rho| \rangle^{\frac{\lambda'}{2}} \delta^{\perp}(\rho) \right) = O_t \left( \langle |\rho| \rangle^{-(1-\lambda')/2} \right), $$

we approximate $\phi^t$ by its differential $d\phi^t_\rho$ at $\rho$, replace $(\phi^t)^\circ$ by $(d\phi^t_\rho)^\circ$. In the next lines, $(V)$, $(R)$ denote respectively vertical and radial gauge defined in Appendix C.2. In the first equality, we subtract the main oscillatory term $e^{-i\theta_\rho(\cdot)}$. In the last equality we use that $\det(d\phi^t) = 1$. We write

$$ \begin{align*} \mathcal{B}_{\tilde{\phi}^t(\rho)} (d\phi^t)^\circ_{\rho} \mathcal{B}_{\rho}^{\dagger} &= e^{i\theta_{\tilde{\phi}^t(\rho)}(\cdot)} \mathcal{B}_{\tilde{\phi}^t(\rho)}^{(V)} (d\phi^t)^\circ_{\rho} \mathcal{B}_{\rho}^{\dagger(V)} e^{-i\theta_{\rho}(\cdot)} \\ &= \mathcal{B}_{\tilde{\phi}^t(\rho)}^{(R)} (d\phi^t)^\circ_{\rho} \mathcal{B}_{\rho}^{\dagger(R)} e^{i\phi_{\rho}} \\ &= e^{-i\phi_{\tilde{\phi}^t(\rho)}} \tilde{\Omega}_{\rho} (d\tilde{\phi}^t_{\rho}) e^{i\phi_{\rho}}. \end{align*} $$

In the approximation replacing $\mathcal{B}_{\tilde{\phi}^t(\rho)} e^{tX} \mathcal{B}_{\rho}^{\dagger}$ by $e^{-i\phi_{\tilde{\phi}^t(\rho)}} \tilde{\Omega}_{\rho} (d\tilde{\phi}^t_{\rho}) e^{i\phi_{\rho}}$, we have neglected the non-linearity of the map $\phi^t$. From Taylor expansion this gives that

$$ |R_2| \stackrel{(2.60)}{\le} C_{N,t} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} \langle |\rho| \rangle^{-(1-\lambda')/2} $$

smaller than previous terms.

(3) For the term $R_3$, one first operation is to pass to the tangent bundle with $\exp^\circ$ and with a truncation $\chi^\lambda$ and past restriction $r_0$. For this, in the linear setting, Lemma C.40 shows that in case of a linear map $A: E \to E$ and Euclidean metric $g$ on $E \oplus E^*$, a cutoff $\chi_\sigma$ gives an error term bounded by $C_N \sigma^{-N}, \forall N$. Here $\sigma = \langle |\rho| \rangle^{\lambda/2}$, so this gives $O_{N,t}((\mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)))^{-N} \langle |\rho| \rangle^{-N})$, smaller than previous terms. Another operation is to pass again to the manifold with the metric $g$ as in step (1). We have seen that it gives $|R_3| \le C_{N,t} \langle \mathrm{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} \langle |\rho| \rangle^{-\frac{1}{2}+\lambda}$. We have checked (2.52) that gives (2.49). $\square$
### 3. MICRO-LOCAL ANALYSIS OF A CONTACT VECTOR FIELD $X$ ON $(M, \mathcal{A})$ NEAR $\Sigma = \mathbb{R}\mathcal{A} \setminus \{0\}$

In this section we pursue the analysis of a smooth non-vanishing vector field $X$ on a closed manifold $M$ done in Section 2) and furthermore we assume that $X$ is a **Reeb vector field**, i.e. there is a smooth contact one form $\mathcal{A}$ on $M$ such that for every $m \in M$, the linear space $\text{Ker}(\mathcal{A}(m))$ endowed with the two form $(d\mathcal{A})(m)$ is a linear symplectic space and

$$ (3.1) \qquad \mathcal{A}(X) = 1, \quad d\mathcal{A}(X, \cdot) = 0. $$

This implies that $\mathcal{A}$ is invariant under the flow $\phi^t$ generated by $X$. Indeed the Lie derivative vanishes: $\mathcal{L}_X \mathcal{A} = \iota_X d\mathcal{A} + d\iota_X \mathcal{A} = d\mathcal{A}(X, \cdot) + d1 = 0$. Also $\text{dim}M = 2d + 1$, with $2d = \text{dim}(\text{Ker}\mathcal{A})$. We will write

$$ (3.2) \qquad dm := \frac{1}{d!} \mathcal{A} \wedge (d\mathcal{A})^{\wedge d} $$

for the corresponding smooth and non-degenerate volume form on $M$ invariant by the flow $\phi^t$.

**Remark 3.1.** For the moment we do not assume that $X$ is Anosov, but later in Section 4, this one form $\mathcal{A}$ will be determined by $\text{Ker}\mathcal{A} = E_u \oplus E_s$ in (4.4). A typical example of contact vector field $X$ on $(M, \mathcal{A})$ is a general geodesic vector field on the unit cotangent bundle $M = (T^*\mathcal{N})_1$ of a Riemannian manifold $\mathcal{N}$ with $\mathcal{A}$ being the Liouville one-form. In $T^*M$ we define

$$ (3.3) \qquad E_0^* := (\text{Ker}\mathcal{A})^{\perp} = \{\rho \in T^*M, \text{ Ker}\rho \supset \text{Ker}\mathcal{A}\} $$

$$ (3.4) \qquad \mathcal{A} = \mathbb{R}\mathcal{A} = \{\omega\mathcal{A}(m), \quad m \in M, \omega \in \mathbb{R}\}. $$

$E_0^*$ is a rank 1 sub-bundle of $T^*M$ over $M$. We have

$$ (3.5) \qquad T^*M = \text{Ker}X \oplus E_0^*, $$

where

$$ (3.6) \qquad \text{Ker}X = \{\rho \in T^*M, \quad \omega(\rho) = \rho(X) = X(\rho) = 0\} $$

is a rank $2d$ sub-bundle of $T^*M$ over $M$.

3.1. **The symplectization $\Sigma = \mathbb{R}\mathcal{A} \setminus \{0\}$**. We first recall the important Lemma 3.2. It shows that $\Sigma := \mathbb{R}\mathcal{A} \setminus \{0\} = E_0^* \setminus \{0\}$ is a symplectic submanifold of $T^*M$, called the symplectization of $M$ [7, Section 11.2], [1, Appendix 4.]. We will denote $\pi: T^*M \to M$ the projection and $\pi^\circ(\text{dm})$ the form $\text{dm}$ (3.2) pulled back on $T^*M$.

**Lemma 3.2.** *The set*

$$ (3.7) \qquad \Sigma := \mathbb{R}\mathcal{A} \setminus \{0\} := \{\omega\mathcal{A}(m) \in T^*M \mid m \in M, \omega \in \mathbb{R} \setminus \{0\}\} $$

is a smooth **symplectic submanifold** of $T^*M$ with $\text{dim}\Sigma = 2(d+1)$. The induced volume form on $\Sigma$ is given by

$$ (3.8) \qquad d\varrho := \frac{1}{(d+1)!} (d\theta)^{\wedge(d+1)} = \omega^d (d\omega) \wedge \pi^\circ(\text{dm}), $$

where $\vartheta$ is the Liouville form (2.11).
Proof. For $m \in M$, $\omega \in \mathbb{R} \setminus \{0\}$, at point $\rho = \omega A(m) \in \Sigma \subset T^*M$ we have

$$ (3.9) \qquad \theta_{\rho} = \omega(\rho) \cdot \pi^{\circ} A, $$

where $\pi^\circ$ is the pull back map on forms. Hence $d\theta = d(\omega(\pi^\circ A)) = d\omega \wedge (\pi^\circ A) + \omega(\pi^\circ dA)$ giving the following volume form on $E_0^*$

$$ \begin{aligned} d\theta := & \frac{1}{(d+1)!} (d\theta)^{\wedge(d+1)} = \frac{1}{(d+1)!} (d+1) \omega^d \cdot d\omega \wedge ((\pi^\circ A) \wedge (\pi^\circ dA)^{\wedge d}) \\ = & \omega^d (d\omega) \wedge \pi^\circ (dm) \end{aligned} $$

which does not vanish on $\Sigma = E_0^* \setminus \{0\}$ since for the line bundle $E_0^* \to M$, $dm$ is a measure on the base $M$ and $d\omega$ is a measure on the fibers. Consequently $d\theta$ restricted to $\Sigma$ is non-degenerate, $\Sigma$ is a smooth submanifold of $T^*M$. $\square$

Since the one form $A$ is invariant under the flow $\phi^t$, the set $\Sigma$ is invariant under the
lifted flow $\tilde{\phi}^t$ in $T^*M$ defined in (2.8). The purpose of this Section 3 is to provide a useful
approximation of the pullback operator $e^{tX}$ restricted to a micro-local neighborhood of
$\Sigma$ (i.e. vicinity of $\Sigma$ in $T^*M$) in terms of the differential map $d\tilde{\phi}_N^t$ that is $d\tilde{\phi}^t$ restricted
to the symplectic normal bundle $N$ of $\Sigma$.

Remark 3.3. Later in Section 4 we will consider a contact Anosov flow and see in Lemma 4.2 that $\Sigma$ is the trapped set or non-wandering set for the lifted flow $\tilde{\phi}^t$. Consequently we will obtain in Theorem 4.6 that the dynamics $e^{tX}$ outside $\Sigma$ is negligible in some sense, so the only important part is indeed a micro-local neighborhood of $\Sigma$ that we consider in this section.

3.2. The symplectic normal bundle $N \to \Sigma$.

3.2.1. *Metaplectic decomposition $K \oplus N$. We have seen in Lemma 3.2 that $\Sigma := E_0^* \setminus \{0\}$ is symplectic. We will denote*

$$ T_{\Sigma} T^{*M} := \{T_{\rho}(T^{*M}) \mid \rho \in \Sigma\} $$

that is the tangent bundle $T(T^*M)$ restricted to the base space $\Sigma$. Lemma 3.4 provides a decomposition of $T_\Sigma T^*M$ into symplectic sub-bundles, invariant under the flow map $d\tilde{\phi}^t: T_\Sigma T^*M \to T_\Sigma T^*M$ with $\tilde{\phi}^t$ defined in (2.8). For the one form $A$ seen as a map $A: M \to T^*M$, we will use its differential map $dA: TM \to TT^*M$. Recall that $\tilde{X}$ was defined in Section 2.3.4.

**Lemma 3.4.** Let $\tilde{E}_0 := \mathbb{R}\tilde{X}$ and $\tilde{E}_0^* := \mathbb{R}\mathcal{A}$ be rank 1 sub-bundles of $T_\Sigma T^*M$. Then

$$ (3.10) \qquad K_0 := \tilde{E}_0 \oplus \tilde{E}_0^* $$

is a symplectic sub-bundle of $T_\Sigma T^*M$ and we have a decomposition of $T_\Sigma T^*M$ as an or-
thogonal sum of symplectic spaces $K, K_0, N$:

$$ \begin{gathered} T_{\Sigma} T^{*M} = \underbrace{K \oplus K_0}_{T^{\Sigma}} \oplus N, \\ \dim K = 2d, \quad \dim K_0 = 2, \quad \dim N = 2d, \end{gathered} $$

invariant under the flow map $d\tilde{\phi}^t: T_\Sigma T^*M \to T_\Sigma T^*M$.
**Remark 3.5.** Notice that $N = (T\Sigma)^{\perp\Omega}$ is the $\Omega$-symplectic orthogonal to $T\Sigma = K \oplus K_0$ in $T_\Sigma T^*M$. To explain the notation of $\tilde{E}_0^*$, notice that $K_0 = \tilde{E}_0 \oplus \tilde{E}_0^*$ is $\Omega$-symplectic, hence $\tilde{E}_0^*$ is isomorphic to the dual of $\tilde{E}_0$.

**Proof.** For fixed $\omega \in \mathbb{R} \setminus \{0\}$, let $T(\omega A) \subset TT^*M$ be the tangent space to the graph of $\omega A$:

$$T(\omega A) := (d(\omega A))(TM).$$

For any $\rho = \omega A(m) \in \Sigma$, $m = \pi(\rho) \in M$, we have that $d\pi : T(\omega A)(\rho) \to TM(m)$ is an isomorphism. We decompose

$$(3.12) \qquad T(\omega A) = K \oplus \tilde{E}_0$$

with $\tilde{E}_0 = \mathbb{R}\tilde{X} = T(\omega A) \cap d\pi^{-1}(\mathbb{R}X)$, $\dim \tilde{E}_0 = 1$, and $K := T(\omega A) \cap d\pi^{-1}(\text{Ker}A)$, $\dim K = 2d$. We have $\tilde{E}_0^* = \mathbb{R}A = T\Sigma \cap d\pi^{-1}(\{0\})$, $\dim \tilde{E}_0^* = 1$ and

$$(3.13) \qquad K_0 = \tilde{E}_0 \oplus \tilde{E}_0^* = T\Sigma \cap d\pi^{-1}(\mathbb{R}X).$$

We have $T\Sigma = T(\omega A) \oplus \tilde{E}_0^* = K \oplus K_0$. We have to prove that $K, K_0$ are symplectic and that $K \perp_\Omega K_0$. We first show Lemma 3.6, where for fixed $\omega$, the one form $\omega A$ is seen as a map $\omega A : M \to T^*M$, with differential $d(\omega A) : TM \to TT^*M$.

**Lemma 3.6.** At point $\omega A(m) \in \Sigma$, the linear map

$$(3.14) \qquad p := (d(\omega A)) \circ (d\pi) \quad : T_\Sigma T^*M \to T_\Sigma T^*M$$

is a projector with

$$(3.15) \qquad \text{Ker} p = \text{Ker}(d\pi), \quad \text{Imp} = T(\text{Im}(\omega A)).$$

We have

$$(3.16) \qquad (d\pi)^{-1}(\text{Ker}A) = (T\mathbb{R}A)^{\perp\Omega} \underset{\text{def}}{=} (\tilde{E}_0^*)^{\perp\Omega}.$$

**Proof.** We have $\pi \circ A = \text{Id}_M$ hence $(d\pi) \circ (dA) = \text{Id}_{TM}$ and

$$p^2 = (d(\omega A)) \underbrace{(d\pi)(d(\omega A))}_{\text{Id}_{TM}} (d\pi) = p.$$

We have

$$(3.17) \qquad \Omega_{/\text{Ker}(d\pi)} \underset{(2.13)}{=} 0.$$

For any $U \in T_{\omega A} T^*M$,

$$\begin{align*}
\omega A((d\pi)(U)) &\underset{(2.14)}{=} \theta((d(\omega A)) \circ (d\pi)(U)) &\underset{(2.16)}{=} \Omega(\mathcal{E}, (d(\omega A)) \circ (d\pi)(U)) \\
&\underset{(3.14)}{=} \Omega(\mathcal{E}, p(U)) = \Omega(\mathcal{E}, (p(U) - U) + U) &\underset{(3.15),(3.17)}{=} \Omega(\mathcal{E}, U).
\end{align*}$$

On $\Sigma = \mathbb{R}A$ we have $\mathbb{R}\mathcal{E} = \mathbb{R}A$, we deduce (3.16). $\square$

<figure><img src="image_6.png" /></figure>

We have an isomorphism

$$d(\omega A) : TM = \text{Ker}A \oplus \mathbb{R}X \to T(\omega A) \underset{(3.12)}{=} K \oplus \tilde{E}_0,$$

and

$$(3.18) \qquad (d(\omega A))^\circ \underset{(2.15)}{=} d(\omega A).$$
Hence

$$d(\omega A) : (\mathrm{Ker}A, \omega dA) \to (K, \Omega)$$

is a symplectomorphism for the respective symplectic structures. Since $\mathrm{Ker}A$ is $dA$-symplectic, this implies that $K$ is $\Omega$-symplectic. This also gives that $\Omega(K, \tilde{E}_0) = \omega dA(\mathrm{Ker}A, \mathbb{R}X) = 0$. By definition we have $\tilde{E}_0^* = \mathbb{R}A$ and $K = T(\omega A) \cap d\pi^{-1}(\mathrm{Ker}A)$. From (3.16) we have that $\Omega(K, \tilde{E}_0^*) = 0$. Since $K_0 = \tilde{E}_0 \oplus \tilde{E}_0^*$, we have obtained that $K \perp_\Omega K_0$ and that $K_0$ is $\Omega$-symplectic. This completes the proof of Lemma 3.4. $\square$

Lemma 3.8 concerns specifically the subspace $K \oplus_\Omega N$ in (3.11). We show that this space is also the sum of canonical Lagrangian spaces $H \oplus V$ (horizontal and vertical). This is illustrated in Figure 3.1. Recall the frequency function $\omega : \rho \in T^*M \to \omega(\rho) = \rho(X) \in \mathbb{R}$ defined in (2.7) and that $(\mathrm{Ker}A, dA)$ is a linear symplectic space from Section 4.1.2.

<figure><img src="image_7.png" /><figcaption>FIGURE 3.1. Illustration of Lemma 3.8 about the vector bundle $\widetilde{\mathrm{Ker}}A \subset T_\Sigma T^*M \xrightarrow{\pi} M$. We have the linear map $\widetilde{\mathrm{Ker}}A \xrightarrow{d\pi} \mathrm{Ker}A \subset TM$ and $\widetilde{\mathrm{Ker}}A$ is a sum of two symplectic subspaces $K \oplus N$ and two Lagrangian subspaces $H \oplus V$. This structure is preserved by the dynamics.</figcaption></figure>

**Definition 3.7.** For $u \in \mathrm{Ker}A$, we define the maps

$$ (3.19) \qquad \begin{aligned} \bar{K}(u) &:= ((d\pi^{-1})(u)) \cap K, & \bar{N}(u) &:= ((d\pi^{-1})(u)) \cap N, \\ \bar{H}(u) &:= \frac{1}{2}(\bar{K}(u) + \bar{N}(u)), & \bar{V}(u) &:= \frac{1}{2}(\bar{K}(u) + \bar{N}(-u)), \end{aligned} $$

that define the horizontal and the vertical spaces:

$$ H := \{\bar{H}(u), u \in \mathrm{Ker}A\}, \\ V := \{\bar{V}(u), u \in \mathrm{Ker}A\} $$

that are subspaces of $K \oplus_\Omega N$.

**Lemma 3.8 (Metaplectic decomposition).** At any point $\omega A \in \Sigma$ with $\omega \neq 0$, let

$$ (3.20) \qquad \widetilde{\mathrm{Ker}}A := (d\pi^{-1}(\mathrm{Ker}A)) \cap \mathrm{Ker}(d\omega). $$
$\widetilde{\text{Ker}}.A$ is a $\Omega$-symplectic subspace of $T_\Sigma T^*M$ and we have the decompositions

$$ (3.21) \qquad \widetilde{\text{Ker}}.A = K \oplus N = H \oplus V, $$

with $\Omega$-symplectic and orthogonal subspaces $K, N$ and $\Omega$-Lagrangian subspaces $H, V$ with $\dim V = \dim H = 2d$.

We have $(d\pi)(H) = \text{Ker}.A$ and $(d\pi)(V) = \{0\}$. The linear maps

$$ (3.22) \qquad \sqrt{\omega} d\pi : (K, \Omega) \to (\text{Ker}.A, dA), $$

$$ (3.23) \qquad \sqrt{\omega} d\pi : (N, \Omega) \to (\text{Ker}.A, -dA) $$

are symplectomorphism for the respective symplectic structures.

*Proof.* We have

$$ (3.24) \qquad \begin{aligned} \widetilde{\text{Ker}}.A &\equiv (d\pi^{-1}(\text{Ker}.A)) \cap \text{Ker}(d\omega) &\equiv (\tilde{E}_0^*)^{\perp\Omega} \cap \text{Ker}(d\omega) \\ &\equiv (K \oplus \tilde{E}_0^* \oplus N) \cap \text{Ker}(d\omega). \end{aligned} $$

We have $d\omega(N) = \Omega(\tilde{X}, N) = \Omega(\tilde{E}_0, N) = 0$. This gives $N \subset \text{Ker}(d\omega)$ and similarly $K \subset \text{Ker}(d\omega)$. We have $\tilde{E}_0^* = \mathbb{R}\mathcal{E}$ and

$$ d\omega(\mathcal{E}) = \Omega(\tilde{X}, \mathcal{E}) = \theta(\tilde{X}) = -\omega \neq 0, $$

hence $\tilde{E}_0^* \cap \text{Ker}(d\omega) = 0$. We deduce the decomposition $\widetilde{\text{Ker}}.A = K \oplus N$ in (3.21). On the space $K$ we have

$$ \begin{aligned} \Omega_{/K} &= p^{\circ} \Omega_{/K} = ((d(\omega A)) \circ (d\pi))^{\circ} \Omega_{/K} \\ &= (d\pi)^{\circ} (d(\omega A))^{\circ} \Omega_{/K} = (d\pi)^{\circ} d(\omega A) = \omega (d\pi)^{\circ} (dA). \end{aligned} $$

This gives (3.22) since $dA$ is bilinear. For $u \in \text{Ker}.A$, we have $(d\pi)(\overline{V}(u))_{(3.19)} = \frac{1}{2}(u-u) = 0$, hence $V$ is vertical hence $\Omega$-Lagrangian. For the map $\overline{V} : \text{Ker}.A \to V$, we have $\overline{V} \circ d\pi_{V \to \text{Ker}.A} = \text{Id}_{/V}$ and $V$ is $\Omega$-Lagrangian hence

$$ 0 = \Omega_{/V} = (d\pi_{V \to \text{Ker}.A})^{\circ} \circ (\overline{V})^{\circ} \Omega = \frac{1}{2} (d\pi_{V \to \text{Ker}.A})^{\circ} \circ ((\overline{K})^{\circ}(\Omega) + (-\text{Id})^{\circ} \circ (\overline{N})^{\circ}(\Omega)). $$

Since $\overline{K} = (d\pi)_{\text{Ker}.A \to K}^{-1}$ and $(-\text{Id})^{\circ} = \text{Id}$ on bilinear forms we deduce

$$ (\overline{N})^{\circ}(\Omega) = -(\overline{K})^{\circ}(\Omega) = -(d\pi)_{\text{Ker}.A \to K}^{-1}(\Omega) = -\omega dA, $$

giving (3.23) because $\overline{N} = (d\pi)_{\text{Ker}.A \to N}^{-1}$. For the map $\overline{H} : \text{Ker}.A \to H$, we have $\overline{H} \circ d\pi_{H \to \text{Ker}.A} = \text{Id}_{/H}$ hence

$$ \begin{aligned} \Omega_{/H} &= (d\pi_{H \to \text{Ker}.A})^{\circ} \circ (\overline{H})^{\circ} \Omega = \frac{1}{2} (d\pi_{H \to \text{Ker}.A})^{\circ} \circ ((\overline{K})^{\circ}(\Omega) + (\overline{N})^{\circ}(\Omega)) \\ &= \frac{1}{2} (d\pi_{H \to \text{Ker}.A})^{\circ} (\omega dA - \omega dA) = 0. \end{aligned} $$

Hence $H$ is $\Omega$-Lagrangian. $\square$

<figure><img src="image_8.png" /></figure>
3.2.2. Modified metric $\tilde{g}$ on the set $\Sigma$. In (3.11), the decomposition $K \oplus_K K_0 \oplus_{\perp g} N$ is orthogonal for the symplectic form $\Omega$. But a priori it is not orthogonal for the metric $g$ given in (2.25). Since this last property would be useful later, we will modify the metric $g$ according to Lemma 3.9 to get $g$-orthogonality as well.

**Lemma 3.9.** On the set $\Sigma$, we construct a modified metric $\tilde{g}$ from the existing metric $g$ as follows. Let $\tilde{g}_N = g_N$, $\tilde{g}_K = g_K$, $\tilde{g}_{K_0} = g_{K_0}$ and assume $K \oplus_{\perp g} K_0 \oplus_{\perp g} N$. Then

(1) $\tilde{g}$ is compatible with $\Omega$.

(2) The decomposition (3.11) is orthogonal with respect to the metric $\tilde{g}$ (and w.r.t. $\Omega$ as before):

$$ (3.25) \qquad T_{\Sigma} T^* M = K \oplus_{\perp \Omega, \tilde{g}} K_0 \oplus_{\perp \Omega, \tilde{g}} N. $$

(3) The modified metric $\tilde{g}$ is equivalent to the metric $g$, uniformly with respect to $\rho \in \Sigma$ for $|\omega| \ge 1$.

*Proof.* By construction we have claims (1), (2). In local coordinates $(x, z, \xi, \omega)$ used in Section 2.4, and for some $\omega_0 \gg 1$, consider the change of coordinates $x' = \omega_0^{1/2} x$, $z' = z$, $\xi' = \omega_0^{-1/2} \xi$, $\omega' = \omega$ so that $\Omega = dx' \wedge d\xi' + dz' \wedge d\omega'$. Let $\rho = \omega A(m) \in \Sigma$ with $|\omega| \asymp \omega_0$. From (2.24) we have $\delta^\perp(\rho) \asymp |\rho|^{-1/2} \asymp |\omega|^{-1/2} \asymp \omega_0^{-1/2}$. Then $g \asymp g_{(2.25)}$ $dx'^2 + d\xi'^2 + dz'^2 + d\omega'^2$, where the equivalences are uniform w.r.t. $|\omega| \ge 1$. Using this new system of coordinates we get that the subspaces $K, K_0, N$ are uniformly apart from each other with respect to $\rho \in \Sigma$, i.e. for the pair of spaces $(K, N)$, we have $\exists 0 < C < 1$, $\forall \rho \in \Sigma, \forall u \in K_\rho, \forall v \in N_\rho$,

$$ (3.26) \qquad |g_{\rho}(u, v)| < C \|u\|_{g_{\rho}} \|v\|_{g_{\rho}}, $$

and similarly for other pair of spaces $(K, K_0), (K_0, N)$. This gives Claim (3). $\square$

<figure><img src="image_9.png" /></figure>

For simplicity of notations, from now on, **we assume that the metric $g$ on $T^*M$ has the properties (1), (2), (3) of Lemma 3.9 on $\Sigma$**.

3.2.3. *Factorization formula for $\tilde{\text{Op}}((d\tilde{\phi}^t)_{/\Sigma})$.* We consider now the operator $\tilde{\text{Op}}(d\tilde{\phi}^t)$ that appears on the right hand side of (2.48). Since $\Sigma$ is invariant by $\tilde{\phi}^t$ and $\tilde{\text{Op}}(d\tilde{\phi}^t)$ is fiber-wise, we can consider its restriction to $S(T_\Sigma T^*M)$. It is denoted and given by

$$ \tilde{\text{Op}}((d\tilde{\phi}^t)_{/\Sigma}) \underset{(2.47)}{=} (Y(d\tilde{\phi}^t))^{1/2} \mathcal{P}(d\tilde{\phi}^t)^{-\circ} \mathcal{P} \quad : S(T_\Sigma T^*M) \to S(T_\Sigma T^*M). $$

The decomposition (3.11) is orthogonal for both the symplectic form and the modified metric $\tilde{g}$, and preserved by the differential of the flow, i.e.

$$ (3.27) \qquad (d\tilde{\phi}^t)_{/\Sigma} = \underbrace{d\tilde{\phi}^t_K + d\tilde{\phi}^t_{K_0}}_{d\tilde{\phi}^t_{T\Sigma}} + d\tilde{\phi}^t_N. $$

**Remark 3.10.** Eq. (3.11) implies that on the set $\Sigma$ we have a natural identification

$$ (3.28) \qquad S(T_\Sigma T^*M) = S(K) \otimes_\Sigma S(K_0) \otimes_\Sigma S(N), $$

where the right hand side is a notation for the space of sections

$$ S(\Sigma; S(K) \otimes S(K_0) \otimes S(N)), $$

i.e. the tensor product is fiber-wise over $\Sigma$ and not global.
From invariant Lagrangian decomposition (3.21), (3.10) and Lemma C.22 we have

$$ \mathcal{P} = \mathcal{P}_K \otimes \mathcal{P}_{K_0} \otimes \mathcal{P}_N. $$

Then (C.36) is used to define $\check{\text{Op}}(d\check{\phi}_K^t) = (Y(d\check{\phi}_K^t))^{1/2} \mathcal{P}_K (d\check{\phi}_K^t)^{-\circ} \mathcal{P}_K$ and similarly $\check{\text{Op}}(d\check{\phi}_{K_0}^t), \check{\text{Op}}(d\check{\phi}_N^t)$.

**Theorem 3.11** (Factorization formula). On the set $\Sigma$, with respect to the identification (3.28) we have

$$ (3.29) \qquad \check{\text{Op}}((d\check{\phi}^t)_{/\Sigma}) = \underbrace{\check{\text{Op}}(d\check{\phi}_K^t) \otimes_{\Sigma} \check{\text{Op}}(d\check{\phi}_{K_0}^t)}_{\check{\text{Op}}(d\check{\phi}_{T\Sigma}^t)} \otimes_{\Sigma} \check{\text{Op}}(d\check{\phi}_N^t). $$

*Proof.* Use previous factorization and (C.19) to get (3.29). □

<figure><img src="image_10.png" /></figure>

**3.3. Equivalent family of operators.** As we have said, we want to describe only the restriction of the operator $e^{tX}$ to a small neighborhood of the set $\Sigma \subset T^*M$, i.e. to consider Schwartz kernel of operators in this neighborhood and not on the whole manifold $T^*M$. For this purpose, we introduce here an equivalence relation between operators denoted $\approx$ that we will often use later. The equivalence $A \approx B$ will mean roughly that the two operators $A, B$ are approximately the same at high frequency and near the set $\Sigma$.

Let $\sigma > 0$ that will be chosen large enough later. Define the following characteristic function $\chi_{\Sigma,\sigma} : T^*M \to \mathbb{R}^+$ as follows. We write $\rho = \rho_* + \rho_0 \in T^*M$ with transverse component $\rho_* \in \text{Ker}X$ and frequency component $\rho_0 \in E_0^*$, see (3.5). We set

$$ (3.30) \qquad \chi_{\Sigma, \sigma}(\rho) := 1_{\{\|\rho_*\|_{g_{\rho}} \le \sigma\}}, $$

and

$$ (3.31) \qquad \text{Op}(\chi_{\Sigma, \sigma}) := \mathcal{T}^{\dagger} \chi_{\Sigma, \sigma} \mathcal{T} : C^{\infty}(M) \to C^{\infty}(M) $$

be the corresponding P.D.O. operator, as defined in [20, def. 4.28] that extracts components at distance less than $\sigma$ from the set $\Sigma$.

Recall $\Psi_{\phi t}^m$ that has been defined in Definition 2.11.

**Definition 3.12.** For two operators $A, B : C^{\infty}(M) \to S'(M)$ we write

$$ (3.32) \qquad A \approx B $$

if there exists $m < 0$, $t \in \mathbb{R}$, such that for any $\sigma > 0$,

$$ \text{Op}(\chi_{\Sigma, \sigma})(A - B) \text{Op}(\chi_{\Sigma, \sigma}) \in \Psi_{\phi t}^m. $$

**Remark 3.13.** From $m < 0$, we recall that consequently we have the estimate (2.37) for $R = \text{Op}(\chi_{\Sigma, \sigma})(A - B) \text{Op}(\chi_{\Sigma, \sigma})$, where the constant $C_t$ depends on $\sigma$ also.

**3.4. Description of the operator $e^{tX}$ near the set $\Sigma$ with the bundle $N$.** We first define some operators. The construction that we pursue below is represented in Figure 1.1. Using the metric $g$ on $T^*M$, and considering the sub-bundle $N \subset T_\Sigma T^*M$, we have the exponential map

$$ (3.33) \qquad \exp_N : N \to T^*M. $$

Due to the slow variation of the metric $g$ at high frequencies, given in (2.59), we have that for any $0 \le \mu < 1$, and any frequency $|\omega| \ge \omega_0$ with $\omega_0$ large enough, $\exp_N$ is a
diffeomorphism on the neighborhood $||v||_g \le \langle\omega\rangle^{\mu/2}$. To express this, let $0 < \mu < 1$ and $\omega_0 \ge 1$ for this property. We introduce the following cutoff function in frequency $\chi_\Sigma^\mu : N \to [0, 1]$ defined as follows. For $\rho \in \Sigma$, $v \in N_\rho$ and $\omega = \omega(\rho)$,

$$ (3.34) \qquad \chi_\Sigma^\mu(v) = 1_{\{||v||_g \le (\omega)^{\mu/2}, |\omega| \ge \omega_0\}} $$

The multiplication operator by $\chi_\Sigma^\mu$ is also denoted $\chi_\Sigma^\mu : S(N) \to S'(N)$. Later, in Theorem 4.37, we will need to take $\mu$ small enough depending in particular on the Hölder exponents of the Anosov dynamics.

Similarly to (2.42), we define the twisted pull back and push forward operators

$$ \chi_\Sigma^\mu(\widetilde{\exp_N}) : S(T^*M) \to S'(N), $$

$$ (\widetilde{\exp_N^{-1}})^\circ \chi_\Sigma^\mu : S(N) \to S'(T^*M). $$

Namely, for $u \in S(T^*M)$, $w \in S(N)$, let $\rho \in T^*M$ with $\rho = \exp_N(v_2)$, $v_2 \in N_{\rho_2}$, $\rho_2 \in \Sigma$, $|\omega(\rho_2)| \ge \omega_0$, and $||v_2||_g \le \langle\omega(\rho_2)\rangle^{\mu/2}$,

$$ (3.35) \qquad (\widetilde{\exp_N})^\circ u(v_2) := e^{i\varphi(v_2)}u(\rho), $$

$$ (3.36) \qquad (\widetilde{\exp_N^{-1}})^\circ w(\rho) := e^{-i\varphi(v_2)}w(v_2), $$

with phase function $\varphi$ defined in (2.43), to pass from vertical gauge to radial gauge. In Definition 3.14, we combine these previous operators together with $\mathcal{T} : C^\infty(M) \to S(T^*M)$ in (2.33).

**Definition 3.14.** We define the operators

$$ (3.37) \qquad \mathcal{T}_N := \chi_\Sigma^\mu(\widetilde{\exp_N})^\circ \mathcal{T} : C^\infty(M) \to S'(N), $$

$$ (3.38) \qquad \mathcal{T}_N^\Delta := \mathcal{T}^\dagger(\widetilde{\exp_N^{-1}})^\circ \chi_\Sigma^\mu : S(N) \to C^\infty(M). $$

Using the operators (3.37), (3.38), Theorem 3.15 gives a good approximation of the operator $e^{tX}$ in a neighborhood of the set $\Sigma$ in terms of the differential map $d\tilde{\phi}_N^t$ on the normal bundle $N$. We will use the notation $\approx$ defined in (3.32) that compares operators near $\Sigma$. We also use the shortened notation

$$ (3.39) \qquad Y_t := Y(d\tilde{\phi}_{T\Sigma}^t), $$

that is the metaplectic correction (C.17) of the differential restricted to $T\Sigma$, $d\tilde{\phi}_{T\Sigma}^t = d\tilde{\phi}_K^t + d\tilde{\phi}_{K_0}^t$ that appears in (3.27). In Theorem 3.15 we view $Y_t^{1/2} > 0$ as a multiplication operator over $\Sigma$.

**Theorem 3.15.** For any $t \in \mathbb{R}$, we have

$$ (3.40) \qquad e^{tX} \approx \mathcal{T}_N^\Delta Y_t^{1/2} \tilde{\text{Op}}(d\tilde{\phi}_N^t) \mathcal{T}_N. $$

*Remark 3.16.* Let us compare formula (3.40) with (2.48). Both formulas express the pull back operator $e^{tX}$ from some quantum (metaplectic) operator $\tilde{\text{Op}}(d\tilde{\phi}^t)$ or $\tilde{\text{Op}}(d\tilde{\phi}_N^t)$ acting on some vector bundle over $T^*M$. The difference is that (2.48) is valid micro-locally on the whole $T^*M$ (but at high frequencies) and general to any non-vanishing vector fields $X$, whereas (3.40) is specific to contact (or Reeb) vector field and valid micro-locally near $\Sigma \subset T^*M$ (and high frequencies).
Remark 3.17. As a particular case, taking $t = 0$ in (3.40) gives that

$$ (3.41) \qquad \mathcal{T}_N^\Delta \mathcal{T}_N \approx \text{Id}. $$

*Proof.* As we already did in the proof of Theorem 2.16, we use $\mathcal{T} : C^\infty(M) \to S(T^*M)$ to lift the analysis to $T^*M$ and consider the Schwartz kernel

$$ \tilde{R} := |\langle \delta_{\rho'} | \mathcal{T} (e^{tX} - \mathcal{T}_N^\Delta Y_t^{1/2} \tilde{\text{Op}}(d\tilde{\phi}_N^t) \mathcal{T}_N) \mathcal{T}^\dagger \delta_\rho \rangle|, $$

that we will estimate for points $\rho', \rho \in T^*M$ close to the set $\Sigma$ as needed by Definition 3.12. For this, let $\sigma > 0$. We write $\rho = \rho_* + \rho_0$ with transverse component $\rho_* \in \text{Ker}X$ and frequency component $\rho_0 \in E_0^*$, see (3.5), similarly for $\rho'$ and we assume that

$$ (3.42) \qquad \|\rho_*\|_{g_\rho} \le \sigma, \quad \|\rho'_*\|_{g_\rho} \le \sigma. $$

We also assume that points $(\rho, \rho') \in T^*M \times T^*M$ are “near” the graph of $\tilde{\phi}^t$, i.e. that (2.57) holds true. Otherwise the proof of Theorem 2.16, step (A), shows that far from the graph, the Schwartz kernel is totally negligible.

As before, the strategy is to approximate the dynamical operator $e^{tX}$ to its linear part, approximate the metric to a Euclidean metric, and then check that formula holds true exactly in this linear and Euclidean setting. For this, we use local coordinates in $\mathbb{R}^{2(2d+1)}$ on $T^*M$ and use the operator $\tilde{\text{Op}}_{\rho}(d\tilde{\phi}_{\rho}^t)$ and phase $\varphi_{\rho}$ that have already been defined in (2.58). We introduce intermediary operators

$$ \begin{align*} O_1 &:= \mathcal{T} e^{tX} \mathcal{T}^\dagger \qquad O_2 := e^{-i\varphi_{\phi^t(\rho)}} \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_{\rho}^t) e^{i\varphi_{\rho}}, \\ O_3 &:= e^{-i\varphi_{\phi^t(\rho)}} \mathcal{T}_{\tilde{\phi}^t(\rho)} (\overline{\exp_N^{-1}})^{\circ} Y_t^{1/2} \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_N^t) (\overline{\exp_N}^{\circ}) \mathcal{T}_{\rho} e^{i\varphi_{\rho}}, \\ O_4 &:= \mathcal{T} \mathcal{T}^\dagger (\overline{\exp_N^{-1}})^{\circ} \chi_\Sigma^\mu Y_t^{1/2} \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_N^t) \chi_\Sigma^\mu (\overline{\exp_N}^{\circ}) \mathcal{T} \mathcal{T}^\dagger \\ &\quad = \mathcal{T} \mathcal{T}_N^\Delta Y_t^{1/2} \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_N^t) \mathcal{T}_N \mathcal{T}^\dagger, \end{align*} $$

and their differences

$$ (3.43) \qquad \begin{aligned} R_1 &:= |\langle \delta_{\rho'} | (O_1 - O_2) \delta_{\rho} \rangle|, & R_2 &:= |\langle \delta_{\rho'} | (O_2 - O_3) \delta_{\rho} \rangle|, \\ R_3 &:= |\langle \delta_{\rho'} | (O_3 - O_4) \delta_{\rho} \rangle|. \end{aligned} $$

So we have

$$ \tilde{R} \le R_1 + R_2 + R_3. $$

For the term $R_1$, in the proof of Theorem 2.16, step (B), we have already shown that for any $0 < \lambda < \frac{1}{2}$,

$$ R_1 \le C_{N,t} \langle \text{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} |\langle |\rho| \rangle|^{-\frac{1}{2}+\lambda}. $$

For the term $R_2$, we decompose in $O_2$ the term

$$ \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_{\rho}^t) = (\tilde{\text{Op}}_{\rho}(d\tilde{\phi}_{K}^t) \otimes \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_{N}^t)) \otimes \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_{K_0}^t). $$
For the transverse part $\tilde{\Phi} = d\tilde{\phi}_K^t + d\tilde{\phi}_N^t$, Lemma C.35 in the appendix gives that (and this is the main relation that underlies Theorem 3.15)

$$ \begin{aligned} \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_K^t) \otimes \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_N^t) &= \tilde{\text{Op}}_{\rho}(\tilde{\Phi}) \\ &= \mathcal{P}_{\tilde{\phi}_t(\rho)}(\widetilde{\exp_N^{-1}})^{\circ} (Y(d\tilde{\phi}_K^t))^{1/2} \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_N^t) (\widetilde{\exp_N})^{\circ} \mathcal{P}_{\rho}. \end{aligned} $$

If we add the neutral component $\tilde{\text{Op}}_{\rho}(d\tilde{\phi}_{K_0}^t)$ that has no effect this gives

$$ \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_{\rho}^t) = \mathcal{P}_{\tilde{\phi}_t(\rho)}(\widetilde{\exp_N^{-1}})^{\circ} Y_t^{1/2} \tilde{\text{Op}}_{\rho}(d\tilde{\phi}_N^t) (\widetilde{\exp_N})^{\circ} \mathcal{P}_{\rho}. $$

This shows that $R_2 \stackrel{(3.43)}{=} 0$. Finally for the term $R_3$, we add the cutoff $\chi_{\Sigma}^{\mu}$ that has no effect from our assumptions on points $\rho, \rho'$, we pass from the Euclidean metric to the metric $g$ repeating arguments with the slow variation of the metric (2.59) and get that $R_3 \le C_{N,t} \langle \text{dist}_g(\rho', \tilde{\phi}^t(\rho)) \rangle^{-N} \langle |\rho| \rangle^{-\frac{1}{2}+\lambda}$ for any $0 < \lambda < \frac{1}{2}$, as well.

From these estimates and assumption (3.42) on $\rho, \rho'$, we can add the cutoff $\text{Op}(\chi_{\Sigma,\sigma})$ and deduce that for any $0 < \lambda < \frac{1}{2}$,

$$ \text{Op}(\chi_{\Sigma,\sigma}) (e^{tX} - \mathcal{T}_N^{\Delta} Y_t^{1/2} \tilde{\text{Op}}(d\tilde{\phi}_N^t) \mathcal{T}_N) \text{Op}(\chi_{\Sigma,\sigma}) \in \Psi_{\tilde{\phi}^t}^{-\frac{1}{2}+\lambda}. $$

From Definition 3.12, this gives (3.40). $\square$

**3.5. Description of the operator $e^{tX}$ near the set $\Sigma$ with the bundle $T_{\Sigma}T^*M$.** In this section we provide another expression for the operator $e^{tX}$ similar to (3.40) but sometimes more convenient, see Remark 3.20. The little difference with (3.40) is that we will use the bundle $T_{\Sigma}T^*M \stackrel{(3.11)}{=} T_{\Sigma} \oplus N \to \Sigma$ instead of the sub-bundle $N \to \Sigma$.

Similarly to (3.34), we introduce the following cutoff function in frequency $\chi_{\Sigma}^{\mu}$ : $T_{\Sigma}T^*M \to [0,1]$ defined as follows. For $\nu \in T_{\Sigma}T^*M$, and $\omega = \omega(\pi(\nu))$,

$$ (3.44) \qquad \chi_{\Sigma}^{\mu}(\nu) = 1_{\{\|\nu\|_g \le (\omega)^{\mu/2}, |\omega| \ge \omega_0\}}. $$

The multiplication operator by $\chi_{\Sigma}^{\mu}$ is also denoted $\chi_{\Sigma}^{\mu} : S(T_{\Sigma}T^*M) \to S'(T_{\Sigma}T^*M)$.

We denote $r_{/\Sigma} : S(TT^*M) \to S(T_{\Sigma}T^*M)$ the restriction operator to the base space $\Sigma \subset T^*M$. We denote

$$ (3.45) \qquad r_N : S(T_{\Sigma}T^*M) \to S(N) $$

the restriction operator to the sub-bundle $N \subset T_{\Sigma}T^*M$.

**Definition 3.18.** We define the operators

$$ (3.46) \qquad \mathcal{T}_{\Sigma} := \chi_{\Sigma}^{\mu} r_{/\Sigma} (\widetilde{\exp^{\circ}}) \mathcal{T} : C^{\infty}(M) \to S'(T_{\Sigma}T^*M), $$

$$ (3.47) \qquad \mathcal{T}_{\Sigma}^{\Delta} := \mathcal{T}^{\dagger} (\widetilde{\exp_N^{-1}})^{\circ} r_N \chi_{\Sigma}^{\mu} : S(T_{\Sigma}T^*M) \to C^{\infty}(M). $$

**Theorem 3.19.** For any $t \in \mathbb{R}$, we have

$$ (3.48) \qquad e^{tX} \approx \mathcal{T}_{\Sigma}^{\Delta} \tilde{\text{Op}}((d\tilde{\phi}^t)_{/\Sigma}) \mathcal{T}_{\Sigma}. $$

*Remark 3.20.* This remark is very similar to Remark 2.18. One advantage of (3.48) compared to (3.40) is that it does not contain the time dependent metaplectic correction $Y_t^{1/2}$, but contains instead, through the decomposition (3.29), the operator $\tilde{\text{Op}}(d\tilde{\phi}_{T_{\Sigma}}^t)$
that itself contains $Y_t^{1/2}$. Later in the proof of Theorem 4.31, we will use and take advantage of some unitary properties of this operator $\tilde{\text{Op}}(d\tilde{\phi}_t^t)$. On the other hand, one advantage of (3.40) compared to (3.48) is that it deals directly with the normal sub-bundle $N$ on which we aim to work, so it is easier at first sight.

*Proof.* The proof follows the same lines as the proof of Theorem 3.15. The only difference is that instead of using Lemma C.35, we use Lemma C.37 in the appendix for the Euclidean and linear case. □

**3.6. Emergence of quantum mechanics near the set $\Sigma$ for a general contact vector field.** A direct consequence of Theorem 3.15 is Theorem 3.21 that shows that at any time $t$ the dynamics of the pull back operator $e^{tX_F}$ is well approximated micro-locally near the set $\Sigma$ (at distance less than $\sigma$) and at high frequencies $\omega$, by a quantum operator $\mathcal{J}_N^\Delta Y_t^{1/2} \tilde{\text{Op}}(d\tilde{\phi}_N^t) \mathcal{J}_N$ (this is a F.I.O., compare with (1.11)).

This result is less precise than Theorem 1.5 that gives an approximation of the full operator $e^{tX_F}$, whereas here we have an approximation for a restriction to a micro-local neighborhood of $\Sigma$. However this may be interesting because $\Sigma$ is an invariant set for the dynamics. We may wonder if with some additional generic assumptions, this result can be improved towards Theorem 1.5.

**Theorem 3.21.** $\forall t \in \mathbb{R}, \exists m < 0, \forall \sigma > 0, \exists C_{t,\sigma} > 0 \forall \omega > 0,$

$$ \| \text{Op}(\chi_{\Sigma,\sigma}) (e^{tX_F} - \mathcal{J}_N^\Delta Y_t^{1/2} \tilde{\text{Op}}(d\tilde{\phi}_N^t) \mathcal{J}_N) \text{Op}(\chi_{\Sigma,\sigma}) (\text{Id} - \text{Op}(\chi_\omega)) \|_{L^2(M;F)} \le C_t \omega^m \underset{\omega \to +\infty}{\to} 0. $$

*Proof.* From Theorem 3.15, Definition 3.12 and Lemma 2.12. □

**4. MICRO-LOCAL ANALYSIS OF A CONTACT ANOSOV VECTOR FIELD $X$ NEAR**

$$ \Sigma = \mathbb{R}A \setminus \{0\} $$

In this section, as a last step, we consider the model of interest for this paper: a smooth contact Anosov vector field $X$ on a closed manifold $M$. In other words, we add the hypothesis of Anosov dynamics to Section 3. We first recall the precise definitions and notations.

**4.1. Definitions and notations.**

**4.1.1. Anosov vector field.** We make the hypothesis that $X$ is a smooth **Anosov vector field** on a closed manifold $M$. This means that for every point $m \in M$ the tangent vector $X(m)$ is non-zero and we have a continuous splitting of the tangent space

$$ (4.1) \qquad T_m M = E_u(m) \oplus E_s(m) \oplus \underbrace{E_0(m)}_{\mathbb{R}X(m)} $$

that is invariant under the action of the differential of the flow map $d\phi^t: TM \to TM$ and there exist $\lambda_{\min} > 0, C > 0$ and a smooth metric $g_M$ on $M$ such that

$$ (4.2) \quad \forall t \ge 0, \forall m \in M, \quad \left\| d\phi_{E_u(m)}^{-t} \right\|_{g_M} \le C e^{-\lambda_{\min}t}, \quad \left\| d\phi_{E_s(m)}^{t} \right\|_{g_M} \le C e^{-\lambda_{\min}t}. $$

See Figure 4.1. The linear subspaces $E_u(m), E_s(m) \subset T_m M$ are called the unstable/stable spaces and the one dimensional space $E_0(m) := \mathbb{R}X(m)$ is called the neutral direction or flow direction. For a general Anosov vector field, the maps $m \to E_u(m), m \to E_s(m)$ and $m \to E_u(m) \oplus E_s(m)$ are only Hölder continuous with (a priori different) Hölder exponents respectively [31].
<figure><img src="image_11.png" /><figcaption>FIGURE 4.1. Anosov flow $\phi^t$ (in solid line) generated by a vector field $X$ on a compact manifold $M$</figcaption></figure>

$$ (4.3) \qquad \beta_u, \beta_s, \beta_0 \in ]0, 1], \quad \beta := \min(\beta_u, \beta_s). $$

Let $\mathcal{A} \in C^0(M; T^*M)$ be the continuous one form on $M$ called **Anosov one form** defined for every $m \in M$ by the conditions

$$ (4.4) \qquad \mathcal{A}(m)(X(m)) = 1 \quad \text{and} \quad \text{Ker}(\mathcal{A}(m)) = E_u(m) \oplus E_s(m). $$

From this definition, $\mathcal{A}$ is preserved$^{12}$ by the flow $\phi^t$ and the map $m \to \mathcal{A}(m)$ is Hölder continuous with exponent $\beta_0$.

4.1.2. Contact Anosov vector field. In addition to Section 4.1.1 we assume in this paper that the Anosov vector field $X$ is **contact**, i.e. that the Anosov one form $\mathcal{A}$ defined in (4.4) is a **smooth contact one form** on $M$. This means that the distribution $E_u(m) \oplus E_s(m)$ is smooth (i.e. $C^\infty$) w.r.t. $m \in M$ (hence with Hölder exponent $\beta_0 = 1$) and that the linear space $\text{Ker}(\mathcal{A}(m)) = E_u(m) \oplus E_s(m)$ endowed with the two form $d\mathcal{A}(m)$ is a linear symplectic space for every $m \in M$.

Since $\mathcal{A}$ is invariant by the flow, for any $u_1, u_2 \in E_s(m)$ we have $\forall t \in \mathbb{R}, (d\mathcal{A})(u_1, u_2) = (d((\phi^t)^*\mathcal{A}))(u_1, u_2) = (d\mathcal{A})(d\phi^t(u_1), d\phi^t(u_2)) \xrightarrow{t \to +\infty} 0$ hence $d\mathcal{A}_{/E_s} = 0$ meaning that $E_s(m)$ is $d\mathcal{A}$-isotropic. Similarly $E_u(m)$ is $d\mathcal{A}$-isotropic hence $E_u(m), E_s(m)$ are $d\mathcal{A}$-Lagrangian and

$$ (4.5) \qquad d := \dim E_u(m) = \dim E_s(m), $$

hence $\dim M = 2d + 1$ is odd.

**Remark 4.1.** The vector field $X$ is the **Reeb vector field** of $\mathcal{A}$, i.e. $X$ is determined from $\mathcal{A}$ by $\mathcal{A}(X) = 1$ and$^{13}$

$$ (4.6) \qquad (d\mathcal{A})(X, \cdot) = 0, $$

<footnote>$^{12}$Notice from (4.2) that $\mathcal{A}$ is the unique continuous one form preserved by the flow with the normalization condition $\mathcal{A}(X) = 1$.</footnote>

<footnote>$^{13}$From Cartan formula and invariance of $\mathcal{A}: 0 = \mathcal{L}_{\chi}\mathcal{A} = \iota_{\chi}d\mathcal{A} + d\iota_{\chi}\mathcal{A}$ and $\iota_{\chi}\mathcal{A} = 1$ give $(d\mathcal{A})(X, \cdot) = 0$.</footnote>
as the setting of Section 3.

4.1.3. Dual decomposition. The dual bundle decomposition of (4.1) is

$$ (4.7) \qquad T^*M = E_u^* \oplus E_s^* \oplus E_0^* $$

with

$$ (4.8) \qquad E_u^* := (E_s \oplus E_0)^\perp = \{\rho \in T^*M, \text{ Ker}\rho \supset E_s \oplus E_0\}, $$

$$ (4.9) \qquad E_s^* := (E_u \oplus E_0)^\perp = \{\rho \in T^*M, \text{ Ker}\rho \supset E_u \oplus E_0\}, $$

$$ (4.10) \qquad E_0^* := (E_u \oplus E_s)^\perp = \{\rho \in T^*M, \text{ Ker}\rho \supset E_u \oplus E_s\} $$

$$ (4.11) \qquad \mathcal{A} = \{\omega\mathcal{A}(m), m \in M, \omega \in \mathbb{R}\}. \qquad (4.4) $$

Hence

$$ \dim_{(4.5)} E_u^*(m) = \dim_{(4.5)} E_s^*(m) = d, \quad \dim_{(4.5)} E_0^*(m) = 1. $$

From (4.3), the map $m \in M \to E_u^*(m)$ is $\beta_s$-Hölder continuous and $m \in M \to E_s^*(m)$ is $\beta_u$-Hölder continuous. However the maps $m \in M \to E_0^*(m) = \mathbb{R}\mathcal{A}(m)$ and $m \in M \to (E_u^* \oplus E_s^*)(m) = E_0^\perp(m) = \text{Ker}(X)(m)$ are smooth and have already been defined in (3.3) and (3.6).

A cotangent vector $\rho \in T^*M$ is decomposed accordingly to the dual decomposition (4.7) as

$$ (4.12) \qquad \rho = \rho_u + \rho_s + \rho_0 $$

with components

$$ \rho_u \in E_u^*, \rho_s \in E_s^*, \quad \rho_0 = \omega\mathcal{A}(m) \in E_0^* $$

$$ \text{with } m = \pi(\rho) \text{ and the frequency } \omega = \omega(\rho) \in \mathbb{R}. \qquad (2.7) $$

By duality, the hyperbolicity assumption (4.2) gives that the components (4.12) of $\rho(t) = \tilde{\phi}^t(\rho)$ satisfy

$$ \exists C > 0, \forall \rho(0) \in T^*M, \forall t \ge 0, \qquad (4.13) $$

$$ \|\rho_u(t)\|_{g_M} \ge \frac{1}{C} e^{\lambda_{\min} t} \|\rho_u(0)\|_{g_M}, \quad \|\rho_s(t)\|_{g_M} \le Ce^{-\lambda_{\min} t} \|\rho_s(0)\|_{g_M}, \quad \omega(t) = \omega(0). $$

See Figure 4.2.
<figure><img src="image_12.png" /><figcaption>FIGURE 4.2. The Anosov flow $\phi^t = \exp(tX)$ on $M$ induces a Hamiltonian flow $\tilde{\phi}^t = \exp(t\tilde{X})$ in cotangent space $T^*M$. The magenta lines on $M$ represent “internal oscillations” of a function $u$, that is a wave packet and is supposed here to have a small support near $\phi^t(m)$. These oscillations correspond to a cotangent vector $\rho \in T_{\phi^t(m)}^*M$. Transported by the flow the oscillations of $e^{tX}u = u \circ \phi^t$ increase and the wave fronts of these oscillations become parallel to $E_s \oplus E_0$ equivalently $\rho(t) = \tilde{\phi}^t(\rho)$ converges to the direction of $E_u^* \subset T^*M$ and remains in the frequency level $\Sigma_\omega := \omega^{-1}(\omega)$. The **trapped set** of the lifted flow $\tilde{\phi}^t$ is the rank one vector bundle $E_0^* = \mathbb{R}A$ (green line) where $A$ is the Anosov one form. From Lemma 3.2, $\Sigma := E_0^* \setminus \{0\}$ is a symplectic submanifold of $T^*M$, with $\dim\Sigma = 2(d+1)$.</figcaption></figure>
4.1.4. **Trapped set.** From (4.13) we deduce Lemma 4.2.

**Lemma 4.2.** The set $E_0^* = \mathbb{R}A$ is the **trapped set** (or non-wandering set) of the flow $\tilde{\phi}^t$ in the sense that
(4.14)

$$E_0^* = \mathbb{R}A \quad (3.7) \quad \Sigma \cup \{0\} = \{\rho \in T^*M \mid \exists \mathcal{K} \subset T^*M \text{ compact, s.t. } \tilde{\phi}^t(\rho) \in \mathcal{K}, \forall t \in \mathbb{R}\},$$

and $E_0^*$ is **transversely hyperbolic**.

*Proof.* From (4.12), the trapped set $E_0^*$ is characterized by $\rho_u = 0, \rho_s = 0$ and from (4.13), $E_0^*$ is transversely hyperbolic. $\square$

**4.2. Anisotropic Sobolev space, decay of norm outside the trapped set and discrete Ruelle spectrum.** In Lemma 4.2 we have observed that the trapped set $E_0^*$ is transversely hyperbolic. This will permit us to define a weight function or escape function $W: T^*M \to \mathbb{R}^+$ that decays outside the trapped set (i.e. that is a Lyapunov function for the flow $\tilde{\phi}^t$), following the constructions in [16] or [20, Def 5.6]. Then we define the anisotropic Sobolev space $H_W(M)$ and obtain that the norm of the pull back operator $e^{tX}$ restricted to the outside of $E_0^*$ decays exponentially fast with an arbitrarily large rate. This is the important property that will enable us to restrict the analysis of the operator $e^{tX}$ for $t > 0$, to a neighborhood of the trapped set $E_0^*$ in the forthcoming sections. In this section we recall all these results.

**4.2.1. Definition and properties of the escape function W.** Following [20, Def 5.6], we consider the positive valued continuous function $W \in C(T^*M; \mathbb{R}^+ \setminus \{0\})$ called **weight function** defined as follows. Let $R \ge 0$, $\gamma \in [1-\beta, 1[$ with $\beta$ given in (4.3). For $\rho \in T^*M$, with stable/unstable components $\rho_s \in E_s^*$, $\rho_u \in E_u^*$ given in (4.12), we define

$$(4.15) \qquad W(\rho) := \frac{\langle h_{\gamma}(\rho) \|\rho_s\|_{g_{\rho}} \rangle^R}{\langle h_{\gamma}(\rho) \|\rho_u\|_{g_{\rho}} \rangle^R}$$

with

$$(4.16) \qquad h_{\gamma}(\rho) := \left\langle \left( \|\rho_u\|_{g_{\rho}}^2 + \|\rho_s\|_{g_{\rho}}^2 \right)^{1/2} \right\rangle^{-\gamma}.$$

*Remark 4.3.* In the previous expressions, the vertical vectors $\rho_u, \rho_s \in T_m^*M$ are naturally identified with vectors in the tangent space $T_\rho(T^*M)$ of $T^*M$ at point $\rho \in T^*M$, so we can get their norm, $\|\rho_u\|_{g_\rho}, \|\rho_s\|_{g_\rho}$.

The function $W: \rho \in T^*M \to W(\rho) \in \mathbb{R}$ in (4.15) is positive and Hölder continuous, since the decomposition (4.12) is Hölder continuous. $W$ has a few important properties given in [20, Thm 5.9] as **decay property** and **slow variation properties**.

**4.2.2. Anisotropic spaces $H_W(M; F)$**. We will define the anisotropic Sobolev spaces $H_W(M; F)$ used in this paper, in order to get Theorem 4.8, saying that the operator $X_F$ has **intrinsic discrete spectrum** in $H_W(M; F)$.
**Remark 4.4.** Let us remark why to get discrete spectrum, we cannot consider the Hilbert space $L^2(M;F)$. If we choose an arbitrary smooth Hermitian metric on the vector bundle $F \to M$ and using the measure $dm$ on $M$, (3.2), we can define the Hilbert space $L^2(M;F)$ of sections of $F$. One can show that the operator $X_F$, Eq. (2.3), acting in $L^2(M;F)$ has only essential spectrum contained in some vertical band (the detailed argument will be explained in a forthcoming paper):

$$ \mathrm{spec}(X_F : L^2(M;F) \to L^2(M;F)) = \{z \in \mathbb{C}, \mathrm{Re}(z) \in [\gamma_{L^2}, \gamma_{L^2}^+]\}, $$

with

$$ \gamma_{L^2}^{\pm} = \lim_{t \to \pm\infty} \log \left\| e^{tX_F} \right\|_{L^{\infty}}^{1/t} . $$

For example, considering the vector field $X$ itself and the trivial bundle $F = M \times \mathbb{C}$, since $\mathrm{div}_{dm}X = 0$, we have that $X = -X^\dagger$ is skew symmetric in $L^2(M)$ and $X$ has continuous spectrum on the imaginary axis $i\mathbb{R}$.

For simplicity of notations, we will forget the vector bundle $F$, that plays no role for this construction, i.e. we first consider only a trivial bundle $F = M \times \mathbb{C}$. For any $u \in C^\infty(M)$ we define its $\mathcal{H}_W(M)$-norm

$$ (4.17) \qquad \|u\|_{\mathcal{H}_W(M)} := \|W\mathcal{T}u\|_{L^2(T^*M)}, $$

where $W$ is used as a multiplication operator on $S(T^*M)$. Let $\mathcal{H}_W(M)$ be the Hilbert space obtained by completion of $C^\infty(M)$ with respect to this norm and called **anisotropic Sobolev space**:

$$ (4.18) \qquad \mathcal{H}_W(M) := \overline{\{u \in C^\infty(M), \ \|u\|_{\mathcal{H}_W(M)} < \infty\}}. $$

In other words $W\mathcal{T} : \mathcal{H}_W(M) \to L^2(T^*M)$ is an isometry. To define $\mathcal{H}_W(M;F)$ we proceed similarly by completion of the space of smooth sections $C^\infty(M;F)$. For more details or information, we refer to [20, def 4.37].

**Remark 4.5.** The function $W$ in (4.15) satisfies

$$ (4.19) \qquad \exists C > 0, \forall \rho \in T^*M, C^{-1} (|\rho|_{g_M})^{-|r|} \le W(\rho) \le C (|\rho|_{g_M})^{|r|}, $$

with

$$ (4.20) \qquad r = \frac{R}{2} (1 - \gamma) > 0. $$

Consequently

$$ (4.21) \qquad H^r(M) \subset \mathcal{H}_W(M) \subset H^{-r}(M), $$

where $H^r(M) := \mathrm{Op}((|\xi|_{g_M})^{-r})(L^2(M))$ is the standard Sobolev space of order $r$.

**4.2.3. Decay of norm outside the trapped set.** Recall the truncation operator $\mathrm{Op}(\chi_{\Sigma,\sigma})$ at distance $\sigma$ from the set $\Sigma$ defined in (3.31).

**Theorem 4.6 (Decay outside the trapped set [20, Thm 5.13]).** For any $\Lambda > 0$, we can choose $R \gg 1$ large enough as a parameter of the function $W$ in (4.15), such that $\exists C > 0, \forall t \ge 0, \exists \sigma_t, \forall \sigma > \sigma_t$ we have

$$ (4.22) \qquad \left\| e^{tX} (\mathrm{Id} - \mathrm{Op}(\chi_{\Sigma, \sigma})) \right\|_{\mathcal{H}_W(M;F)} \le C e^{-\Lambda t}, $$
$$ (4.23) \qquad \left\| (\mathrm{Id} - \mathrm{Op}(\chi_{\Sigma,\sigma})) e^{tX} \right\|_{\mathcal{H}_W(M;F)} \le C e^{-\Lambda t}. $$

**Remark 4.7.** Consider the decomposition $e^{tX} = e^{tX}(\mathrm{Id} - \mathrm{Op}(\chi_{\Sigma,\sigma})) + e^{tX}\mathrm{Op}(\chi_{\Sigma,\sigma})$. Theorem 4.6 means that the first component $e^{tX}(\mathrm{Id} - \mathrm{Op}(\chi_{\Sigma,\sigma}))$ decays exponentially fast as $O(e^{-\Lambda t})$ with an arbitrarily large rate $\Lambda \gg 1$. We deduce that in order to describe the dominant effect of the operator $e^{tX}$ and Ruelle eigenvalues on the spectral domain $\mathrm{Re}(z) > -\Lambda$, one has to study the second component $e^{tX}\mathrm{Op}(\chi_{\Sigma,\sigma})$ namely the dynamics $e^{tX}$ restricted to a neighborhood of the trapped set $\Sigma = E_0^* \setminus \{0\}$. Then, from Theorem 3.15, we only need to study the operator $\tilde{\mathrm{Op}}(d\tilde{\phi}_N^t)$ acting on the normal bundle over the trapped set $\Sigma$. This is the subject of Section 4.3.

**4.2.4.** *Discrete Ruelle spectrum in anisotropic Sobolev spaces $\mathcal{H}_W(M;F)$. In paper [20, thm 5.14], Theorem 4.6 is used to prove Theorem 4.8. Recall that the function $W$ in (4.15) depends on the exponent $R$ and that the order $r = \frac{R}{2}(1-\gamma)$ is defined from $R$ in (4.20).*

**Theorem 4.8** (“Group property and discrete spectrum” [20, thm 2.11]). For a general Anosov vector field $X$ on $M$, the family of operators

$$ e^{tX_F} : \mathcal{H}_W(M;F) \to \mathcal{H}_W(M;F), \quad t \in \mathbb{R}, $$

forms a **strongly continuous group** and the generator $X_F$ has **discrete spectrum denoted** spect $(X_F)$, on $C - r\lambda_{\min} \le \mathrm{Re}(z) \le C'$ with $C, C'$ independent of the parameter $R$. This spectrum is **intrinsic** (i.e. does not depend on $W$) and called **future Ruelle spectrum** in [20, Thm 2.11].

See Figure 1.2.

**4.3. Description of the operator $e^{tX}$ near the set $\Sigma$ with the bundle $N_s$**

**4.3.1. Decomposition $N = N_u \oplus N_s$.** From the Anosov property, we have the decomposition $\mathrm{Ker}A = E_u \oplus E_s$ into Hölder continuous sub-bundles, that gives a refined decomposition of (3.11) into isotropic subspaces:

**Lemma 4.9.** Let

$$ N_u := N \cap d\pi^{-1}(E_u), \quad N_s := N \cap d\pi^{-1}(E_s), $$

$$ (4.24) \qquad N = N_u \oplus N_s, $$

$$ (4.25) \qquad \begin{gathered} K_u := K \cap d\pi^{-1}(E_u), \quad K_s := K \cap d\pi^{-1}(E_s), \quad K = K_u \oplus K_s, \\ \dim K_u = \dim K_s = \dim N_u = \dim N_s = d, \quad \dim K_0 = 2. \end{gathered} $$

Each component $K_u, K_s, \tilde{E}_0, \tilde{E}_0^*$, $N_u, N_s$ is a $\Omega$-isotropic vector space, invariant under the dynamics $d\tilde{\phi}^t$. The indices $u, s, 0$ denote respectively instability, stability or neutrality. See Figure 1.1.

**Remark 4.10.**

* Notice that due to (2.10) that reverses the direction of dynamics, $N_u$ is stable for $(d\tilde{\phi}^t)_{t \ge 0}$ and $N_s$ is unstable.

* The symplectic normal bundle $N \to \Sigma$ is smooth, but the isotropic sub-bundle $N_s \to \Sigma$ is only $\beta_s$-Hölder continuous.
*Proof.* From the fact explained in Section 4.1.2 that $E_s, E_u$ are $dA$-isotropic and that $d\pi$ is a symplectomorphism in (3.22) and (3.23), we deduce that $N_u, N_s, K_u, K_s$ are $\Omega$-isotropic. $\square$

**4.3.2. Transfer operators on $N_s$.** We start from Theorem 3.15. We have the splitting $N = N_s \oplus N_u$ in (4.24) of the symplectic bundle $N$ over $\Sigma$ into Lagrangian sub-bundles. We consider the operator $\check{\text{Op}}(d\tilde{\phi}_N^t)$ on the right hand side of (3.40) and our purpose is to express it as an operator acting only on the Lagrangian sub-bundle $N_s$. The bundles $N_u, N_s$ are invariant under the dynamics $d\tilde{\phi}^t$, hence the flow map splits accordingly (recall Remark 4.10)

$$d\tilde{\phi}_N^t = d\tilde{\phi}_{N_u}^t + d\tilde{\phi}_{N_s}^t \quad : N \to N.$$

**Notation 4.11.** The bundle $N_s \to \Sigma$ defined in (4.25) is Hölder continuous with exponent $\beta$. For simplicity we denote

$$ (4.26) \qquad S_{\beta}(N_s) := C^{\beta}(\Sigma; S(N_s(\cdot))) $$

as the set of functions $f \in C^\beta(\Sigma; S(N_s(\cdot)))$, $\beta$-Hölder continuous on $\Sigma$ with fast decay in frequency $\omega$, valued in the Schwartz space of functions $S(N_s(\rho))$ for each $\rho \in \Sigma$. This notation has already been introduced in (1.8) and will also concern later other space of sections over $\Sigma$. Notice that $S(N_s(\cdot)) \to \Sigma$ is an infinite rank bundle vector over $\Sigma$.

**4.3.3. The bundle $\mathcal{F}(N_s) \to \Sigma$.** Let us denote $|\det N_s|^{-1/2}$ the dual of the half density bundle in $N_s$ [28, p.509].

**Definition 4.12.** Let

$$ (4.27) \qquad \mathcal{F}(N_s) := |\det N_s|^{-1/2} \otimes S_{\beta}(N_s) $$

and

$$ (4.28) \qquad X_{\mathcal{F}} : C^{\beta}(\Sigma; \mathcal{F}(N_s)) \to C^{\beta}(\Sigma; \mathcal{F}(N_s)) $$

be the derivation of sections of $\mathcal{F}$ over the vector field $\tilde{X}$, that is generator of the group of operators, $\forall t \in \mathbb{R}$,

$$ (4.29) \qquad e^{tX_{\mathcal{F}}} := |\det(d\tilde{\phi}_{N_s}^t)|^{-1/2} (d\tilde{\phi}_{N_s}^t)^{-\circ} : C^{\beta}(\Sigma; \mathcal{F}(N_s)) \to C^{\beta}(\Sigma; \mathcal{F}(N_s)). $$

**Remark 4.13.** The derivation $X_{\mathcal{F}}$ over $\tilde{X}$ in (4.28) is well defined since the bundle $\mathcal{F}(N_s)$ is constructed from the bundle $N_s$ and the given vector field $X$ on $M$ induces a natural derivation on $N_s$ hence $X_{\mathcal{F}}$ on $\mathcal{F}(N_s)$. Recall from Remark 4.10 that the linear map $d\tilde{\phi}_{N_s}^t$ is expanding on $N_s$, hence $|\det(d\tilde{\phi}_{N_s}^t)| > 1$. The operator $X_{\mathcal{F}}$ has a domain that includes sections smooth along the flow direction. Recall that for simplicity of notation we have ignored the bundle $F$ in (4.27).

Now we will relate the operator $\check{\text{Op}}(d\tilde{\phi}_N^t)$ on the right hand side of (3.40) to the operator $e^{tX_{\mathcal{F}}}$ defined in (4.29). In (C.8) we define the Bargman transform

$$ \mathcal{B}_{N_s} : S_{\beta}(N_s) \to S(N_s \oplus N_s^*). $$

As explained in Lemma C.16, we have an $(\Omega, g)$-isomorphism

$$ \Psi : N = N_s \oplus N_s^{\perp g} \to N_s \oplus N_s^*. $$
But $N_u \neq N_s^{\perp g}$ in general, so we introduce $S_{N_s,N_u} : N \to N$ the linear (shear) symplectic map defined as identity on $N_s$ and any vector of $N_s^{\perp g}$ is mapped to its projection on $N_u$ parallel to $N_s$:

$$S_{N_s,N_u}(N_s) = N_s, \quad S_{N_s,N_u}(N_s^{\perp g}) = N_u.$$

Notice that the bundle of maps $S_{N_s,N_u}, \Psi$ is uniformly bounded over $\Sigma$. The operator $\tilde{\text{Op}}(S_{N_s,N_u}\Psi^{-1}) : S(N_s \oplus N_s^*) \to S(N)$ is defined as in (C.36) and we set

$$(4.30) \qquad \mathcal{B}_{N_s,N_u} := \tilde{\text{Op}}(S_{N_s,N_u}\Psi^{-1}) \mathcal{B}_{N_s} : S_{\beta}(N_s) \to S(N).$$

**Lemma 4.14.** We have

$$(4.31) \qquad \tilde{\text{Op}}(d\tilde{\phi}_N^t) = \mathcal{B}_{N_s,N_u} e^{tX_f} \mathcal{B}_{N_s,N_u}^\dagger.$$

*Proof.* Let $\Phi_{N_s} := (d\tilde{\phi}_{N_s}^t) \oplus (d\tilde{\phi}_{N_s}^t)^{-1} : N_s \oplus N_s^* \to N_s \oplus N_s^*$. From Lemma C.31 with the setting $\phi = (d\tilde{\phi}_{N_s}^t)^{-1}$, we have

$$(4.32) \qquad \tilde{\text{Op}}(\Phi_{N_s}) = \det(d\tilde{\phi}_{N_s}^t)^{-1/2} \mathcal{B}_{N_s} (d\tilde{\phi}_{N_s}^t)^{-\phi} \mathcal{B}_{N_s}^\dagger \\ = \mathcal{B}_{N_s} e^{tX_f} \mathcal{B}_{N_s}^\dagger : S(N_s \oplus N_s^*) \to S(N_s \oplus N_s^*).$$

By construction, we have the conjugation

$$d\tilde{\phi}_N^t = (S_{N_s,N_u}\Psi^{-1}) \circ \Phi_{N_s} \circ (S_{N_s,N_u}\Psi^{-1})^{-1} \quad : N \to N$$

that gives

$$(4.33) \qquad \tilde{\text{Op}}(d\tilde{\phi}_N^t) = \tilde{\text{Op}}(S_{N_s,N_u}\Psi^{-1}) \mathcal{B}_{N_s} e^{tX_f} \mathcal{B}_{N_s}^\dagger \tilde{\text{Op}}(S_{N_s,N_u}\Psi^{-1})^\dagger \\ = \mathcal{B}_{N_s,N_u} e^{tX_f} \mathcal{B}_{N_s,N_u}^\dagger.$$

□

We finish this section with the description of the operator $e^{tX}$ near the set $\Sigma$ using the bundle $N_s$.

**Definition 4.15.** We define the operators

$$(4.34) \qquad \mathcal{J}_{N_s} := \mathcal{B}_{N_s,N_u}^\dagger \mathcal{J}_N = \mathcal{B}_{N_s,N_u}^\dagger \chi_\Sigma^\mu (\widetilde{\exp_N})^\circ \mathcal{J} : C^\infty(M) \to C^\beta(\Sigma; \mathcal{F}(N_s)),$$

$$(4.35) \qquad \mathcal{J}_{N_s}^\Delta := \mathcal{J}_N^\Delta \mathcal{B}_{N_s,N_u} = \mathcal{J}^\dagger (\widetilde{\exp_N}^{-1})^\circ \chi_\Sigma^\mu \mathcal{B}_{N_s,N_u} : C^\beta(\Sigma; \mathcal{F}(N_s)) \to C^\infty(M).$$

**Theorem 4.16.** For any fixed $t \in \mathbb{R}$, we have

$$(4.36) \qquad e^{tX} \approx \mathcal{J}_{N_s}^\Delta Y_t^{1/2} e^{tX_f} \mathcal{J}_{N_s}.$$

Recall that the metaplectic correction $Y_t^{1/2} > 0$ defined in (3.39) is a function on $\Sigma$ considered as a multiplication operator.
*Proof.* We write

$$ (4.37) \qquad \begin{aligned} e^{tX} &\approx_{(3.40)} \mathcal{J}_N^\Delta Y_t^{1/2} \tilde{\Omega}_p (d\tilde{\phi}_N^t) \mathcal{J}_N \\ &=_{(4.31),(4.34),(4.35)} \mathcal{J}_{N_s}^\Delta Y_t^{1/2} e^{tX_\mathcal{F}} \mathcal{J}_{N_s}. \end{aligned} $$

□

**4.4. The bundle map $e^{tX_\mathcal{F}}$ and sub-bundle $\mathcal{F}_k(N_s)$.** The expression (4.36) will be convenient to get some properties of the dynamical operator $e^{tX}$ from the study of properties of the operator $e^{tX_\mathcal{F}}$. For example, one important property that we will first exploit is that $d\tilde{\phi}_{N_s}^t : N_s \to N_s$ is a linear map hence the operators $(d\tilde{\phi}_{N_s}^t)^{-\circ}$ and $e^{tX_\mathcal{F}}$ in (4.29) preserve the finite rank sub-bundle of homogeneous polynomials of degree $k \in \mathbb{N}$. This property is responsible for the band structure of the spectrum.

**4.4.1. Taylor projectors $T_k$.** Since $d\tilde{\phi}_{N_s}^t : N_s \to N_s$ is a vector bundle homomorphism, the push-forward operator $(d\tilde{\phi}_{N_s}^t)^{-\circ}$ in (4.29) keeps invariant the space of homogeneous polynomial functions of order $k \in \mathbb{N}$ denoted $\text{Pol}_k(N_s) \subset S'_\beta(N_s)$. To express this property, we first consider the vector bundle $E_s \to M$ and the bundle map of finite rank “Taylor projectors” over $\text{Id}$, from Definition C.38, denoted

$$ (4.38) \qquad T_k : S_\beta(E_s) \to \text{Pol}_k(E_s) \subset S'_\beta(E_s) $$

and

$$ (4.39) \qquad T_{[0,K]} := \sum_{k=0}^{K} T_k, \qquad T_{\ge (K+1)} = \text{Id} - T_{[0,K]}, $$

satisfying

$$ (4.40) \qquad T_k T_{k'} = \delta_{k=k'} T_k. $$

Using the projection map $d\pi : N_s \to E_s$ in (3.23), we lift these operators to the vector bundle $S_\beta(N_s) \to \Sigma$:

$$ (4.41) \qquad \tilde{T}_k : S_\beta(N_s) \to \text{Pol}_k(N_s) \subset S'_\beta(N_s). $$

We define $\tilde{T}_{[0,K]}$ and $\tilde{T}_{\ge (K+1)}$ similarly to (4.39). We have (in the next equation and after, a strict notation should be $\text{Id}_{|\det N_s|^{-1/2}} \otimes \tilde{T}_k$ instead of $\tilde{T}_k$)

$$ (4.42) \qquad [e^{tX_\mathcal{F}}, \tilde{T}_k] = 0. $$

**Definition 4.17.** We denote

$$ (4.43) \qquad \mathcal{F}_k(N_s) := |\det N_s|^{-1/2} \otimes \text{Pol}_k(N_s) $$

that is a finite rank bundle over $\Sigma$. The space of sections $C^\beta(\Sigma; \mathcal{F}_k(N_s))$ is the image of the bundle map $\tilde{T}_k : C^\beta(\Sigma; \mathcal{F}(N_s)) \to C^\beta(\Sigma; \mathcal{F}_k(N_s))$.

We denote $X_{\mathcal{F}_k}$ the generator $X_\mathcal{F}$ in (4.28) restricted to the invariant space of sections $C^\beta(\Sigma; \mathcal{F}_k(N_s))$.

More generally, for $0 \le k_1 \le k_2$, we denote the direct sum bundle

$$ (4.44) \qquad \mathcal{F}_{[k_1, k_2]}(N_s) := \bigoplus_{k=k_1}^{k_2} \mathcal{F}_k(N_s), $$
and denote $X_{\mathcal{F}[k_1,k_2]}$ the generator in $C^\beta(\Sigma; \mathcal{F}[k_1,k_2](N_s))$.

4.4.2. Spectrum of the derivation $X_{\mathcal{F}_k}$ in $L^2(\Sigma; \mathcal{F}_k(N_s))$. The operators $(e^{tX_{\mathcal{F}_k}})_{t \in \mathbb{R}}$ in (4.29) form a group of point-wise linear bundle maps over $\tilde{\phi}^t: \Sigma \to \Sigma$ that preserves the volume. Consequently all $L^p$ norms $\|e^{tX_{\mathcal{F}_k}}\|_{L^p(\Sigma; \mathcal{F}_k)}$ for $p \in [1, \infty]$ are equal, as it is for general multiplication operators.

**Definition 4.18.** For any $k \in \mathbb{N}$, let

$$ (4.45) \qquad \gamma_k^{\pm} := \lim_{t \to \pm\infty} \log \left\| e^{tX_{\mathcal{F}_k}} \right\|_{L^2(\Sigma; \mathcal{F}_k(N_s))}^{1/t} . $$

**Remark 4.19.** The values $\gamma_k^{\pm}$ defined in (4.45) with $L^2(\Sigma; \mathcal{F}_k(N_s))$ coincide with the Definition (1.6) using $L^{\infty}(M; \mathcal{F}_k(E_s))$.

**Lemma 4.20.** We have the estimates $\forall \epsilon > 0, \exists C_{\epsilon} > 0, \forall t \ge 0$,

$$ (4.46) \qquad \left\| e^{tX_{\mathcal{F}_k}} \right\|_{L^2(\Sigma; \mathcal{F}_k(N_s))} \le C_{\epsilon} e^{t(\gamma_k^+ + \epsilon)}, \qquad \left\| e^{-tX_{\mathcal{F}_k}} \right\|_{L^2(\Sigma; \mathcal{F}_k(N_s))} \le C_{\epsilon} e^{-t(\gamma_k^- - \epsilon)}. $$

Equivalently $\forall \epsilon > 0, \exists C_{\epsilon} > 0, \forall z \in \mathbb{C}, \mathrm{Re}(z) > \gamma_k^+ + \epsilon \text{ or } \mathrm{Re}(z) < \gamma_k^- - \epsilon$,

$$ (4.47) \qquad \left\| (z - X_{\mathcal{F}_k})^{-1} \right\|_{L^2(\Sigma; \mathcal{F}_k(N_s))} \le C_{\epsilon}. $$

Consequently the spectrum of the operator $X_{\mathcal{F}_k}$ in $L^2(\Sigma; \mathcal{F}_k(N_s))$ is contained in the vertical band

$$ (4.48) \qquad B_k := \{ \mathrm{Re}(z) \in [\gamma_k^-, \gamma_k^+] \}. $$

**Remark 4.21.** We will explain in a forthcoming paper that the spectrum of $X_{\mathcal{F}_k}$ in $L^2(\Sigma; \mathcal{F}_k(N_s))$ consists only of essential spectrum that fills the whole band $B_k$.

**Proof.** Eq. (4.45) implies (4.46) and also (4.47) and claim for (4.48) by writing the convergent expression for $\mathrm{Re}(z) > \gamma_k^+ + \epsilon$:

$$ (z - X_{\mathcal{F}_k})^{-1} = \int_0^{\infty} e^{t(X_{\mathcal{F}_k} - z)} dt $$

that gives, if $\epsilon' < \epsilon$,

$$ \left\| (z - X_{\mathcal{F}_k})^{-1} \right\| \le \int_0^\infty e^{-\mathrm{Re}(z)t} \left\| e^{tX_{\mathcal{F}_k}} \right\| dt \le \int_0^\infty e^{-\mathrm{Re}(z)t} C_{\epsilon'} e^{t(\gamma_k^+ + \epsilon')} dt \le C_{\epsilon}. $$

Similarly for $\mathrm{Re}(z) < \gamma_k^- - \epsilon$, we have $(z - X_{\mathcal{F}_k})^{-1} = - \int_{-\infty}^0 e^{t(X_{\mathcal{F}_k} - z)} dt$, and get also (4.47). $\square$

4.4.3. Sobolev space $H_W(N_s)$ and remainder term. The operator $e^{tX_{\mathcal{F}}}$ acts in the infinite rank bundle $\mathcal{F}(N_s)$ in (4.27). We explain here how to “extract” the finite codimension part $e^{tX_{\mathcal{F}}} \tilde{T}_{\ge (K+1)}$. We define a weight function $W: N \to \mathbb{R}^+$ similar to $W: T^*M \to \mathbb{R}^+$ in (4.15), with the same parameters $0 < \gamma < 1, R > 0$, as follows. For $\rho \in \Sigma$, $v = v_u + v_s \in N(\rho) = N_u(\rho) \oplus N_s(\rho)$ with $v_u \in N_u(\rho), v_s \in N_s(\rho)$, let

$$ (4.49) \qquad W(v) := \frac{\langle h_{\gamma}(v) ||v_s||_{g_{\rho}} \rangle^R}{\langle h_{\gamma}(v) ||v_u||_{g_{\rho}} \rangle^R} $$
with

$$h_{\gamma}(v) = \langle \|v\|_{g_{\rho}} \rangle^{-\gamma}.$$

The metric g on $T^*M$ is moderate [20, Lemma 4.12]. This implies that under the exponential map $\exp_N: N \to T^*M$ defined in (3.33), the functions $W$ and $W$ are equivalent in the neighborhood of the trapped set defined in (3.34), i.e. $\exists C \ge 1, \exists \omega_0 > 0, \forall v \in N$ with $\|v\|_g \le \langle\omega\rangle^{\mu/2}, \omega = \omega(\pi(v)), |\omega| \ge \omega_0$, we have

$$(4.50) \qquad \frac{1}{C} W(v) \le W(\exp_N(v)) \le CW(v).$$

We define the Sobolev norm of $u \in S_\beta(N_s)_{(4.26)} = C^\beta(\Sigma; S(N_s))$ by

$$\|u\|_{H_W(N_s)} := \|W B_{N_s, N_u} u\|_{L^2(N)},$$

and the anisotropic Sobolev space bundle by completion

$$(4.51) \qquad \mathcal{H}_W(N_s) := \overline{\{u \in S_\beta(N_s), \|u\|_{\mathcal{H}_W(N_s)} < \infty\}}.$$

**Lemma 4.22.** For any $\varepsilon > 0$, $K \in \mathbb{N}$, we can choose $R$ in (4.49) large enough and $\omega_0 > 0$ so that $\exists C_{K,\varepsilon} > 0, \forall t \ge 0$,

$$(4.52) \qquad \left\| e^{tX_f} \tilde{T}_{\ge (K+1)} \chi_{|\omega| \ge \omega_0} \right\|_{H_W(N_s)} \le C_{K,\varepsilon} e^{(\gamma_{K+1}^+ + \varepsilon)t}.$$

*Proof.* Let $K \in \mathbb{N}$, $\varepsilon > 0$ and $t \ge 0$. Recall that $e^{tX_f}$ is a bundle map over $\tilde{\phi}^t: \Sigma \to \Sigma$. Let $\rho \in \Sigma$ with $\omega(\rho) \ge \omega_0$ and $m = \pi(\rho) \in M$. We first observe that the operator

$$e^{tX_f} \tilde{T}_{\ge (K+1)} : S(N_s(\rho)) \to S(N_s(\tilde{\phi}^t(\rho)))$$

is isomorphic to

$$e^{tX_f} \tilde{T}_{\ge (K+1)} : S(E_s(m)) \to S(E_s(\phi^t(m))),$$

with $m \in M$ on a compact space providing uniformity over $\Sigma$. We apply Proposition D.3 in the appendix giving that $\forall \varepsilon > 0, \exists C_{K,\varepsilon} > 0, \forall t \ge 0, \forall u \in S_\beta(N_s)$,

$$\begin{aligned} \left\| e^{tX_f} \tilde{T}_{\ge (K+1)} \chi_{|\omega| \ge \omega_0} u \right\|_{H_W(N_s)}^2 &= \int_{\Sigma} \left\| (e^{tX_f} \tilde{T}_{\ge (K+1)} \chi_{|\omega| \ge \omega_0} u)(\rho) \right\|_{H_W(N_s)}^2 d\rho \\ &\le (C_{K,\varepsilon} e^{(\gamma_{K+1}^+ + \varepsilon)t})^2 \int_{\Sigma} \left\| (\chi_{|\omega| \ge \omega_0} u)(\rho) \right\|_{H_W(N_s)}^2 d\rho \\ &\le (C_{K,\varepsilon} e^{(\gamma_{K+1}^+ + \varepsilon)t})^2 \|u\|_{H_W(N_s)}^2, \end{aligned}$$

giving (4.52). $\square$

<figure><img src="image_13.png" /></figure>

**Proposition 4.23.** For any $K \in \mathbb{N}$, we can choose $R$ in (4.49) large enough so that for every $k \in [0, K]$, there exists $C_k$ such that

$$(4.53) \qquad \left\| \tilde{T}_k \right\|_{H_W(N_s)} \le C_k.$$

*Proof.* As in the proof of Lemma 4.22, we use the fact that we have a bundle map and at each point of the base, we use Lemma D.6 that is uniform over $\Sigma$. $\square$

<figure><img src="image_14.png" /></figure>
4.5. **Symbols and F.I.O. on Σ.** Recall $\mathcal{F}_k(E_s) = |\det E_s|^{-1/2} \otimes \text{Pol}_k(E_s)$ that is a $C^\beta$ continuous bundle over $M$ and more generally $\mathcal{F}_{[k_1, k_2]}(E_s)$ in (4.44).

**Definition 4.24.** A symbol is a function on $M$ operator valued in $L(\mathcal{F}_{[0,K]}(E_s))$ (or $L(\mathcal{F}(E_s))$):

$$a \in C^\beta (M; L(\mathcal{F}_{[0,K]}(E_s))).$$

Using the projection map $\sqrt{\omega}d\pi : N_s \to E_s$ in (3.23) that is invertible, we obtain a
lifted function denoted

$$\tilde{a} \in C^\beta (\Sigma; L(\mathcal{F}_{[0,K]}(N_s))),$$

considered below as a multiplicative operator.

**Remark 4.25.** Here is precisely how we get $\tilde{a}$ from $a$: for $\rho \in \Sigma$, $m = \pi(\rho) \in M$, one has $(\sqrt{\omega}d\pi_\rho)^\circ : S(E_s(m)) \to S(N_s(\rho))$ and set

$$\tilde{a} : \rho \in \Sigma \to \tilde{a}(\rho) := (\sqrt{\omega}d\pi_\rho)^\circ a (\sqrt{\omega}d\pi_\rho)^{-\circ}.$$

**Example 4.26.** Later we will use Definition (4.55) only for symbols of the form (4.54)

$$a = T_k \text{ (giving } \tilde{a} = \tilde{T}_k), \text{ or } a = T_{[0,K]} \text{ or } a \in \text{End}(\mathcal{F}_{[0,K]}) \text{ or } a = T_{\geq (K+1)} \text{ or } a = \text{Id}_{\mathcal{F}}.$$

Recall the operators $\mathcal{J}_{N_s} : C^\infty(M) \to C^\beta(\Sigma; \mathcal{F}(N_s))$ and $\mathcal{J}_{N_s}^\Delta : C^\beta(\Sigma; \mathcal{F}(N_s)) \to C^\infty(M)$ defined in (4.34) and (4.35).

**Definition 4.27** (F.I.O., quantum operator). For a symbol $a \in C^\beta(M; L(\mathcal{F}(E_s)))$ and $t \in \mathbb{R}$, the **quantization** of the symplectic bundle map $e^{tX_\beta} \tilde{a} : C^\beta(\Sigma; \mathcal{F}(N_s)) \to C^\beta(\Sigma; \mathcal{F}(N_s))$ is the F.I.O. operator

$$(4.55) \qquad \text{Op}_\Sigma(e^{tX_\beta} \tilde{a}) := \mathcal{J}_{N_s}^\Delta Y_t^{1/2} e^{tX_\beta} \tilde{a} \mathcal{J}_{N_s} : C^\infty(M) \to C^\infty(M).$$

**Remark 4.28.** The term F.I.O. stands for Fourier integral operator, that is the usual name for this kind of operator.

Using the new notation (4.55), Theorem 4.16 can be rephrased as follows.

**Theorem 4.29** (Approximation of the dynamics by a quantum operator). For any $t \in \mathbb{R}$,

$$(4.56) \qquad e^{tX} \approx \text{Op}_\Sigma(e^{tX_\beta}).$$

4.5.1. Continuity theorem, boundness estimates with respect to time.

**Remark 4.30.** In the rest of this section we take any $K \in \mathbb{N}$ and fix it. Then we choose a weight function $W$ with exponent $R$ such that (4.52) holds true and consider symbols $a \in C^\beta(M; L(\mathcal{F}_{[0,K]}(E_s)))$.

**Theorem 4.31** (Continuity theorem for F.I.O.). There exists $C > 0$, such that for any symbol $a \in C^\beta(M; L(\mathcal{F}_{[0,K]}(E_s)))$ and any $t \in \mathbb{R}$,

$$(4.57) \qquad \left\| \text{Op}_\Sigma(e^{tX_\beta} \tilde{a}) \right\|_{\mathcal{H}_W(M)} \leq C \left\| e^{tX_\beta} \tilde{a} \right\|_{\mathcal{H}_W(\mathcal{F})}.$$

*Proof.* We first give an equivalent expression for the operator as we already saw in Theorem 3.19. This expression is more complicated in appearance but more useful for the proof to work uniformly w.r.t. time $t$.
**Lemma 4.32 (Equivalent expression).** We have
(4.58)

$$
\operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}}\tilde{a}) \underset{(4.55)}{=} \mathcal{J}_{N_s}^{\Delta} Y_t^{1/2} e^{tX_{\mathcal{F}}} \tilde{a} \mathcal{J}_{N_s} \approx \mathcal{J}_{\Sigma}^{\Delta} (\tilde{\operatorname{Op}}(d\tilde{\phi}_{T\Sigma}^{t}) \otimes (\mathcal{B}_{N_s, N_u}(e^{tX_{\mathcal{F}}}\tilde{a}) \mathcal{B}_{N_s, N_u}^{\dagger})) \mathcal{J}_{\Sigma}.
$$

Proof. We have already obtained

$$
\begin{align*}
\mathcal{J}_{N_s}^{\Delta} Y_t^{1/2} e^{tX_f} \mathcal{J}_{N_s} &\approx \mathcal{J}_{\Sigma}^{\Delta} \tilde{\operatorname{Op}} \left( (d\tilde{\phi}^t)_{/\Sigma} \right) \mathcal{J}_{\Sigma} \\
&= \mathcal{J}_{\Sigma}^{\Delta} (\tilde{\operatorname{Op}} (d\tilde{\phi}_{T\Sigma}^t) \otimes_{\Sigma} \tilde{\operatorname{Op}} (d\tilde{\phi}_N^t)) \mathcal{J}_{\Sigma} \\
&= \mathcal{J}_{\Sigma}^{\Delta} (\tilde{\operatorname{Op}} (d\tilde{\phi}_{T\Sigma}^t) \otimes (\mathcal{B}_{N_s, N_u} e^{tX_f} \mathcal{B}_{N_s, N_u}^{\dagger})) \mathcal{J}_{\Sigma}.
\end{align*}
$$

By inserting the bundle map $\tilde{a}$ we obtain (4.58).

<figure><img src="image_15.png" /></figure>

We have defined $\mathcal{H}_W(N_s)$ in (4.51). Similarly, we can define $\mathcal{H}_W(N) := L^2(N; W^2)$ with the weight $W$ on $N$ in (4.49) and define $\mathcal{H}_W(T_\Sigma T^*M)$ by the weight $W$ extended trivially to $T_\Sigma T^{*M} = T\Sigma \oplus N$ by $W(u, v) = W(v)$ for any $u \in T_\rho\Sigma, v \in N_\rho$.

**Lemma 4.33.** $\exists C > 0$,

$$
(4.59) \quad \left\| \mathcal{J}_{\Sigma} \right\|_{\mathcal{H}_W(M) \to \mathcal{H}_W(T_\Sigma T^*M)} \le C, \quad \left\| \mathcal{J}_{\Sigma}^{\Delta} \right\|_{\mathcal{H}_W(T_\Sigma T^*M) \to \mathcal{H}_W(M)} \le C.
$$

Proof. Recall the expression $\mathcal{J}_{\Sigma} = \chi_{\Sigma}^{\mu} r_{/\Sigma} (\widetilde{\exp}^{\circ}) \mathcal{T}$ where, from (4.50), the operator $\chi_{\Sigma}^{\mu} r_{/\Sigma} (\widetilde{\exp}^{\circ})$ satisfies $\exists C > 0$,

$$
\left\| \chi_{\Sigma}^{\mu} r_{/\Sigma} (\widetilde{\exp}^{\circ}) \right\|_{L^2(T^*M; W) \to \mathcal{H}_W(T_\Sigma T^*M)} \le C.
$$

Since $\mathcal{T} : \mathcal{H}_W(M) \to L^2(T^*M; W)$ is an isometry, we deduce the first bound in (4.59).
Similarly from the expression $\mathcal{J}_{\Sigma}^{\Delta} = \mathcal{T}^{\dagger}(\overline{\exp_N^{-1}})^{\circ} r_N \chi_{\Sigma}^{\mu}$, we deduce the second claim.
$\square$

<figure><img src="image_16.png" /></figure>

**Lemma 4.34.** We have

$$
(4.60) \quad \left\| \tilde{\operatorname{Op}}(d\tilde{\phi}_{T\Sigma}^t) \otimes (\mathcal{B}_{N_s, N_u}(e^{tX_f}\tilde{a}) \mathcal{B}_{N_s, N_u}^\dagger) \right\|_{\mathcal{H}_W(T_\Sigma T^{*M})} = \left\| e^{tX_f}\tilde{a} \right\|_{\mathcal{H}_W(\mathcal{F})}.
$$

*Proof.* From the definition of the norms above, we have

$$
\begin{align*}
& \left\| \tilde{\operatorname{Op}}(d\tilde{\phi}_{T\Sigma}^t) \otimes (\mathcal{B}_{N_s, N_u}(e^{tX_f}\tilde{a}) \mathcal{B}_{N_s, N_u}^\dagger) \right\|_{\mathcal{H}_W(T_\Sigma T^{*M})} \\
&= \left\| \tilde{\operatorname{Op}}(d\tilde{\phi}_{T\Sigma}^t) \right\|_{L^2} \left\| (\mathcal{B}_{N_s, N_u}(e^{tX_f}\tilde{a}) \mathcal{B}_{N_s, N_u}^\dagger) \right\|_{\mathcal{H}_W(N)} \\
&= \left\| e^{tX_f}\tilde{a} \right\|_{\mathcal{H}_W(\mathcal{F})}.
\end{align*}
$$

<figure><img src="image_17.png" /></figure>

We come back to the proof of Theorem 4.31. Then

$$
\left\| \mathrm{Op}_{\Sigma} (e^{tX_f} \tilde{a}) \right\|_{\mathcal{H}_W(M) (4.58), (4.59), (4.60)} \le \frac{C}{(4.59)} \left\| e^{tX_f} \tilde{a} \right\|_{\mathcal{H}_W(\mathcal{F})}
$$

giving (4.57).

<figure><img src="image_18.png" /></figure>

Here are some examples of Theorem 4.31 that we will use later.
**Corollary 4.35.** We have $\forall \epsilon > 0, \exists C_{K,\epsilon} > 0, \forall t \ge 0$, (positive time)

$$ (4.61) \qquad \left\| \mathrm{Op}_{\Sigma} (e^{tX_{\mathcal{F}}} \tilde{T}_{\ge (K+1)}) \right\|_{\mathcal{H}_{W}(M)} \le C_{K,\epsilon} e^{t(\gamma_{K+1}^{+} + \epsilon)} $$

and for any $0 \le k_1 \le k_2 \le K, \forall \epsilon > 0, \exists C_{K,\epsilon} > 0, \forall t \ge 0,$

$$ (4.62) \qquad \left\| \mathrm{Op}_{\Sigma} (e^{tX_{\mathcal{F}}} \tilde{T}_{[k_1, k_2]}) \right\|_{\mathcal{H}_{W}(M)} \le C_{K,\epsilon} e^{t(\gamma_{k_1}^{+} + \epsilon)}. $$

Also $\forall t \le 0$, (negative time)

$$ (4.63) \qquad \left\| \mathrm{Op}_{\Sigma} (e^{tX_{\mathcal{F}}} \tilde{T}_{[k_1, k_2]}) \right\|_{\mathcal{H}_{W}(M)} \le C_{K,\epsilon} e^{t(\gamma_{k_2}^{-} - \epsilon)}. $$

*Proof.* To get (4.61) we have that $\exists C, \forall \epsilon > 0, \exists C_{K,\epsilon} > 0, \forall t \ge 0,$

$$ \begin{aligned} \left\| \mathrm{Op}_{\Sigma} (e^{tX_{\mathcal{F}}} \tilde{T}_{\ge (K+1)}) \right\|_{\mathcal{H}_{W}(M)} &\le C \left\| e^{tX_{\mathcal{F}}} \tilde{T}_{\ge (K+1)} \right\|_{\mathcal{H}_{W}(\mathcal{F})} \\ &\le C_{K,\epsilon} e^{(\gamma_{K+1}^{+} + \epsilon)t}. \end{aligned} $$

To get (4.62), in the equations below we will use (*) that $\gamma_{k+1}^{+} \le \gamma_k^{+}, \forall k \in \mathbb{N}$. We have that $\exists C, \forall \epsilon > 0, \exists C_K, C_{K,\epsilon} > 0, \forall t \ge 0,$

$$ \begin{aligned} \left\| \mathrm{Op}_{\Sigma} (e^{tX_{\mathcal{F}}} \tilde{T}_{[k_1, k_2]}) \right\|_{\mathcal{H}_{W}(M)} &\le C \left\| e^{tX_{\mathcal{F}}} \tilde{T}_{[k_1, k_2]} \right\|_{\mathcal{H}_{W}(\mathcal{F})} \le C \left\| e^{tX_{\mathcal{F}}[k_1, k_2]} \right\|_{L^2(\Sigma; \mathcal{F}[k_1, k_2])} \\ &\le C_{K,\epsilon} \max_{k \in [k_1, k_2]} e^{t(\gamma_k^{+} + \epsilon)} = C_{K,\epsilon} e^{t(\gamma_{k_1}^{+} + \epsilon)}. \end{aligned} $$

Finally to get (4.63), we repeat the same first lines as above and write $\forall \epsilon > 0, \exists C_{K,\epsilon} > 0, \forall t \ge 0,$

$$ \begin{aligned} \left\| \mathrm{Op}_{\Sigma} (e^{-tX_{\mathcal{F}}} \tilde{T}_{[k_1, k_2]}) \right\|_{\mathcal{H}_{W}(M)} &\le C \left\| e^{-tX_{\mathcal{F}}[k_1, k_2]} \right\|_{L^2(\Sigma; \mathcal{F}[k_1, k_2])} \\ &\le C_{K,\epsilon} \max_{k \in [k_1, k_2]} e^{-t(\gamma_k^{-} - \epsilon)} = C_{K,\epsilon} e^{-t(\gamma_{k_2}^{-} - \epsilon)}. \end{aligned} $$

□

*Remark 4.36.* In particular, taking $t = 0$ we get for any $k \in [0, K]$,

$$ (4.64) \qquad \left\| \mathrm{Op}_{\Sigma} (\tilde{T}_k) \right\|_{\mathcal{H}_{W}(M)} \le C_K. $$

4.5.2. *Composition formula.* Recall $\mathcal{F}_{[0,K]}$ defined in (4.44). Recall that the operators $\mathcal{F}_{N_s}, \mathcal{F}_{N_s}^{\Delta}$ in Definition 3.14 depend on the exponent $0 < \mu < 1$ introduced in the cut-off (3.34). Consequently the quantization $\mathrm{Op}_{\Sigma}(\cdot)$ defined in (4.55) also depends on $\mu$. Recall that $\beta$ is the Hölder exponent of $E_u, E_s$ in (4.3).

**Theorem 4.37 (Composition formula).** Take $\mu < \beta$. For any symbols $a, b \in C^{\beta}(M; L(\mathcal{F}_{[0,K]}(E_s)))$, any $t, t' \in \mathbb{R}$, we have

$$ (4.65) \qquad \mathrm{Op}_{\Sigma} (e^{tX_{\mathcal{F}}} \tilde{a}) \mathrm{Op}_{\Sigma} (e^{t'X_{\mathcal{F}}} \tilde{b}) \approx \mathrm{Op}_{\Sigma} (e^{tX_{\mathcal{F}}} \tilde{a} e^{t'X_{\mathcal{F}}} \tilde{b}) $$

(where the right hand side term has the factor $Y_{t+t'}^{1/2}$ in its Definition (4.55)).
*Proof.* For convenience, we first naturally extend Definition 3.12 of the equivalence
≈ in (3.32) to operators on $S(T^*M)$ and also $C^\beta(\Sigma; \mathcal{F})$ (instead of $C^\infty(M)$ only). We
define

$$
(4.66) \qquad
\begin{aligned}
\tilde{J}_{N_s} &:= B_{N_s, N_u}^{\dagger} \chi_{\Sigma}^{\mu} \widehat{(\exp_N)}^{\circ} & : S(T^*M) &\to C^{\beta}(\Sigma; \mathcal{F}), \\
\tilde{J}_{N_s}^{\Delta} &:= (\widehat{\exp_N}^{-1})^{\circ} \chi_{\Sigma}^{\mu} \mathcal{B}_{N_s, N_u} & : C^{\beta}(\Sigma; \mathcal{F}) &\to C(T^*M).
\end{aligned}
$$

These operators enter in the following expressions

$$
(4.67) \qquad \mathcal{T}_{N_s} \underset{(4.34)}{=} \tilde{\mathcal{J}}_{N_s} \mathcal{T}, \qquad \mathcal{T}_{N_s}^{\Delta} \underset{(4.35)}{=} \mathcal{T}^{\dagger} \tilde{\mathcal{J}}_{N_s}^{\Delta}.
$$

For an operator $A : C(\Sigma; \mathcal{F}) \to C(\Sigma; \mathcal{F})$, we denote

$$
(4.68) \qquad \tilde{A} := \tilde{J}_{N_s}^{\Delta} A \tilde{J}_{N_s} : S(T^*M) \to S'(T^*M).
$$

**Definition 4.38.** For two operators $A, B : S(T^*M) \to S'(T^*M)$ we write

$$
(4.69) \qquad A \approx B
$$

if there exists $m < 0$, $t \in \mathbb{R}$, for any $N > 0$, any $\sigma > 0$, there exists a constant $C_{\sigma,N,t} > 0$
such that for any $\rho, \rho' \in T^*M$,

$$
|\langle \delta_{\rho'} | \chi_{\Sigma, \sigma} (A - B) \chi_{\Sigma, \sigma} \delta_{\rho} \rangle| \le C_{\sigma, N, t} \langle \mathrm{dist}_g (\rho', \phi^t(\rho)) \rangle^{-N} |\langle |\rho| \rangle|^m.
$$

For two operators $A, B : C(\Sigma; \mathcal{F}) \to C(\Sigma; \mathcal{F})$ we write $A \approx B$ if $\tilde{A} \approx \tilde{B}$.

Using operators $\mathcal{T}_{N_s}, \mathcal{T}_{N_s}^{\Delta}$ defined in (4.34), (4.35), let

$$
(4.70) \qquad \mathcal{P}_{N_s} := \mathcal{T}_{N_s} \mathcal{T}_{N_s}^{\Delta} \quad : C^{\beta}(\Sigma; \mathcal{F}) \to C^{\beta}(\Sigma; \mathcal{F}).
$$

**Lemma 4.39.** We have

$$
(4.71) \qquad \mathcal{P}_{N_s}^2 \approx \mathcal{P}_{N_s}, \qquad \mathcal{P}_{N_s} \mathcal{J}_{N_s} \approx \mathcal{J}_{N_s}.
$$

Proof. Taking $t=0$ in (4.36) gives

$$
(4.72) \qquad \mathcal{T}_{N_s}^{\Delta} \mathcal{T}_{N_s} \approx \mathrm{Id}.
$$

Then

$$
\mathcal{P}_{N_s}^2 = \mathcal{T}_{N_s} (\mathcal{T}_{N_s}^{\Delta} \mathcal{T}_{N_s}) \mathcal{T}_{N_s}^{\Delta} \approx_{(4.72)} \mathcal{T}_{N_s} \mathcal{T}_{N_s}^{\Delta} = \mathcal{P}_{N_s}.
$$

$$
\mathcal{P}_{N_s} \mathcal{T}_{N_s} = \mathcal{T}_{N_s} (\mathcal{T}_{N_s}^{\Delta} \mathcal{T}_{N_s}) \approx_{(4.72)} \mathcal{T}_{N_s}.
$$

□

We show Lemma 4.40 (similar to [20, Lemma 4.47])

**Lemma 4.40 (Basic lemma).** For any symbol $a \in C^\beta(M; L(\mathcal{F}_{[0,K]}(E_s)))$, we have for
$\mu < \beta$ that

$$
(4.73) \qquad [\tilde{a}, \mathcal{P}_{N_s}] \approx 0.
$$

*Proof.* Observe that

$$
(4.74) \qquad \tilde{J}_{N_s} \tilde{J}_{N_s}^{\Delta} \approx_{(4.66)} \mathrm{Id}, \qquad \tilde{J}_{N_s}^{\Delta} \tilde{J}_{N_s} \approx_{(4.66)} \mathcal{P}_N.
$$
The notation ≈ has been defined for operators. This definition can be understood also for Schwartz kernel of operators lifted on $T^*M$ and we use it below. We have $\forall N \ge 0, \exists C_N > 0, \forall \rho, \rho' \in T^*M$,

$$ (4.75) \quad \begin{aligned} \left|\langle \delta_{\rho'} | \tilde{\mathcal{P}}_{N_s} \delta_{\rho} \rangle\right| &= \left|\langle \delta_{\rho'} | \mathcal{J}_{N_s}^{\Delta} \mathcal{P}_{N_s} \mathcal{J}_{N_s} \delta_{\rho} \rangle\right| &&= \left|\langle \delta_{\rho'} | \mathcal{J}_{N_s}^{\Delta} \mathcal{J}_{N_s} \mathcal{J}^{\dagger} \mathcal{J}_{N_s}^{\Delta} \delta_{\rho} \rangle\right| \\ &\approx &&\left|\langle \delta_{\rho'} | \mathcal{P} \delta_{\rho} \rangle\right| \le C_N \langle \mathrm{dist}_g(\rho', \rho) \rangle^{-N}. \end{aligned} $$

Then

$$ (4.76) $$

$$ \begin{aligned} \left|\langle \delta_{\rho'} | \mathcal{J}_{N_s}^{\Delta} [\tilde{a}, \mathcal{P}_{N_s}] \mathcal{J}_{N_s} \delta_{\rho} \rangle\right| &\approx \left|\langle \delta_{\rho'} | [\tilde{a}, \tilde{\mathcal{P}}_{N_s}] \delta_{\rho} \rangle\right| \\ &= \left|\int_{\rho'' \in T^*M} \langle \delta_{\rho'} | \tilde{a} \delta_{\rho''} \rangle \langle \delta_{\rho''} | \tilde{\mathcal{P}}_{N_s} \delta_{\rho} \rangle - \langle \delta_{\rho'} | \tilde{\mathcal{P}}_{N_s} \delta_{\rho''} \rangle \langle \delta_{\rho''} | \tilde{a} \delta_{\rho} \rangle\right|. \end{aligned} $$

The bundle $E_s$ is Hölder continuous with some exponent $\beta > 0$ and consequently $\tilde{a}$ satisfies the following property (called slowly varying property in [20, def 4.40]), using any local trivialization, for any $\rho, \rho' \in \Sigma$, setting $x = \pi(\rho), x' = \pi(\rho') \in M$, we assume $\omega = \omega(\rho) \asymp \omega(\rho')$, we use $|x' - x| \lesssim (\omega)^{-1/2} \mathrm{dist}_g(\rho', \rho)$, and we have

$$ (4.77) \quad \|\tilde{a}(\rho') - \tilde{a}(\rho)\| \asymp \|a(x') - a(x)\| \lesssim |x' - x|^{\beta} \asymp \langle\omega\rangle^{-\beta/2} (\mathrm{dist}_g(\rho', \rho))^{\beta}. $$

Hence for $\rho, \rho' \in T^*M$, due to the cutoff $\chi_\Sigma^\mu$ in (3.44) at distance $\langle\omega\rangle^{\mu/2}$ from $\Sigma$, writing $\rho \equiv (\rho, v) \in T^*M$ with $\rho \in \Sigma, v \in N_\rho$, $\exp_N(v) = \rho$, we have for $\mathrm{dist}_g(\rho', \rho) \lesssim 1$ and $\mathrm{dist}_g(\rho'', \rho) \lesssim 1$ that

$$ (4.78) \quad \left| \langle \delta_{\rho'} | \tilde{a} \delta_{\rho''} \rangle - \langle \delta_{\rho''} | \tilde{a} \delta_{\rho} \rangle \right| \le \langle \omega \rangle^{\mu/2} |\tilde{a}(\rho') - \tilde{a}(\rho)| \le \langle \omega \rangle^{\mu/2} \langle \omega \rangle^{-\beta/2}. $$

We deduce

$$ \left| \langle \delta_{\rho'} | \mathcal{J}_{N_s}^{\Delta} [\tilde{a}, \mathcal{P}_{N_s}] \mathcal{J}_{N_s} \delta_{\rho} \rangle \right| \le C \langle \omega \rangle^{-(\beta-\mu)/2} C_N \langle \mathrm{dist}_g(\rho', \rho) \rangle^{-N}. $$

From Definition 4.38, we get that $[\tilde{a}, \mathcal{P}_{N_s}] \approx 0$ if $\beta > \mu$. $\square$

<figure><img src="image_19.png" /></figure>

Observe that

$$ (4.79) \quad \mathrm{Op}_\Sigma(e^{tX_f}) \mathrm{Op}_\Sigma(e^{t'X_f}) \underset{(4.56)}{\approx} e^{tX_f} e^{t'X_f} = e^{(t+t')X_f} \underset{(4.56)}{\approx} \mathrm{Op}_\Sigma(e^{(t+t')X_f}). $$

Also

$$ (4.80) \quad \mathrm{Op}_\Sigma(\tilde{a}) \mathrm{Op}_\Sigma(\tilde{b}) = \mathcal{J}_{N_s}^\Delta \tilde{a} \mathcal{P}_{N_s} \tilde{b} \mathcal{J}_{N_s} \underset{(4.73),(4.71)}{\approx} \mathcal{J}_{N_s}^\Delta \tilde{a} \tilde{b} \mathcal{J}_{N_s} = \mathrm{Op}_\Sigma(\tilde{a} \tilde{b}), $$

and similarly

$$ (4.81) \quad \mathrm{Op}_\Sigma(e^{tX_f}) \mathrm{Op}_\Sigma(\tilde{a}) \approx \mathrm{Op}_\Sigma(e^{tX_f} \tilde{a}), $$

$$ (4.82) \quad \mathrm{Op}_\Sigma(\tilde{a}) \mathrm{Op}_\Sigma(e^{tX_f}) \approx \mathrm{Op}_\Sigma(\tilde{a} e^{tX_f}). $$
From these relations, we deduce (4.65) as follows. We set $\tilde{a}' = e^{-t'X_{\mathcal{F}}} \tilde{a} e^{t'X_{\mathcal{F}}}$.

$$
\begin{align*}
& \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}'} \tilde{a}) \operatorname{Op}_{\Sigma}(e^{t'X_{\mathcal{F}}'} \tilde{b}) \\
& \quad \approx_{(4.81), (4.82), (4.80)} \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}}) \operatorname{Op}_{\Sigma}(e^{t'X_{\mathcal{F}}}) \operatorname{Op}_{\Sigma}(\tilde{a}'\tilde{b}) \\
& \quad \approx_{(4.79)} \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}} e^{t'X_{\mathcal{F}}}) \operatorname{Op}_{\Sigma}(\tilde{a}'\tilde{b}) \approx_{(4.81), (4.82), (4.80)} \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}} \tilde{a} e^{t'X_{\mathcal{F}}} \tilde{b}).
\end{align*}
$$

□

Corollary 4.41 follows as special cases of (4.65).

**Corollary 4.41.** *Take $\mu < \beta$. We have*

*   **Egorov formula:** for a symbol $a \in C^\beta(M; \text{End}(\mathcal{F}_{[0,K]}(E_s)))$, $t \in \mathbb{R}$,
(4.83) $$ \operatorname{Op}_\Sigma(e^{tX_\mathcal{F}}) \operatorname{Op}_\Sigma(\tilde{a}) \approx \operatorname{Op}_\Sigma(e^{t\operatorname{ad}X_\mathcal{F}} \tilde{a}) \operatorname{Op}_\Sigma(e^{tX_\mathcal{F}}), $$ with the natural action $e^{t\operatorname{ad}X_\mathcal{F}} \tilde{a} = e^{t[X_\mathcal{F}, \cdot]} \tilde{a} := e^{tX_\mathcal{F}} \tilde{a} e^{-tX_\mathcal{F}}$.

*   **Approximate projectors:** for any $k, k' \le K$,

$$
(4.84) \qquad \operatorname{Op}_{\Sigma}(\tilde{T}_k) \operatorname{Op}_{\Sigma}(\tilde{T}_{k'}) \approx \delta_{k=k'} \operatorname{Op}_{\Sigma}(\tilde{T}_k).
$$

*   **Decomposition of the dynamics in components:** for any $k, k' \le K$,

$$
(4.85) \quad \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}} \tilde{T}_k) \operatorname{Op}_{\Sigma}(e^{t'X_{\mathcal{F}}} \tilde{T}_{k'}) \approx \delta_{k=k'} \operatorname{Op}_{\Sigma}(e^{(t+t')X_{\mathcal{F}}} \tilde{T}_k).
$$

*Remark 4.42.* The algebraic structure of (4.85) will manifest itself later in the band structure of the Ruelle spectrum.

*Proof.* To get (4.83), we write

$$
\begin{align*}
\operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}}) \operatorname{Op}_{\Sigma}(\tilde{a}) &\approx_{(4.65)} \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}}) \tilde{a} = \operatorname{Op}_{\Sigma}((e^{t\operatorname{ad}X_{\mathcal{F}}}) e^{tX_{\mathcal{F}}}) \\
&\approx_{(4.65)} \operatorname{Op}_{\Sigma}(e^{t\operatorname{ad}X_{\mathcal{F}}}) \tilde{a} \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}}).
\end{align*}
$$

To get (4.84), we write

$$
\operatorname{Op}_{\Sigma}(\tilde{T}_k) \operatorname{Op}_{\Sigma}(\tilde{T}_{k'}) \approx_{(4.65)} \operatorname{Op}_{\Sigma}(\tilde{T}_k \tilde{T}_{k'}) \approx_{(4.40)} \delta_{k=k'} \operatorname{Op}_{\Sigma}(\tilde{T}_k).
$$

To get (4.85), we write

$$
\begin{align*}
\operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}} \tilde{T}_k) \operatorname{Op}_{\Sigma}(e^{t'X_{\mathcal{F}}} \tilde{T}_{k'}) &\approx_{(4.65)} \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}} \tilde{T}_k e^{t'X_{\mathcal{F}}} \tilde{T}_{k'}) \\
&= \operatorname{Op}_{\Sigma}(e^{tX_{\mathcal{F}}} e^{t'X_{\mathcal{F}}} \tilde{T}_k \tilde{T}_{k'}) = \delta_{k=k'} \operatorname{Op}_{\Sigma}(e^{(t+t')X_{\mathcal{F}}} \tilde{T}_k).
\end{align*}
$$

□

4.5.3. *Trace formula.* We present some trace formula for P.D.O. (Pseudo-Differential Operators), i.e. F.I.O. (Fourier Integral Operators) at $t = 0$. We will use this formula in Section 7.1. It is also possible to express $\operatorname{Tr}(\operatorname{Op}_{\Sigma}(1_{[\omega,\omega']})e^{tX_{\mathcal{F}}})$ for $t \neq 0$, but we don't need it in this paper.

**Theorem 4.43** (Trace formula). *Take $\mu < \beta$. There exists $C > 0$, for any $\omega \ge \delta \ge 1$, for any symbol $a \in C^\beta(M; L(\mathcal{F}_{[0,K]}(E_s)))$,*

$$
(4.86) \quad \left\| \operatorname{Op}_{\Sigma} (1_{\omega(\rho) \in [\omega - \delta, \omega + \delta]} \tilde{a}) \right\|_{\operatorname{Tr}_{\mathcal{H}_W(M)}} \le C \omega^d \delta.
$$
$$
(4.87) \quad \mathrm{Tr} (\mathrm{Op}_{\Sigma} (1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a})) = \left( \frac{\omega^d}{(2\pi)^{d+1}} (2\delta) \right) \int_M \mathrm{Tr}(a(m)) dm + r(\omega, \delta),
$$

with remainder

$$
|r(\omega, \delta)| \le C\omega^d \delta \left( |\omega|^{-(\beta-\mu)/2} + \frac{\delta}{\omega} \right) + C\omega^d.
$$

Remark 4.44. Later in (7.39) we will use (4.87) as follows. For any small $c' > 0$ we will take $\delta \gg 1$ and $\omega \gg \delta$ large enough so that the remainder $r(\omega, \delta)$ is smaller than $c'\omega^d\delta$, i.e. negligible w.r.t. the first term.

*Proof.* Since the symbol $1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a}$ is fiber-wise, we have

$$
(4.88) \qquad \begin{aligned}
\mathrm{Op}_{\Sigma} (1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a}) & \stackrel{(4.55)}{=} \mathcal{T}_{N_s}^{\Delta} 1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a} \mathcal{T}_{N_s} \\
& = \int_{\rho \in \Sigma} A(\rho) 1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a}(\rho) B(\rho) \frac{d\rho}{(2\pi)^{d+1}}
\end{aligned}
$$

with for $\rho \in \Sigma$, the finite dimensional vector space $\mathcal{F}_{[0,K]}(N_s(\rho))$ and the finite rank
linear operators

$$
(4.89) \qquad A(\rho) := \mathcal{T}_{N_s}^{\Delta} \tilde{T}_{[0,K]}(\rho) : \mathcal{F}_{[0,K]}(N_s(\rho)) \to C^{\infty}(M),
$$

$$
(4.90) \qquad B(\rho) := \tilde{T}_{[0,K]}(\rho) \mathcal{T}_{N_s} : C^{\infty}(M) \to \mathcal{F}_{[0,K]}(N_s(\rho)),
$$

where $\tilde{T}_{[0,K]}(\rho)$ is the bundle map projector onto $\mathcal{F}_{[0,K]}(N_s(\rho))$, defined in (4.41). We
have $\exists C > 0, \forall \rho \in \Sigma$,

$$
\|A(\rho)\| \le C, \quad \|B(\rho)\| \le C,
$$

and

$$
(4.91) \qquad \left\| 1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a}(\rho) \right\|_{\mathrm{Tr}_{\mathcal{H}_W(M)}} \le C.
$$

For $\omega \ge \delta > 0$, we have

$$
(4.92) \qquad \int_{\Sigma} \frac{1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]}}{(3.8)} d\rho \stackrel{\text{Vol}(M)}{=} \int_{\omega-\delta}^{\omega+\delta} \omega'^d d\omega' = \text{Vol}(M) \omega^d (2\delta) \left(1 + O\left(\frac{\delta}{\omega}\right)\right).
$$

Hence

$$
\begin{align*}
\| \mathrm{Op}_{\Sigma} (1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a}) \|_{\mathrm{Tr}_{\mathcal{H}_W(M)}} & \le C \int_{\Sigma} \| 1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a}(\rho) \|_{\mathrm{Tr}_{\mathcal{H}_W(M)}} \frac{d\rho}{(2\pi)^{d+1}} \\
& \le C \omega^d \delta, \tag{4.91), (4.92)
\end{align*}
$$

giving (4.86). Then

$$
\mathrm{Tr} ( \mathrm{Op}_{\Sigma} ( 1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a} ) ) \underset{(4.88)}{=} \int_{\Sigma} \mathrm{Tr} ( 1_{\omega(\rho) \in [\omega-\delta, \omega+\delta]} \tilde{a}(\rho) B(\rho) A(\rho) ) \frac{d\rho}{(2\pi)^{d+1}},
$$

where

$B(\rho)A(\rho)$

$$
\begin{align*}
&= \frac{\tilde{T}_{[0,K]}(\rho) B_{N_s, N_u}^{\dagger} \chi_{\Sigma}^{\mu} (\overline{\exp_N})^{\circ} \mathcal{J} \mathcal{J}^{\dagger} (\overline{\exp_N^{-1}})^{\circ} \chi_{\Sigma}^{\mu} B_{N_s, N_u} \tilde{T}_{[0,K]}(\rho)}{(4.89), (4.90), (4.35), (4.34)} \\
&= \tilde{T}_{[0,K]}(\rho) + O\left(\langle\omega\rangle^{-(\beta-\mu)/2}\right),
\end{align*}
$$
where $\omega(\rho)^{-\frac{1}{2}(\beta-\mu)}$ comes from Hölder irregularity as in (4.78). Hence

$$ \mathrm{Tr}(\mathrm{Op}_{\Sigma}(1_{\omega(\rho)\in[\omega-\delta, \omega+\delta]}\tilde{a})) = \int_{\Sigma} \mathrm{Tr}(1_{\omega(\rho)\in[\omega-\delta, \omega+\delta]}\tilde{a}(\rho)) \frac{d\rho}{(2\pi)^{d+1}} + R $$

with remainder

$$ \begin{aligned} |R| & \le C \int_{\Sigma} 1_{\omega(\rho)\in[\omega-\delta, \omega+\delta]} \omega(\rho)^{-(\beta-\mu)/2} d\rho + C\omega^d \\ & \le C\omega^d \delta\omega^{-(\beta-\mu)/2} + C\omega^d. \end{aligned} $$

$C\omega^d$ is due to sharp cutoff in frequency, and main term

$$ \int_{\Sigma} \mathrm{Tr}(1_{\omega(\rho)\in[\omega-\delta, \omega+\delta]} \tilde{a}(\rho)) \frac{d\rho}{(2\pi)^{d+1}} = \frac{\omega^d (2\delta)}{(3.8), (4.92)} (1 + O(\frac{\delta}{\omega})) \int_M \mathrm{Tr}(a(m)) dm. \quad \square $$

We have obtained (4.87).

### 5. PROOF OF THEOREM 1.7 (BAND SPECTRUM)

We will follow the strategy presented in the proof of [20, Lemma 5.16]. Let $K \in \mathbb{N}$, $\epsilon > 0$ be fixed and consider $z \in \mathbb{C}$ in the spectral plane such that

$$ (5.1) \qquad \gamma_{K+1}^+ + \epsilon < \mathrm{Re}(z) < \gamma_K^- - \epsilon. $$

We put

$$ (5.2) \qquad \omega = \mathrm{Im}(z). $$

Let $\delta > 0$. We consider the frequency intervals

$$ (5.3) \qquad J_{\omega} := [\omega - 1, \omega + 1], \qquad J'_{\omega, \delta} := [\omega - 1 - \delta, \omega + 1 + \delta], $$

so that $J_{\omega} \subset J'_{\omega, \delta}$. Let $\sigma > 0$. We consider the following partition of the cotangent bundle

$$ (5.4) \qquad T^*M = \Omega_0 \cup \Omega_1 \cup \Omega_2 $$

with, as in (3.30),

$$ (5.5) \qquad \Omega_0 := \{\rho \in T^*M \mid \|\rho_u + \rho_s\|_{g_{\rho}} \le \sigma \text{ and } \omega(\rho) \in J'_{\omega, \delta}\}, $$

$$ (5.6) \qquad \Omega_1 := \{ \rho \in T^*M \mid \|\rho_u + \rho_s\|_{g_{\rho}} > \sigma \text{ and } \omega(\rho) \in J'_{\omega, \delta} \}, \\ \Omega_2 := \{ \rho \in T^*M \mid \omega(\rho) \notin J'_{\omega, \delta} \}. $$

For $j = 0, 1, 2$, we define $1_{\Omega_j} : T^*M \to [0, 1]$ as being the characteristic function of the set $\Omega_j$ above and as in (3.31), we set

$$ \mathrm{Op}(1_{\Omega_j}) := \mathcal{T}^\dagger 1_{\Omega_j} \mathcal{T} \quad : C^\infty(M) \to C^\infty(M), $$

giving a resolution of identity

$$ \mathrm{Id}_{\mathcal{H}_W(M)} = \mathrm{Op}(1_{\Omega_0}) + \mathrm{Op}(1_{\Omega_1}) + \mathrm{Op}(1_{\Omega_2}) $$

$$ (5.7) \qquad = \mathrm{Op}_{\Sigma}(T_{[0,K]}) \mathrm{Op}(1_{\Omega_0}) + \mathrm{Op}_{\Sigma}(T_{\ge (K+1)}) \mathrm{Op}(1_{\Omega_0}) + \mathrm{Op}(1_{\Omega_1}) + \mathrm{Op}(1_{\Omega_2}) $$

$$ (5.8) \qquad + (\mathrm{Id} - \mathrm{Op}_{\Sigma}(\mathrm{Id}_{\mathcal{T}})) \mathrm{Op}(1_{\Omega_0}). $$