INVARIANCE PRINCIPLE and DYNAMICS

François Ledrappier

► To cite this version:

François Ledrappier. INVARIANCE PRINCIPLE and DYNAMICS. 2026. [hal-05526016]

HAL Id: hal-05526016

https://hal.science/hal-05526016v1

Preprint submitted on 24 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
INVARIANCE PRINCIPLE IN DYNAMICS

FRANÇOIS LEDRAPPIER

ABSTRACT. This is a survey of some invariances that arise in dynamical sys-
tems theory in the presence of zero Lyapunov exponents.

# 1. INTRODUCTION

An active aspect of the theory of dynamical systems is the study of asymptotic
properties of successive iterations of a transformation. This is performed through
the description of global (often statistical) properties and objects.

For instance, the Lyapunov exponents were introduced by A.M. Lyapunov to
study the stability of dynamical properties under perturbations. In the late six-
ties, in fundamental paper about smooth diffeomorphisms and flows on compact
manifolds, D.V. Anosov and S. Smale showed that uniform absence of vanishing
exponents also induces some form of stability (see [M88] for the converse). On
the other hand, at about the same time, D.V. Oseledets showed that Lyapunov
exponents always exist in a probabilistic sense, almost everywhere for any invariant
probability measure. The systems with no vanishing Lyapunov exponents in this
sense, also have a rich theory, see [Y13], [W17]. The question so arises of under-
standing vanishing Lyapunov exponents of invariant probability measures. It turns
out that, often, vanishing Lyapunov exponents implies a non-trivial invariance of...
something. These results are nicknamed as the action of an ‘Invariance Principle’
<fnref n="1" /> and the goal of this paper is to survey the ergodic theory behind some examples.

The common feature of these examples is the introduction of some entropy, typi-
cally an average of Liebner-Küllback information, that quantifies how some measure
is distorted by the dynamics. Then the vanishing Lyapunov exponent forces this
entropy to vanish as well (see the basic inequalities propositions 2.5, 2.11, 2.20,3.5).
The vanishing of this entropy forces the desired invariance by an application of the
equality case in Jensen inequality.

In the first section, we recall more or less classical results that led to the formula-
tion of the Invariance Principle. The story starts with H. Furstenberg criterion for
a random walks on SL($d$, $\mathbb{R}$) to have non-zero exponents ([F63]), and the general-
ization by S. Kotani and B. Simon ([S83]) for the study of Schrödinger operators.
An analog of Furstenberg result in the non-linear setting was observed by P. Bax-
endale and A. Carverhill and is the prototype of results about random products of
diffeomorphisms. A further recent example is a result of A. Tahzibi and J. Yang

<footnote n="1">¹Our Invariance Principle is not related with the one in Large Deviations theory. Ours came
later, but it is more of a principle that has something to do with some invariance.</footnote>
([**TY19**]) about dynamically coherent partially hyperbolic systems with compact center leaves.

The IP is the realization that all these results are manifestations of the same phenomenon, that we explain in Section 3. This follows [AV10], who understood and popularized that connection. We give a few examples in Section 4. Section 5 is an incomplete laundry list of other related results. There is a considerable overlap with another survey on the same topic (and with almost the same title) by Katrina Marin and Mauricio Pelotti ([MP25]), which presents a synthetic view of the current research. I tried to present the ergodic theoretic background of the Invariance Principle and to sketch the main proofs. [MP25] is more precise and complete on the examples of applications. We also refer to Marcelo Viana's book ([V14]) about Lyapunov exponents for more background and more details about some of the results mentioned here.

This survey is the extended version of a mini-course I gave in the September 25 Meeting of the AnR DynAtrois in Dijon. I thank the organizers for the invitation and the audience for their attention and remarks. I also thank Y. Coudène, R. Dujardin and A. Wilkinson for pertinent suggestions.

## 2. EXAMPLES OF APPLICATIONS OF AN INVARIANCE PRINCIPLE

### 2.1. Product of independent 2 × 2 matrices. Exponents. Furstenberg criterion.

Let $\mu$ be a probability measure on $SL(2, \mathbb{R})$, with $\int \log \|A\| d\mu(A) < +\infty$.

We form the *independent product* of matrices with distribution $\mu$. Formally, set $(\overline{X}, \overline{\mathcal{A}}, \overline{m})$ for the product space $\overline{X} := (SL(2, \mathbb{R}))^{\otimes \mathbb{N}}$, with the $\sigma$-algebra $\overline{\mathcal{A}}$ generated by the coordinates $\{x_n\}_{n \ge 0}$, and the product measure $\overline{m} := \mu^{\otimes \mathbb{N}}$. For $\{x_n\}_{n \ge 0} \in \overline{X}, n \ge 1$ set

$$Z_n := x_{n-1} \dots x_0.$$

**Theorem 2.1.** [**FK60**] There is a number $\gamma \ge 0$ such that, for $\overline{m}$-a.e. $x \in \overline{X}$,

$$\gamma = \lim_{n \to \infty} \frac{1}{n} \log \|Z_n\| = \lim_{n \to \infty} \frac{1}{n} \log \|Z_n^{-1}\|.$$

Theorem 2.1 anticipates the subadditive ergodic theorem. We have $\gamma \ge 0$ because $Z_n Z_n^{-1} = \mathbb{I}$. The original Invariance Principle is

**Theorem 2.2.** [Furstenberg [F63]] If $\gamma = 0$, then there exists a probability $\nu$ on $\mathbb{P}^1(\mathbb{R})$ which is invariant under $\mu$-a.e. matrix $A$, where $A \in SL(2, \mathbb{R})$ acts naturally on $\mathbb{P}^1(\mathbb{R}) : A_*\nu = \nu$ for $\mu$-a.e. $A$.

By discussing how different matrices may have the same invariant probability measure for their actions on $\mathbb{P}^1(\mathbb{R})$, we see that if $\gamma = 0$, then either there is a finite set of directions invariant under $\mu$-a.e. matrix $A$, or the closed subgroup generated by the support of $\mu$ is compact.
*Proof.* The probability measure $\nu$ on $\mathbb{P}^1(\mathbb{R})$ is called stationary if

$$ \nu = \int A_* \nu d\mu(A). $$

Observe that there always exist stationary probability measures $\nu$ on $\mathbb{P}^1(\mathbb{R})$. The proof consists in showing that if $\gamma = 0$, then stationary measures have to be invariant. The tool is Furstenberg entropy.

**Definition 2.3.** Let $\mu$ be a probability measure on SL(2, $\mathbb{R}$), $\nu$ a stationary probability measure on $\mathbb{P}^1(\mathbb{R})$. The Furstenberg entropy $\kappa(\mu, \nu)$ is defined by

$$ \kappa(\mu, \nu) := \int \log \frac{dA_* \nu}{d\nu}(A\xi) d\nu(\xi) d\mu(A). $$

We set $\kappa(\mu, \nu) = +\infty$ if, for a set of matrices $A$ of positive $\mu$-measure, the measure $A_* \nu$ is not absolutely continuous with respect to $\nu$. We also have, a priori, $\kappa(\mu, \nu) = +\infty$ if the function $\log \frac{dA_* \nu}{d\nu}(A\xi)$ is not integrable (by Jensen inequality, the integral makes sense). Again by Jensen inequality, we have:

**Proposition 2.4.** We have $\kappa(\mu, \nu) \ge 0$ and $\kappa(\mu, \nu) = 0$ if, and only if, $\nu$ is invariant under $\mu$-a.e. matrix $A$.

*Proof.* Indeed, there is equality in Jensen inequality if, and only if, $\frac{dA_* \nu}{d\nu}(\xi) = 1 (d(A_* \nu)(\xi)d\mu(A))$-a.e.. $\square$

Theorem 2.2 clearly follows from propositions 2.4 and 2.5:

**Proposition 2.5.** [Basic inequality] Let $\mu$ be a probability measure on SL(2, $\mathbb{R}$), with $\int \log \|g\| d\mu(g) < +\infty$, $\nu$ a stationary probability measure on $\mathbb{P}^1(\mathbb{R})$. Then,

$$ \kappa(\mu, \nu) \le 2\gamma. $$

$\square$

*Proof.* (Proof of proposition 2.5)<fnref n="2" /> Consider $(\overline{\mathcal{E}}, \overline{M}) := (\overline{X} \times \mathbb{P}^1(\mathbb{R}), \overline{m} \times \nu)$. We have

$$ \kappa(\mu, \nu) := \int \log \frac{d(x_0)_* \nu}{d\nu}(x_0 \xi) d\overline{M}(x, \xi). $$

Consider the transformation $\overline{F} : \overline{\mathcal{E}} \to \overline{\mathcal{E}}$, defined by $\overline{F}(x, \xi) := (\overline{f}(x), x_0 \xi)$, where $\overline{f}$ is the shift on the product space $\overline{X}$. We have $\overline{F}^j(x, \xi) = (\overline{f}^j x, Z_j(x)\xi)$.

Observe that the measure $\overline{M}$ is invariant under the transformation $\overline{F}$ : for all $\varphi \in C(\overline{\mathcal{E}})$, we can write, using the stationarity relation,

$$ \int \varphi \circ \overline{F} d\overline{M} = \int \varphi(x_1, \dots, x_0 \xi) d\nu(\xi) d\mu(x_0) d\mu(x_1) \dots = \int \varphi d\overline{M}. $$

<footnote n="2">²Another proof, closer to the original one, consists in approximating $\mu$ by measures $\mu_n$ with an absolutely continuous stationary measure $\nu_n$. There is equality when the stationary measure is absolutely continuous and the inequality goes to the limit (see [L84]). The proof we present uses the ideas of the proof of proposition 3.5 in a simpler setting. If one considers the action of $\mathbb{P}$ SL(2, $\mathbb{R}$) on the hyperbolic space by isometries, then the exponent $\gamma$ is also given by $\gamma = \lim_{n \to \infty} \frac{1}{n} d(o, Z_n o)$ and then the basic inequality goes back to Guivarc'h ([G80]) if $\mu$ is supported on a discrete group.</footnote>
By a general result of Breiman about Markov chains (cf. e.g. [D26] Théorème 5.6 in section XIII) the ergodic decomposition of $\overline{M}$ is given by the extremal decomposition of $\nu$ into stationary measures. So we may assume that the measure $\overline{M}$ is ergodic for $\overline{F}$ and that the measure $\nu$ is extremal among stationary measures.

Fix $\varepsilon > 0$. Write, for $x \in \overline{X}, \xi \in \mathbb{S}^1$,

$$g(x, \xi) := \frac{d(x_0^{-1})_* \nu}{d\nu}(\xi)$$

and, for $N$ large, $g^N(x, \xi) := \sup\{g(x, \xi), e^{-N}\}$. The function $-\log g^N$ is $\overline{M}$-integrable, $-\log g \ge -\log g^N$ and, by monotone limit,

$$(2.1) \qquad \kappa(\mu, \nu) = \lim_{N \to \infty} - \int \log g^N(x, \xi) d\overline{M}(x, \xi).$$

Choose $\delta > 0$ such that, setting

$$g_{\delta}(x, \xi) := \frac{\int_{B(\xi, \delta)} g(x, \eta) d\nu}{\nu(B(\xi, \delta))} = \frac{\nu(x_0(B(\xi, \delta)))}{\nu(B(\xi, \delta))},$$

$$g_{\delta}^{N}(x, \xi) := \frac{\int_{B(\xi, \delta)} g^{N}(x, \eta) d\nu(\eta)}{\nu(B(\xi, \delta))}, \quad g_{\delta}^{*} := \sup_{\delta' \le \delta} g_{\delta'}^{N},$$

we have $\int \log \frac{g_{\delta}^{*}}{g^{N}} d\overline{M} \le \varepsilon$ (see proposition 6.4). Observe that, for $q \le \delta$,
$$-\log g^{N} \le -\log g_{q} + \log g_{q} - \log g_{q}^{N} + \log g_{\delta}^{*} - \log g^{N} \le -\log g_{q} + \log g_{\delta}^{*} - \log g^{N}.$$

Set, for $j \ge 0$, $q_j := \frac{\delta}{\|x_0\|^2 \dots \|x_{j-1}\|^2}$. In particular, $q_j \le \delta$ for all $j \ge 0$.

$$\begin{align*} - \int \log g^N d\overline{M} &= \lim_{n \to \infty} \frac{1}{n} \sum_{j=0}^{n-1} - \log g^N(\overline{F}^j(x, \xi)) \\ &\le \liminf_{n \to \infty} \frac{1}{n} \sum_{j=0}^{n-1} - \log g_{q_j}(\overline{F}^j(x, \xi)) + \int \log \frac{g_{\delta}^*}{g^N} d\overline{M} \\ &\le \liminf_{n \to \infty} \frac{1}{n} \sum_{j=0}^{n-1} - \log \frac{\nu(x_j(B(Z_j(x)\xi, q_j)))}{\nu(B(Z_j(x)\xi, q_j))} + \varepsilon. \end{align*}$$

Observe that $x_j(B(Z_j(x)\xi, q_j)) \supset B(Z_{j+1}(x)\xi, \frac{q_j}{\|x_j\|^2}) = B(Z_{j+1}(x)\xi, q_{j+1})$. The sum telescopes and we get, at $\overline{M}$-a.e. $(x, \xi)$,

$$- \int \log g^N d\overline{M} \le \liminf_{n \to \infty} -\frac{1}{n} \log \frac{\nu(B(Z_n(x)\xi, q_n))}{\nu(B(\xi, \delta))} + \varepsilon,$$

By proposition 6.4, there are constants $C$ and $\gamma$ such that the set $D$ of points in $\mathbb{S}^1$ such that $\nu(B(\xi, \gamma')) \ge C\gamma'$ for all $0 < \gamma' \le \gamma$ has $\nu$-measure at least $1 - \varepsilon$. Therefore, for $\overline{M}$-a.e. $(x, \xi)$, there are infinitely many indices $n$ such that $q_n < \gamma$ and $\overline{F}^{(n)}(x, \xi) = (\overline{f}^n x, Z_n(x)\xi) \in \overline{X} \times D$. It follows that, for all $\varepsilon > 0$, for $\overline{M}$-a.e. $(x, \xi)$,

$$\liminf_{n \to \infty} -\frac{1}{n} \log \frac{\nu((B(Z_n(x)\xi), q_n))}{\nu(B(\xi, \delta))} \le \lim_{n \to \infty} -\frac{1}{n} \log q_n = 2 \int \log \|A\| d\mu(A).$$
Letting $\varepsilon \to 0$ and then using (2.1), we obtain

$$ \kappa(\mu, \nu) \le 2 \int \log \|A\| d\mu(A). $$

Let $\mu^{*L}$ be the distribution of $Z_L$. The measure $\nu$ is stationary for $\mu^{*L}$ as well and $\kappa(\mu^{*L}, \nu) = L\kappa(\mu, \nu)$. It follows that

$$ \kappa(\mu, \nu) \le 2 \lim_{L \to \infty} \frac{1}{L} \int \log \|Z_L\| d\mu^{*L} = 2\gamma. $$

□

Proposition 2.5 and Theorem 2.2 readily extend to SL($d, \mathbb{R}$), for all $d \ge 2$: we have $\kappa(\mu, \nu) \le (d-1)(\gamma^+ - \gamma^-)$, where, for $\bar{m}$-a.e. $x \in (\text{SL}(d, \mathbb{R})^N,$

$$ \gamma^+ := \lim_{n \to \infty} \frac{1}{n} \log \|Z_n(x)\|, \quad \gamma^- := - \lim_{n \to \infty} \frac{1}{n} \log \|Z_n^{-1}(x)\|. $$

**Corollary 2.6.** Let $\mu$ be a probability measure on SL(2, $\mathbb{R}$), such that $\int \log \|A\| d\mu(A) < +\infty$, $\nu$ a stationary probability measure on $\mathbb{S}^1$. Then, for $\mu$-a.e. $A$, the measure $(A^{-1})_*\nu$ is absolutely continuous with respect to $\nu$, and $\int \log \frac{d(A^{-1})_*\nu}{d\nu}(\xi) d\nu(\xi) > -\infty$.

## 2.2. Application to discrete 1-d Schrödinger equation.
One of the first applications of Furstenberg theorem 2.2 is to one-dimensional Anderson model in theoretical physics.

Let $\{V_n\}_{n \in \mathbb{Z}}$ be a sequence of bounded independent real variables with distribution $\tau$, $(X, m) := (\mathbb{R}^Z, \tau^{\otimes Z})$. The one-dimensional Schrödinger operator at $x = \{V_n\}_{n \in \mathbb{Z}} \in X$ is the self-adjoint operator $H_x$ of $\ell^2(\mathbb{Z})$ given for $u = \{u_n\}_{n \in \mathbb{Z}} \in \ell^2(\mathbb{Z})$, for all $n \in \mathbb{Z}$, by

$$ (H_x(u))_n := u_{n+1} + u_{n-1} + V_n u_n. $$

The operator $H_x$ admits a spectral measure $\sigma_x$ on $\mathbb{R}$, given by its moments $\int E^n d\sigma_x(E) = \frac{1}{2}((H_x(\delta_0))_n + (H_x(\delta_1))_n)$. By ergodicity of the shift transformation $f$ on $X$, the type and the support of $\sigma_x$ are $m$-a.e. constant.

In the one-dimensional Anderson model, $\exp(itH_x)$ describes the quantum evolution of the quantum object $u$. Eigenfunctions (i.e. the discrete part of $\sigma_x$) correspond to **localization**, the absolutely continuous part of $\sigma_x$ to **diffusion**.

**Theorem 2.7 ([173]).** For $m$-a.e. $x \in X$, the spectral measure $\sigma_x$ is singular with respect to the Lebesgue measure on $\mathbb{R}$.

*Proof.* (See [P80]) Consider the measure $\Sigma$ on $X \times \mathbb{R}$ given by $\Sigma(dx, dE) := \sigma_x(dE)m(dx)$. We want to show that the measures $m \times \text{Leb.}$ and $\Sigma$ are mutually singular. We exhibit two disjoint sets of full measure for respectively $m \times \text{Leb.}$ and $\Sigma$.

Fix $x \in X$. A sequence $\{u_n\}_{n \in \mathbb{Z}} \in \mathbb{C}^Z$ satisfies $(H_x u)_n = Eu_n$ iff for all $n \in \mathbb{Z}$,

$$ (2.2) \qquad \begin{pmatrix} u_{n+1} \\ u_n \end{pmatrix} = \begin{pmatrix} E - V_n & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} u_n \\ u_{n-1} \end{pmatrix}. $$
On the one hand, by Furstenberg theorem 2.2, as soon as $\tau$ is not supported on a point, the exponent $\gamma(E)$ is positive. Then, by Oseledets theorem 6.9, for $m$-a.e. $x \in X$, a solution of (2.2) satisfies

$$ \lim_{n \to +\infty} \frac{1}{n} \log \max\{\|u_n\|, \|u_{-n}\|\} = \gamma(E). $$

On the other hand, the spectral theorem yields an unitary isomorphism $U$ of $\ell^2(\mathbb{Z})$ with a Hilbert integral $\int_{\mathbb{R}}^{\oplus} V_E d\sigma_x(E)$ that conjugates $H_x$ with the multiplication by $E$. Since $H_x^n \delta_0, n \in \mathbb{Z}$ and $H_x^m \delta_1, m \in \mathbb{Z}$, generate $\ell^2(\mathbb{Z})$, the dimension of the Hilbert spaces $V_E$ is either 1 or 2. Assume $\text{Dim } V_E = 2$ $\sigma_x$-a.e. (the case $\text{Dim } V_E = 1$ $\sigma_x$-a.e is similar). Choose $a(E), b(E) \in \ell^2(\mathbb{Z})$ such that, for $\sigma_x$-a.e. $E$, $a(E), b(E)$ form an orthonormal basis of $V_E$ and define $u_n(E), v_n(E)$ by

$$ U(\delta_n) = u_n(E)a(E) + v_n(E)b(E). $$

We have, for all $n \in \mathbb{Z}$, $\int (|u_n(E)|^2 + |v_n(E)|^2) d\sigma_x(E) = 1$. Moreover, by applying $U H_x U^{-1}$, we see that the sequences $\{u_n(E)\}_{n \in \mathbb{Z}}$, $\{v_n(E)\}_{n \in \mathbb{Z}}$ are solutions of (2.2) for $\sigma_x$-a.e. $E$. Therefore, for $\Sigma$-a.e. $(x, E)$, there exists a solution of (2.2) with $\sum_{n \in \mathbb{Z}} \frac{\|u_n\|^2}{n^2} < +\infty$. $\square$

It was shown later that $\sigma_x$ is a discrete measure for $m$-a.e. $x$ (see [CKM87]).

A wide extension of Furstenberg theorem 2.2 was suggested by the more general one-dimensional Anderson model. Consider a topological dynamical system $(X, \mathcal{A}, m, f)$, where $X$ is a compact metric space, $\mathcal{A}$ the Borel $\sigma$-algebra, $f$ a homeomorphism and $m$ an ergodic $f$-invariant measure.

Let $V: X \to \mathbb{R}$ be a continuous function on $X$. The Schrödinger operator at $x \in X$ is the self-adjoint operator $H_x$ of $\ell^2(\mathbb{Z})$ given for $u = \{u_n\}_{n \in \mathbb{Z}} \in \ell^2(\mathbb{Z})$, for all $n \in \mathbb{Z}$, by

$$ (H_x(u))_n := u_{n+1} + u_{n-1} + V(f^n x)u_n. $$

Fix $E \in \mathbb{R}$. Consider the field of matrices $B_E(x) := \begin{pmatrix} E - V(x) & -1 \\ 1 & 0 \end{pmatrix}$ and the exponent

$$ \gamma(x, E) := \lim_{n \to \infty} \frac{1}{n} \log \|B_E(f^{n-1}x) \dots B_E(x)\|. $$

By ergodicity, there is a constant $\gamma(E)$ such that $\gamma(x, E) = \gamma(E)$ for $m$-a.e. $x$.

The independent case corresponds to $X = (\text{supp.}\tau)^{\otimes \mathbb{Z}}$, $m = \tau^{\otimes \mathbb{Z}}$, $f$ the shift transformation on $X$ and $V(x) = x_0$.

**Theorem 2.8.** $[\mathbf{S83}]$ Assume that the Lebesgue measure of $\{E \in \mathbb{R} : \gamma(E) = 0\}$ is positive. Then $V$ is deterministic, meaning: $V$ is measurable with respect to the $\sigma$-algebra generated by $V(f^n x)$, $n > 0$.

S. Kotani had a similar result for the continuous Schrödinger equation. B. Simon gave a direct proof for the discrete case. Of course, in the independent case, $V$ is deterministic only if the measure $\tau$ is supported on a single point $E_0$. Then, for all $E \in \mathbb{R}$, the matrix $\begin{pmatrix} E - E_0 & -1 \\ 1 & 0 \end{pmatrix}$ preserves a probability measure on $\mathbb{P}^1$.
More generally, if the function $V$ is NOT deterministic, then $\gamma(E) > 0$ Leb.-a.e. and the above Pastur argument applies:

**Corollary 2.9.** If the function $V$ is NOT deterministic, then for $m$-a.e. $x \in X$, the spectral measure $\sigma_x$ is singular with respect to the Lebesgue measure on $\mathbb{R}$.

**2.3. Random diffeomorphisms in dimension 1.** We now describe a non-linear version of section 2.1. Let $\mu$ be a probability measure on $\text{Diff}^2(\mathbb{S}^1)$ such that $\int \log|T'| d\mu(T) < +\infty$ and $\int \log|(T')^{-1}| d\mu(T) < +\infty$. Consider $(\bar{X}, \bar{\mathcal{A}}, \bar{m})$ the product space $\bar{X} := (\text{Diff}^2(\mathbb{S}^1))^{\otimes\mathbb{N}}$, with the $\sigma$-algebra $\bar{\mathcal{A}}$ generated by the coordinates $\{T_n\}_{n\ge 0}$, and the product measure $\bar{m} := \mu^{\otimes\mathbb{N}}$. For $x = \{T_n\}_{n\ge 0} \in \bar{X}$, $n \ge 1$ set

$$ \Phi_n(x) := T_{n-1} \dots T_0. $$

For $\nu$ stationary probability measure on $\mathbb{S}^1$, we can define $(\bar{\mathcal{E}}, \bar{\mathcal{M}}) := (\bar{X} \times \mathbb{S}^1, \bar{m} \times \nu)$. and the transformation $\bar{F} : \bar{\mathcal{E}} \to \bar{\mathcal{E}}$, defined by $\bar{F}(x, \xi) := (\bar{f}(x), T_0\xi)$, where $\bar{f}$ is the shift on the product space $\bar{X}$. As before, the measure $\bar{M}$ is $\bar{F}$-invariant, $\bar{F}$-ergodic if, and only if, the measure $\nu$ is extremal among stationary probability measures. It appears (see [LJ87], [K93]) that there is a familiar dichotomy: either there is a probability measure which is invariant under $\mu$-a.e. $T$, or, for large $n$, the typical iterate $\Phi_n$ is a North-South diffeomorphism of $\mathbb{S}^1$, with a finite number of sinks and sources. It was observed by A. Carverhill ([C87], P. Baxendale ([B89]) and H. Crauel ([C90]) that this phenomenon is very much related to theorem 2.2.

By the subadditive ergodic theorem, the following exponents $\gamma^+$, $\gamma^-$ are defined for $M$-a.e. $x \in \bar{X}$ and non-negative:

$$ \gamma^+ = \lim_{n \to \infty} \frac{1}{n} \log |\Phi'_n|, \quad \gamma^- = \lim_{n \to \infty} \frac{1}{n} \log |(\Phi_n^{-1})'|. $$

**Theorem 2.10.** Let $\mu$ be a probability measure on $\text{Diff}^2(\mathbb{S}^1)$ such that $\int \log|T'| d\mu(T) < +\infty$ and $\int \log|(T')^{-1}| d\mu(T) < +\infty$. If $\gamma^- = 0$, then all stationary probability measures are invariant under $\mu$-a.e. $T$.

*Proof.* For a stationary probability measure $\nu$, we define the Furstenberg entropy

$$ \kappa(\mu, \nu) := \int \log \frac{dT_*\nu}{d\nu}(T\xi) d\nu(\xi) d\mu(T) $$

(cf. definition 2.3). We still have $\kappa(\mu, \nu) \ge 0$, with equality if, and only if, the measure $\nu$ is invariant under $\mu$-a.e. $T$. We have another basic inequality

**Proposition 2.11.** Let $\nu$ be an extremal stationary probability measure. If $\gamma^- = 0$, then $\kappa(\mu, \nu) = 0$.

The proof of proposition 2.11 is completely analogous to the proof we gave of proposition 2.5: we set, for $j \ge 0$, $q_j := \frac{\delta}{\|(T_0^{-1})'\| \dots \|(T_{j-1}^{-1})'\|}$, so that

$$ T_j(B(\Phi_j(x)\xi, q_j)) \supset B(\Phi_{j+1}(x)\xi, \frac{q_j}{\|(T_j^{-1})'\|}) = B(\Phi_{j+1}(x)\xi, q_{j+1}). $$

Indeed, if $A \in \text{SL}(2, \mathbb{R})$, the absolute value of the derivative of the action of $A^{-1}$ on $\mathbb{S}^1$ is at most $\|A^{-1}\|^2 = \|A\|^2$. Actually, we adapted Crauel's proof (see [C90]) of
proposition 2.11 to the linear case. The proof of proposition 3.5 below will follow
the same scheme.

By proposition 2.11, if $\gamma^{-} = 0$, then all extremal stationary measure are invariant. But any stationary measure is the average of the measures in its extremal decomposition. $\square$

Theorem 2.10 has been extended to Homeo($\mathbb{S}^1$) by D. Malicet (see [M17]). See also [DKN07]. Random pseudo-groups of homeomorphisms occur naturally when studying the transversal properties of foliated Brownian motion. Analogs of theorem 2.10 have been proven independently by C. Bonatti and X. Gomez-Mont ([BGM01]) and by B. Deroin and V. Klepstyn ([DK07]) for codimension 1 foliations with hyperbolic leaves.

## 2.4. Compact extension of hyperbolic systems.

**Definition 2.12.** Let $Y$ be a compact connected Riemannian manifold. The $C^2$ diffeomorphism $\varphi$ of $Y$ is called partially hyperbolic if there exist a continuous decomposition $y \mapsto E_y^u \oplus E_y^c \oplus E_y^s$ of $T_yY$ and a Riemannian metric such that

* for $v^s \in E_y^s, v^c \in E_y^c$ and $v^s \in E_y^s, \|v^s\| = 1$ for $*=s, c, u$, we have

$$ \|D_y\varphi(v^s)\| < \|D_y\varphi(v^c)\| < \|D_y\varphi^{-1}(v^u)\| $$

* and $\|D_y\varphi|_{E^s}\| < 1, \|D_y\varphi^{-1}|_{E^u}\| < 1$.

Let $\varphi$ be a partially hyperbolic diffeomorphism of the compact manifold $Y$. By compactness, the following stable manifold theorem applies to $f = \varphi, V = E^s$ and to $f = \varphi^{-1}, V = E^u$.

**Proposition 2.13 ([A67]).** Assume $X$ is a compact manifold, $f: X \to X$ a partially hyperbolic $C^{1+\alpha}$ diffeomorphism and $m$ an ergodic $f$-invariant probability measure. Let $\|D_y\varphi|_{E^s}\| < \rho < \|(D_y\varphi|_{E^c})^{-1}\|^{-1} \le 0$. Then,

* for $x \in X$,

$$ W_x^\rho := \{y \in X, \limsup_{n \to \infty} \frac{1}{n} \log d(f^n y, f^n x) \le \log \rho\} $$

is an embedded space $\mathbb{R}^d$ with $T_x W_x^\rho = V(x)$ for all $x \in X$. Moreover,

* the $W_x^\rho, x \in X$ form a Hölder continuous lamination $W^\rho$. In particular, for $y \in W_x^\rho$, there exists a locally Hölder continuous one-to-one holonomy map $h_{x,y}^\rho$ from a neighborhood of $x$ in any small transversal to $W^\rho$ to a neighborhood of $y$ in any small transversal to $W^\rho$.

We denote $W^u$ (resp. $W^s$) the unstable (resp. stable) foliations associated to the fields $E^u$ (resp. $E^s$), $W_y^s, W_y^u, y \in Y$, the individual leaf at $y$, $h_{y,\cdot}^s, h_{y,\cdot}^u$, the corresponding local holonomy maps, defined in local charts of the foliations.

**Definition 2.14.** Assume that $(Y, \varphi)$ is partially hyperbolic with $\dim E^c = 0$. A measure $\nu$ is called a $(s, u)$-product if, in a common local chart for both foliations,
* the conditional $\nu_y^u$ of the measure $\nu$ with respect to the partition in local $W^u$-leaves are $h^s$-invariant or, equivalently,

* the conditional $\nu_y^s$ of the measure $\nu$ with respect to the partition in local $W^s$-leaves are $h^u$-invariant.

The equivalence of the two definitions is a direct application of Fubini Theorem. Famously, for $(Y, \varphi)$ hyperbolic (i.e. partially hyperbolic with $\dim E^c = 0$), G. Margulis ([M69]) constructs an invariant probability measure which is a $(s, u)$-product and R. Bowen ([B75]) shows that Margulis measure is the unique measure of maximal entropy for $\varphi$. Later, R. Bowen and B. Marcus ([BM77]) show that Margulis measure is the only $(s, u)$-product probability measure. Extending these discussions to general partially hyperbolic systems is natural when the system is dynamically coherent.

**Definition 2.15.** A partially hyperbolic system $(Y, \varphi)$ is called **dynamically coherent** if there exist continuous foliations $W^{cs}, W^{cs}$ such that, at every $y \in Y$, $T_y W_y^{cs} = E_y^s \oplus E_y^c$, $T_y W_y^{cu} = E_y^u \oplus E_y^c$.

Partially hyperbolic systems with compact center leaves, perturbations of time-one maps of geodesic flows in negative curvature, non hyperbolic ergodic automorphisms of the torus are dynamically coherent. Not all partially hyperbolic systems are dynamically coherent, even in dimension 3 (see [RHRHU16], [BGHP20]). If $(Y, \varphi)$ is partially hyperbolic and dynamically coherent, then the lamination $W^c$ with leaves the connected components of

$$W_x^c := W_x^{cs} \cap W_x^{cu}$$

is Hölder continuous and integrates the direction $E_x^c$. The lamination $W^c$ is globally invariant under $\varphi$. Let $h_{\cdot, \cdot}^c$ be the local holonomy mappings between transversals to $W^c$. The mappings $h_{\cdot, \cdot}^c$ preserve the local $W^{cs}, W^{cs}$ leaves. We extend the definition 2.14 as follows:

**Definition 2.16.** Assume that $(Y, \varphi)$ is partially hyperbolic and dynamically coherent. A measure $\nu$ is called a partial $(s, c, u)$-product if

* the conditionals $\nu_y^{cs}$ of the measure $\nu$ with respect to the partition in local $W^{cs}$-leaves are $(s, c)$-products and

* the conditionals $\nu_y^{cu}$ of the measure $\nu$ with respect to the partition in local $W^{cu}$-leaves are $(u, c)$-products.

**Definition 2.17.** Let $(Y, \varphi)$ be a partially hyperbolic, dynamically coherent system with compact central leaves, $(Z, \psi)$ the quotient system made of the space of leaves. The system $(Y, \varphi)$ is called a compact extension of hyperbolic if the system $(Z, \psi)$ is a Smale space in the sense of [R04], Chapter 7.

Let $(Y, \varphi)$ be a partially hyperbolic system, $\nu$ be a $\varphi$-invariant ergodic probability measure on $Y$. We can define the upper and lower center Lyapunov exponents by the following $\nu$-a.e. limits

$$\gamma_c^+ := \lim_{n \to +\infty} \frac{1}{n} \log \|D_x \varphi^n|_{E^c}\|, \quad \gamma_c^- := \lim_{n \to +\infty} -\frac{1}{n} \log \|D_x \varphi^{-n}|_{E^c}\|.$$

A. Tahzibi and J. Yang (see [TY19]) observe the following invariance principle.
**Theorem 2.18.** [**[TY19]**] Let $(Y, \varphi)$ be a compact extension of hyperbolic system
Let $\nu$ be an invariant ergodic probability measure and assume that $\gamma_c^+ = \gamma_c^- = 0$.
Then, $\nu$ is a partial $(s, c, u)$-product measure.

Proof. <fnref n="3" /> Let $Z$ be the space of compact leaves, endowed with a quotient metric, $\pi: Y \to Z$ the quotient map. By hypothesis, the action of $\varphi$ on $Z$ is a Hölder continuous hyperbolic map $\psi$. It has a pair of continuous laminations $\mathcal{Z}^s, \mathcal{Z}^u$ with local leaves $Z_{\text{loc}}^s(x), Z_{\text{loc}}^u(x)$ such that for all $y \in Y$ and some $\rho < 1, \varepsilon > 0$,

$$
\begin{align*}
Z_{\text{loc}}^{s}(x) &:= \{y \in X, d(\psi^{n}y, \psi^{n}x) \le \varepsilon\rho^{n} \text{ for all } n \ge 0\}, \\
Z_{\text{loc}}^{u}(x) &:= \{y \in X, d(\psi^{-n}y, \psi^{-n}x) \le \varepsilon\rho^{n} \text{ for all } n \ge 0\}.
\end{align*}
$$

By [R04] chapter 7.5, for any $\varepsilon > 0$ there exists a Markov partition $\mathcal{R} = \{R_j, 1 \le j \le J\}$, such that the rectangles $R_j$ have diameter smaller than $\varepsilon$. We can also arrange that the boundaries of the rectangles $R_j$ have 0 $\pi_*\nu$-measure.

We show that for $\nu$-a.e. $y$, the conditionals of the measure $\nu$ with respect to the partition in local $W^{cu}$-leaves are $(u, c)$-products. The proof is the same for the conditionals associated to the local central stable leaves. We set, for $z \in Z, y \in Y$,
$$ \zeta^u(z) := \cap_{n \ge 0} \psi^{-n} \mathcal{R}(\psi^n(z)), \quad \xi^{cu}(y) := \pi^{-1}(\zeta^u(\pi(y))) \text{ and } \xi^u(y) := \xi^{cu}(y) \cap W_{\text{loc}}^u(y). $$

The elements of the partitions $\zeta^u, \xi^u$ and $\xi^{cu}$ have open interior and negligible boundaries in the local leaves, respectively $Z_{\text{loc}}^u, W_{\text{loc}}^u$ and $W_{\text{loc}}^{cu}$. The partitions $\xi^{cu}$ and $\xi^u$ are measurable, let $\nu_y^u, \nu_y^{cu}$ be the associated conditional measures. For a measurable partition $\xi$ with conditional measures $\nu_y, y \in Y$ and a finite partition $C = \{C_\ell, 1 \le \ell \le L\}$, write the conditional entropy $H(C|\xi)$ as:

$$
H(C|\xi) := \int_{1}^{\ell} - \sum_{\ell \le L} \nu_y(C_{\ell}) \log \nu_y(C_{\ell}) d\nu(y).
$$

The following proposition is classical, we state it as the analog of proposition 2.4
in our setting

**Proposition 2.19.** Let $C_k = C_{k,l}, 1 \le l \le L_k$ be a finite partition of $Z$. We have:

$$
H(\pi^{-1}C_k|\xi^u) \le H(\pi^{-1}C_k|\xi^{cu}).
$$

We have equality for finer and finer partitions $C_k$ such that, for $\nu$-a.e. $y \in Y$, $C_k$, $k \to \infty$, generate the Borel $\sigma$-algebra of $\zeta^u(\pi(y))$ if, and only if, for $\nu$-a.e. $y \in Y$, the conditionals $\nu_y^{cu}$ are $(u, c)$-products.

Proof. We may write, for all $k, \ell, 1 \le \ell \le L_k$, $\nu$-a.e. $y \in Y$,

$$
\nu_y^{cu}(\pi^{-1}C_{k,\ell}) = \int \nu_w^u(\pi^{-1}C_{k,\ell}) d\nu_y^{cu}(w).
$$

Using Jensen inequality with the function $t \mapsto -t \log t$, we obtain, for all $k, \ell, 1 \le \ell \le L_k$, $\nu$-a.e. $y \in Y$,

$$
- \int \nu_w^u(\pi^{-1}C_{k,\ell}) \log[\nu_w^u(\pi^{-1}C_{k,\ell})] d\nu_y^{cu}(w) \le -\nu_y^{cu}(\pi^{-1}C_{k,\ell}) \log[\nu_y^{cu}(\pi^{-1}C_{k,\ell})].
$$

<footnote n="3">³We sketch the proof from [TY19], underlining the similarities with the proof of theorems 2.2 and 2.10. We hope that it makes the later extensions more clear.</footnote>
For fixed $k$, the announced inequality follows by summing in $\ell, 1 \le \ell \le L_k$ and integrating in $y$. In case of equality, we have, for all $\ell, 1 \le \ell \le L_k$, $\nu$-a.e. $y \in Y$, and $\nu_y^{cu}$-a.e. $w \in \xi^{cu}(y)$, $\nu_w^u(\pi^{-1}C_{k,\ell}) = \nu_y^{cu}(\pi^{-1}C_{k,\ell})$.

So, in case of equality, we have, for $\nu$-a.e. $y \in Y$ and for $\nu_y^{cu}$-a.e. $w \in \xi^{cu}(y)$, $\pi_*(\nu_w^u) = \pi_*(\nu_y^{cu})$ on $C_k$. As $k \to \infty$, we obtain that for $\nu$-a.e. $y \in Y$ and $\nu_y^{cu}$-a.e. $w \in \xi^{cu}(y)$, $\pi_*(\nu_w^u) = \pi_*(\nu_y^{cu})$ on $\zeta^u(\pi(y))$. The proposition follows since the $\pi_*(\nu_y^{cu})$, $y \in Y$, are the conditional measures associated with the partition $\zeta^u$. $\square$

In order to prove theorem 2.18, we apply proposition 2.19 to the partitions $C_k$ defined by $C_k(z) := \cap_{n=1}^k \psi^n \mathcal{R}(\psi^{-n}(z))$. Since $\zeta^u(z) := \cap_{n \ge 0} \psi^{-n} \mathcal{R}(\psi^n(z))$, the $C_k$ form an increasing family that generates the Borel $\sigma$-algebra of $\zeta^u(z)$ for all $z \in \mathcal{Z}$.

Observe that, by construction, the partition $\xi^u$ (resp. $\xi^{cu}$) is an increasing partition made, up to measure 0, of open subsets of local unstable (resp. central unstable) leaves. Furthermore, we have

$$H(\pi^{-1}(C_k)|\xi^u) = H(\varphi^{-k}\xi^u|\xi^u), \quad H(\pi^{-1}(C_k)|\xi^{cu}) = H(\varphi^{-k}\xi^{cu}|\xi^{cu}).$$

Theorem 2.18 follows from the analog of the basic inequality in this setting:

**Proposition 2.20** ([LY85], Proposition 5.1 and Corollary 7.2.2). Let $\varphi$ be a $C^2$ diffeomorphism of a compact Riemannian manifold $Y$, $\nu$ an invariant ergodic probability measure, $W^u$ the $\nu$-a.e. defined Pesin unstable foliation. Let $\eta^u$ be an increasing partition made, up to 0 measure, of open bounded subsets of unstable leaves. Then

$$H(\varphi^{-1}\eta^u|\eta^u) = \sup_{\xi, \varphi^{-1}\xi < \xi} H(\varphi^{-1}\xi|\xi).$$

<figure><img src="image_2.png" /></figure>

See A. Brown ([B22]) for this form of the basic inequality if $\varphi$ is only $C^{1+\alpha}$, S. Gan, Y. Tong and J. Yang ([GTY]) for the $C^1$ case in the presence of dominated splitting and a dimension 1 center direction.

### 3. INVARIANCE PRINCIPLE

**3.1. Basic setting.** Consider an measurable ergodic dynamical system $(X, \mathcal{A}, m, f)$, where $(X, \mathcal{A}, m)$ is a Lebesgue space (see ), $f$ a bimeasurable bijection preserving $m$. Let $\Pi: \mathcal{E} \to X$ be a measurable fiber bundle over $X$ with fibers $\mathcal{E}_x$ compact Riemannian manifolds of bounded dimensions and diameters.

Let $F: \mathcal{E} \to \mathcal{E}$ be a measurable bundle of $C^2$-diffeomorphisms $F_x: \mathcal{E}_x \to \mathcal{E}_{f(x)}$. We assume that $\|D_\xi F_x\|$, $\|D_\xi(F_x)^{-1}\|$ and $\|D_\xi^2(F_x)^{-1}\|$ are uniformly bounded. Write

$$F^n = \{F_x^{(n)}, x \in X\}, \text{ where } F_x^{(n)} := F_{f^{n-1}x} \circ \dots \circ F_x.$$

Let $M$ be an ergodic $F$-invariant probability measure on $\mathcal{E}$ such that $\Pi_*M = m$ and write $M = \int M_x dm(x)$ the disintegration of $M$ for $\Pi$. Observe that by
invariance, for $m$-a.e. $x$,

$$ (3.1) \qquad (F_x)_*M_x = M_{fx}. $$

By the subadditive ergodic theorem, there are numbers $\gamma^+, \gamma^-$, with $\gamma^+ \ge \gamma^-$, such that, at $M$-a.e. $(x, \xi), \xi \in \mathcal{E}_x$,

$$ \gamma^+ := \lim_{n \to \infty} \frac{1}{n} \log \|D_\xi F_x^{(n)}\|, \quad \gamma^- := - \lim_{n \to \infty} \frac{1}{n} \log \|(D_\xi F_x^{(n)})^{-1}\|. $$

Let $(X, \mathcal{A}, m, f)$ be as above. We say that a sub-$\sigma$-algebras $\mathcal{B}$ is finer than a sub-$\sigma$-algebra $\mathcal{C} \subset \mathcal{A}$ (noted $\mathcal{C} \subset \mathcal{B}$) if for any set $C \in \mathcal{C}$, there is a set $B \in \mathcal{B}$ with $m(B\Delta C) = 0$. We say that the sub-$\sigma$-algebra $\mathcal{B} \subset \mathcal{A}$ is non-increasing if $f^{-1}\mathcal{B} \subset \mathcal{B}$, generating if the $\sigma$-algebra generated by all $\{f^n\mathcal{B}, n \in \mathbb{Z}\}$ is finer than $\mathcal{A}$.

**Theorem 3.1.** *([AV10])* With the above notations$^4$, assume there is a generating non-increasing sub-$\sigma$-algebra $\mathcal{B}$, that the mappings $x \mapsto \mathcal{E}_x, x \mapsto F_x$ are $\mathcal{B}$-measurable, and that $\gamma^- \ge 0$. Then, the mapping $x \mapsto M_x$ is $\mathcal{B}$-measurable.

In the case of i.i.d. $2 \times 2$ matrices, $X = (\text{SL}(2, \mathbb{R}))^{\otimes \mathbb{Z}}$, $m = \mu^{\otimes \mathbb{Z}}$, $\mathcal{E} = X \times \mathbb{P}^1(\mathbb{R})$. Then, consider $F(x, \xi) = (fx, x_0\xi)$, $F^{-1}(x, \xi) = (f^{-1}x, (x_{-1})^{-1}\xi)$. If the exponent of the product of matrices is 0, for any ergodic $F$-invariant probability measure $M$ on $\mathcal{E}$, $\gamma^+ = \gamma^- = 0$. Applying the IP to $F$ for $\mathcal{B} := \sigma(x_n, n \ge 0)$ and to $F^{-1}$ for $\mathcal{B}' := \sigma(x_n, n < 0)$, we conclude that $x \mapsto M_x$ is measurable with respect to both $\mathcal{B}$ and $\mathcal{B}'$. By independence, there is $\nu$ on $\mathbb{P}^1$ such that $M_x = \nu$ for $m$-a.e. $x$. By (3.1), we have $x_0\nu = \nu$ for $\mu$-a.e. $x_0$.

3.2. **Proof of theorem 3.1.** By section 6.1, there is an essentially unique Lebesgue space $(\overline{X}, \overline{\mathcal{A}}, \overline{m})$ and a projection $\pi : X \to \overline{X}$ such that $\pi^{-1}\overline{A} = \mathcal{B}$ and $\pi_*m = \overline{m}$. Since $\mathcal{B}$ is non-increasing, there is a $\overline{m}$-preserving mapping (non-invertible in general) $\overline{f}$ such that $\pi(fx) = \overline{f}(\pi x)$.

The measurability hypotheses ensure that there exist a fiber bundle $\overline{\Pi} : \overline{\mathcal{E}} \to \overline{X}$ with fibers compact manifolds, a fibered projection $\overline{\pi} : \overline{\mathcal{E}} \to \overline{\mathcal{E}}$ and a quotient mapping $\overline{F}$ on $\overline{\mathcal{E}}$ so that $\overline{F} \circ \overline{\pi} = \overline{\pi} \circ \overline{F}$ and $\overline{\Pi} \circ \overline{F} = \overline{f} \circ \overline{\Pi}$. The measure $\overline{M} := \overline{\pi_*M}$ is a $\overline{F}$-invariant ergodic probability measure on $\overline{\mathcal{E}}$. Let $\overline{M} = \int \overline{M}_{\overline{x}} d\overline{m}(\overline{x})$ be a disintegration of $\overline{M}$ with respect to $\overline{\Pi}$.

We work with the family of measures $\overline{M}_{\overline{x}}$, defined for $\overline{m}$-a.e. $\overline{x} \in \overline{X}$. We have to prove that, for $m$-a.e. $x \in X$, $\overline{\pi}_*M_x = \overline{M}_{\pi(x)}$. We define the entropy $\kappa$:

**Definition 3.2.**

$$ \kappa := \int \left( \int \log \frac{d(\overline{F}_{\overline{x}})_* \overline{M}_{\overline{x}}}{d\overline{M}_{\overline{f}\overline{x}}} (\overline{F}_{\overline{x}}\overline{\xi}) d\overline{M}_{\overline{x}} \right) d\overline{m}_{\overline{x}}. $$

The first step is analogous to proposition 2.4, with the same proof:

**Proposition 3.3.** We have $\kappa \ge 0$, with equality only if, for $\overline{m}$-a.e. $\overline{x} \in \overline{X}$,

$$ (\overline{F}_{\overline{x}})_* \overline{M}_{\overline{x}} = \overline{M}_{\overline{f}\overline{x}}. $$

<footnote>$^4$In particular, for the sake of the proof we give, we assume that $\|D_\xi F_x\|$, $\|D_\xi(F_x)^{-1}\|$ and $\|D_\xi^2(F_x)^{-1}\|$ are uniformly bounded. It should be possible to weaken these hypotheses.</footnote>
The second step is the observation that the family $M_x, x \in X$ can be recovered from the family of measures $\overline{M}_{\bar{x}}$.

**Proposition 3.4.** For m-a.e. $x$, $M_x = \lim_{n \to +\infty} ((\overline{F}_{\bar{x}}^{(n)})^{-1})_* \overline{M}_{\bar{f}^n \pi x}$.

*Proof.* For $n \ge 0$, let $X_n$ be the space associated to $f^{-n}\mathcal{B} \supset \mathcal{B}$, $\pi_n$ the associated projection. There are a fiber bundle $\Pi_n : \mathcal{E}_n \to X_n$ and a fibered projection $\bar{\pi}_n : \mathcal{E} \to \mathcal{E}_n$. Since the $\sigma$-algebra $\mathcal{B}$ is non-increasing and $x \mapsto \mathcal{E}_x$ is $\mathcal{B}$ measurable, the fibers of $\mathcal{E}_n$ are identified with the fibers $\mathcal{E}_{\pi_n(x)}$, $x \in X$ for all $n \ge 0$. Since $\mathcal{B}$ is generating, by the martingale theorem, for m-a.e. $x$, the disintegrations $M_x, x \in X$ are given by the weak*-limits of the disintegrations $M_{\pi_n(x)}^n$ associated to $\Pi_n$. By invariance (3.1),

$$M_{\pi_n x}^n = ((F_{\pi_n x}^{(n)})^{-1})_* M_{f^n(\pi_n x)}^0.$$

With our notations, $\pi_0(x) = \bar{x}$, $f^n(\pi_n x) = \overline{f^n x} = \overline{f^n} \bar{x}$ and $M_{\bar{x}}^0 = \overline{M}_{\bar{x}}$, so that

$$M_x = \lim_{n \to \infty} ((F_{\pi_n x}^{(n)})^{-1})_* \overline{M}_{\bar{f}^n \bar{x}}.$$

Since $x \mapsto F_x$ is $\mathcal{B}$-measurable, $F_{\pi_n x}^{(n)} = F_{\pi_0 x}^{(n)} = \overline{F}_{\bar{x}}^{(n)}$.

□

The third step is a basic inequality:

**Proposition 3.5.** Let $d$ be the maximal dimension of the fibers $\mathcal{E}_x$. Then,

$$\kappa \le -d \min\{\gamma^{-}, 0\}.$$

The IP follows from propositions 3.3, 3.4 and 3.5: assume $\gamma^{-} \ge 0$. Then, by Proposition 3.5, $\kappa = 0$. By proposition 3.3, this implies that, for $\overline{m}$-a.e. $\bar{x} \in \overline{X}$, $(\overline{F}_{\bar{x}})_* \overline{M}_{\bar{x}} = \overline{M}_{\overline{f}\bar{x}}$, i.e. for $m$-a.e. $x \in X$, $\overline{M}_{\pi x} = ((\overline{F}_{\pi x})^{-1})_* \overline{M}_{\overline{f}\pi x}$. Iterating and using proposition 3.4, we find that $M_x = \overline{M}_{\pi x}$. □

*Proof.* (Proof of proposition 3.5)

Assume first that $\gamma^{-} > 0$. Then, by Pesin theory (see proposition ??), $\overline{M}$-almost every point $(\bar{x}, \xi) \in \overline{\mathcal{E}}$ has a local unstable manifold which is an open neighborhood of $\xi$ in $\mathcal{E}_{\bar{x}}$. The measure $\overline{M}_{\bar{x}}$ is therefore discrete for $\overline{m}$-a.e. $\bar{x}$. The individual masses are uniform on each $\mathcal{E}_x$ and constant by ergodicity. Therefore, there exists an integer $k \ge 1$ so that, for $\overline{M}$-a.e. $(\bar{x}, \xi)$, $\overline{M}_{\bar{x}}(\{\xi\}) = 1/k$. So, if $\gamma^{-} > 0$, we have $\kappa = 0 = -d \min\{\gamma^{-}, 0\}$.

Assume now that $\gamma^{-} \le 0$. By the subadditive ergodic theorem,

$$-\gamma^{-} = \inf_n \frac{1}{n} \int \log \|(D_{\xi} F_{\bar{x}}^{(n)})^{-1}\| \, d\overline{M}(\bar{x}, \xi)$$

and given $\varepsilon > 0$, one can find $L$ such that the transformation $\overline{F}^L$ is ergodic for $\overline{M}$ and such that

$$-L\gamma^{-} \le \int \log \|(D_{\xi} F_{\bar{x}}^{(L)})^{-1}\| \, d\overline{M}(\bar{x}, \xi) \le -L(\gamma^{-} - \varepsilon/2).$$
Fix $\varepsilon > 0$ and choose $L$ with these properties. Write, for $\bar{x} \in \bar{X}, \xi \in \mathcal{E}_{\bar{x}}$,

$$g(\bar{x}, \xi) := \frac{d((\bar{F}_{\bar{x}}^{(L)})^{-1}) * \bar{M}_{\bar{f}^{L}\bar{x}}(\xi)}{d\bar{M}_{\bar{x}}}$$

and, for $N$ large, $g^N(\bar{x}, \xi) := \sup\{g(\bar{x}, \xi), e^{-N}\}$. The function $-\log g^N$ is $\bar{M}$-integrable, $-\log g \ge -\log g^N$ and, by monotone limit,

$$(3.2) \qquad L_{\kappa}(\mu, \nu) = \lim_{N \to \infty} - \int \log g^N(\bar{x}, \xi) d\bar{M}(\bar{x}, \xi).$$

**Lemma 3.6.** Fix $N, \varepsilon$ and $L$ and assume that there exists $\beta > 0$ and $J \in \mathbb{N}$ such that, for $\bar{M}$-a.e. $(\bar{x}, \xi) \in \bar{\mathcal{E}}$, there exists a sequence $n_i(\bar{x}, \xi), i > 0$, of upper density at least $\beta$, and for each $i$, a sequence $q(j, n_i, \bar{x}, \xi), 0 \le j \le n_i$, of measurable positive functions on $\bar{\mathcal{E}}$ with the property that for all $i$, for $\bar{M}$-a.e. $(\bar{x}, \xi)$,

$$(3.3) \qquad q(n_i, n_i, \bar{x}, \xi) = e^{(\gamma^{-} - 2\varepsilon)Ln_i},$$

$$(3.4) \qquad q(j, n_i, \bar{x}, \xi) \le e^{(\gamma^{-} - 2\varepsilon)Lj}, \quad \text{for all } j \le n_i \text{ and}$$

$$(3.5) \qquad F_{\bar{f}^{L_j}\bar{x}}^{(L)} B(F_{\bar{x}}^{(Lj)}\xi, q(j, n_i, \bar{x}, \xi)) \supset B(F_{\bar{x}}^{(L(j+1))}\xi, q(j+1, n_i, \bar{x}, \xi)),$$

for all $J \le j < n_i$. Then, $-\int \log g^N(\bar{x}, \xi) d\bar{M}(\bar{x}, \xi) \le dL(-\gamma^{-} + 3\varepsilon)$.

*Proof.* Choose $\delta > 0$ such that, setting

$$g_{\delta}(\bar{x}, \xi) := \frac{\int_{B(\xi, \delta)} g(\bar{x}, \eta) d\bar{M}_{\bar{x}}(\eta)}{\bar{M}_{\bar{x}}(B(\xi, \delta))} = \frac{\bar{M}_{\bar{f}^{L}\bar{x}}(F_{\bar{x}}^{(L)}(B(\xi, \delta)))}{\bar{M}_{\bar{x}}(B(\xi, \delta))},$$

$$g_{\delta}^{N}(\bar{x}, \xi) := \frac{\int_{B(\xi, \delta)} g^{N}(\bar{x}, \eta) d\bar{M}_{\bar{x}}(\eta)}{\bar{M}_{\bar{x}}(B(\xi, \delta))}, \quad g_{\delta}^{*} := \sup_{\delta' \le \delta} g_{\delta'}^{N},$$

we have $\int \log \frac{g_{\delta}^{*}}{g_{\delta}^{N}} d\bar{M} \le \varepsilon$ (see proposition 6.4). Observe that, for $q \le \delta$,
$$-\log g^N \le -\log g_q + \log g_q - \log g_q^N + \log g_{\delta}^* - \log g^N \le -\log g_q + \log g_{\delta}^* - \log g^N.$$

We may thus write, at $\bar{M}$-a.e. $(\bar{x}, \xi)$, for $J$ such that $e^{(\gamma^{-} - 2\varepsilon)JL} < \delta$,

$$-\int \log g^N d\bar{M} \le \liminf_{i \to \infty} \frac{1}{n_i} \sum_{j=J}^{n_i-1} -\log g_{q(j, n_i, \bar{x}, \xi)} (\bar{F}^{jL}(\bar{x}, \xi)) + \varepsilon \\ \le \liminf_{i \to \infty} -\frac{1}{n_i} \sum_{j=J}^{n_i-1} \log \frac{\bar{M}_{\bar{f}^{L(j+1)}\bar{x}}(F_{f^{jL}\bar{x}}^{(L)} B(F_{\bar{x}}^{(Lj)}\xi, q(j, n_i, \bar{x}, \xi)))}{\bar{M}_{\bar{f}^{L_j}\bar{x}}(B(F_{\bar{x}}^{(Lj)}\xi, q(j, n_i, \bar{x}, \xi)))} + \varepsilon \\ \le \liminf_{i \to \infty} -\frac{1}{n_i} \log \frac{\bar{M}_{\bar{f}^{L n_i}\bar{x}}(B(F_{\bar{x}}^{(L n_i)}\xi, q(n_i, n_i, \bar{x}, \xi)))}{\bar{M}_{\bar{f}^{L J}\bar{x}}(B(F_{\bar{x}}^{(L J)}\xi, q(J, n_i, \bar{x}, \xi)))} + \varepsilon$$

We use the ergodic theorem to write the first line. The last line follows by (3.5) and a telescoping argument. To finish the estimate, on the one hand, we have $\bar{M}_{\bar{f}^{L J}\bar{x}}(B(F_{\bar{x}}^{(L J)}\xi, q(J, n_i, \bar{x}, \xi))) \le 1$. On the other hand, by proposition ??, there are constants $C_{\beta}$ and $\gamma_{\beta}$ such that the set $A_{\beta}$ of points in $\bar{\mathcal{E}}$ such that $\bar{M}_{\bar{x}}(B(\xi, \gamma')) \ge$
$C_β(γ')^d$ for all $0 < γ' ≤ γ_β$ has $\overline{M}$-measure at least $1 - β/2$. Therefore, for $\overline{M}$-a.e.($\overline{x}, ξ$), there are infinitely many indices $i$ such that $e^{(γ' - 2ε)Ln_i(\overline{x}, ξ)} < γ_β$ and $\overline{F}^{(Ln_i(\overline{x}, ξ))}(\overline{x}, ξ) ∈ A_β$. It follows that

$$ \liminf_{i \to \infty} -\frac{1}{n_i} \log \overline{M}_{\overline{f}^{Ln_i} \overline{x}} \left( B \left( F_{\overline{x}}^{(Ln_i)} \xi, e^{(\gamma' - 2\epsilon)Ln_i(\overline{x}, \xi)} \right) \right) \le -dL(\gamma' - 2\epsilon). $$

$$ \text{Therefore, } -\int \log g^N(\bar{x}, \xi) d\bar{M}(\bar{x}, \xi) \le -dL(\gamma' - 3\epsilon). \quad \square $$

Next, we find $β > 0$, $J ∈ N$ and, for $\overline{M}$-a.e.($\overline{x}, ξ$), a sequence $n_i(\overline{x}, ξ)$, with upper density at least $β$, of measurable positive functions $q_ε(j, n_i, \overline{x}, ξ)$, $0 ≤ j ≤ n_i$, on $\overline{\mathcal{E}}$ satisfying (3.3), (3.4) and (3.5). We are given $ε > 0$ and $L ∈ N$ such that

$$ L\gamma^{-} \geq -\int \log \|(D_{\xi} F_{\bar{x}}^{(L)})^{-1}\| d\bar{M}(\bar{x}, \xi) \geq L(\gamma^{-} - \epsilon/2). $$

We can consider the constant $K$ defined by

$$ K := \max_{(\bar{x}, \eta)} |\log \|(D_{\eta} F_{\bar{x}}^{(L)})^{-1}\| |. $$

We use Pliss lemma (see corollary 6.7) to find, for $\overline{M}$-a.e.($\overline{x}, ξ$), a sequence $n_i(\overline{x}, ξ)$, $1 ≤ i$, of upper density at least $Lε/(2K + 2L)) =: β$ such that, for any $0 ≤ j ≤ n_i$,

$$ (3.6) \qquad \sum_{k=j}^{n_i-1} -\log \|(D_{F_{\bar{x}}^{(jL)} \xi} F_{\bar{j}L_{\bar{x}}}^{(L)})^{-1}\| \geq (n_i - j)L(\gamma^{-} - \frac{3\epsilon}{2}). $$

For $n_i$ as above, we define the sequence $q(j, n_i, \bar{x}, ξ)$ as follows:

* $q(n_i, n_i, \bar{x}, ξ) = e^{(\gamma' - 2ε)Ln_i}$ and,

* for $0 ≤ j < n_i$,

$$ q(j, n_i, \bar{x}, \xi) := e^{(\gamma' - 2\epsilon)Ln_i + \frac{\epsilon}{2}L(n_i - j)} \prod_{k=j}^{n_i-1} \|(D_{\bar{F}_{\bar{x}}^{(kL)} \xi} F_{\bar{j}L_{\bar{x}}}^{(L)})^{-1}\|. $$

Property (3.3) is verified by definition and, by (3.6), we can write

$$ -\frac{1}{L} \log q(j, n_i, \bar{x}, \xi) \ge -(\gamma^{-} - 2\epsilon)n_i - \frac{\epsilon}{2}(n_i - j) + (n_i - j)(\gamma^{-} - \frac{3\epsilon}{2}) = -j(\gamma^{-} - 2\epsilon), $$
which is property (3.4).

We choose $J$ so that relation (3.5) follows from Taylor theorem. Indeed, we have

$$ q(j + 1, n_i, \bar{x}, \xi) = e^{-L\epsilon/2} \|(D_{\bar{F}_{\bar{x}}^{(jL)} \xi} F_{\bar{j}L_{\bar{x}}}^{(L)})^{-1}\|^{-1} q(j, n_i, \bar{x}, \xi). $$

Since $\|D_{\bar{F}_{\bar{x}}^{(jL)} \xi} F_{\bar{j}L_{\bar{x}}}^{(L)}(v, w)\|$ is bounded from below for vectors $v, w$ of norm 1 in $T_{\bar{F}_{\bar{x}}^{(jL)} \xi} \mathcal{E}_{\bar{j}L_{\bar{x}}}$, there is $δ'$ such that, if $ξ, η$ satisfy $\pi(\bar{F}_{\bar{x}}^{(jL)} \xi) = \pi(\bar{F}_{\bar{x}}^{(jL)} \eta) = \bar{j}L_{\bar{x}}$ with $d_{\mathcal{E}_{\bar{j}L_{\bar{x}}}}(\bar{F}_{\bar{x}}^{(jL)} \xi, \bar{F}_{\bar{x}}^{(jL)} \eta) \le δ'$, we have

$$ \frac{\|(D_{\bar{F}_{\bar{x}}^{(jL)} \eta} F_{\bar{j}L_{\bar{x}}}^{(L)})^{-1}\|^{-1}}{\|(D_{\bar{F}_{\bar{x}}^{(jL)} \xi} F_{\bar{j}L_{\bar{x}}}^{(L)})^{-1}\|^{-1}} \ge e^{-L\epsilon/2}. $$
and therefore the relation (3.5),

$$ (F_{\bar{x}}^{(L)}) B(F_{\bar{x}}^{(Lj)} \xi, q(j, n_i, \bar{x}, \xi))) \supset B(F_{\bar{x}}^{(L(j+1))} \xi, q(j+1, n_i, \bar{x}, \xi))), $$

holds as soon as $q(j, n_i, \bar{x}, \xi) \le \delta'$. But $q(j, n_i, \bar{x}, \xi) \le e^{jL(\gamma^{-} - 2\epsilon)}$ by (3.4)). We can choose $J$ so that $q(j, n_i, \bar{x}, \xi) \le \delta'$ for $j > J$.

We have proven that for all $\epsilon$, $L = L(\epsilon)$ and $N$,

$$ -\int \log g^N(\bar{x}, \xi) d\overline{M}(\bar{x}, \xi) \le dL(-\gamma^{-} + 3\epsilon). $$

The proposition follows in by letting $N \to \infty$ (see (3.2)), dividing by $L(\epsilon)$ and letting $\epsilon \to 0$. $\square$

### 3.3. Remarks and complements.

#### 3.3.1. Non-ergodic case.
If the probability measure $M$ is not ergodic, the same conclusion holds if $\int \min\{\gamma^{-}, 0\} dM = 0$. Indeed, in that case, write $M = \int M_t dn(t)$ for the decomposition of $M$ into ergodic measures, $\gamma_t^{-}$ the corresponding exponents. Since $\int \min\{\gamma^{-}, 0\} dM = 0$, we have $\gamma_t^{-} \ge 0$ for $n$-a.e. $t$. By theorem 3.1, the disintegrations of $n$-a.e. $M_t$ associated to the projection on $X$ are $\mathcal{B}$ measurable. But since $m$ is ergodic, the disintegrations of $M$ are given by the integrals of the disintegrations of $M_t$. The conclusion follows.

#### 3.3.2. Perturbations.
In the setting of the Invariance Principle, assume that the space $X$ is a compact metric space, that $f$ is a homeomorphism of $X$ and that the bundle objects $\mathcal{E}_x, F_x$ depend continuously on $x$. Consider a sequence $\{M_k\}_{k \ge 0}$ of $F$-invariant probability measures on $\mathcal{E}$ with $\pi_* M_k = m$ for all $k$. Assume there is a generating decreasing sub-$\sigma$-algebra $\mathcal{B}$, that the mappings $x \mapsto \mathcal{E}_x, x \mapsto F_x$ are $\mathcal{B}$-measurable, that there is a $F$-invariant probability measure $M$ such that $M_k$ weak* converge to $M$ and that $\int \min\{\gamma^{-}, 0\} dM_k \to 0$ as $k \to \infty$. Then, the disintegration mapping $x \mapsto M_x$ is $\mathcal{B}$-measurable.

**Proof.** We want to show that the entropy $\kappa$ associated to $M$ vanishes. By proposition 3.5 the entropies $\kappa_k$ associated to $M_k$ converge to 0. Let $\mathcal{C}_n, n \ge 1$, be a family of finer and finer finite partitions of $\mathcal{E}$ that generate the Borel $\sigma$-algebra. For each $n$ we can define approximate entropies $\kappa_{k,n}$ by

$$ \kappa_{k,n} := \int \left( \int -\log \mathbb{E} \left[ \frac{d(\overline{F_{\bar{x}}})_*^{-1} \overline{M}_{\overline{F_{\bar{x}}}}^k}{d\overline{M}_{\overline{x}}^k} (\xi) \middle| \mathcal{C}_n \right] d\overline{M}_{\overline{x}}^k \right) d\overline{m}(\overline{x}). $$

By a classical result of Dobrushin ([Dob59]), the entropies $\kappa_{k,n}, \kappa_{\infty,n}$ are nondecreasing in $n$ and

$$ \kappa_k = \lim_{n \to \infty} \kappa_{k,n}, \quad \kappa = \lim_{n \to \infty} \kappa_{\infty,n}, $$

The remark follows by choosing the elements of the partitions $\mathcal{C}_n$ with negligible boundaries for all $n$ so that $\kappa_{\infty,n} = \lim_{k \to \infty} \kappa_{k,n}$ for all $n$. $\square$
3.3.3. Deformations. Let $\pi : \mathcal{E} \to X, f, F = \{F_x, x \in X\}, m, M, \pi_*M = m$ be as in the setting of the Invariance Principle. Assume that there is a measurable transformation $H : \mathcal{E} \to \mathcal{E}$ of the form $H(x, \xi) = (x, H_x(\xi))$ such that $H_x^{-1}$ is uniformly Hölder continuous:

there exist $C, \beta$ such that, for $x \in M$, $\xi, \eta \in \mathcal{E}_x$, $d_{\mathcal{E}}(\xi, \eta) \le C(d_{\mathcal{E}}(H_x\xi, H_x\eta))^\beta$.

We say that $\tilde{F} := H \circ F \circ H^{-1}$ is a deformation of $F$ if there is a family $\tilde{F}_x, x \in M$ such that $\tilde{F}(x, \xi) = (fx, \tilde{F}_x(\xi))$.

**Proposition 3.7.** Assume there is a generating decreasing sub-$\sigma$-algebra $\mathcal{B}$, that $\tilde{F}$ is a deformation of $F$ and that the mappings $x \mapsto \mathcal{E}_x, x \mapsto \tilde{F}_x$ are $\mathcal{B}$-measurable. Assume that $\int \min\{\gamma^-, 0\} dM = 0$ where $\gamma^-$ is the original exponent. Then the mapping $x \mapsto \tilde{M}_x := (H_x)_*M_x$ is $\mathcal{B}$-measurable.

Observe that the $\tilde{M}_x$ are the disintegrations of a $\tilde{F}$-invariant probability measure $\tilde{M}$. The basic inequality pertains now to the $(\tilde{F}, \tilde{M})$ entropy $\tilde{\kappa}$.

$$ (3.7) \qquad \tilde{\kappa} \le -\frac{d}{\beta} \int \min\{\gamma^-, 0\} d\tilde{M}. $$

*Proof.* (Proof of (3.7)) By the above argument, we can reduce to the ergodic case. We then follow the proof of proposition 3.5. For the same reason, we can assume that the exponent is non-positive. Then, we have to estimate, for a suitable $L$,

$$ L\tilde{\kappa} = - \int \log \tilde{g}(\bar{x}, \eta) d\tilde{M}_{\bar{x}}(\eta) d\bar{m}(\bar{x}), $$

where

$$ \tilde{g}(\bar{x}, \eta) := \frac{d((\tilde{F}_{\bar{x}}^{(L)})^{-1})_* \tilde{M}_{\tilde{f}^L \bar{x}}(\eta)}{d\tilde{M}_{\bar{x}}} = \frac{d((\bar{F}_{\bar{x}}^{(L)})^{-1})_* \bar{M}_{\bar{f}^L \bar{x}}(H_{\bar{x}}^{-1}\eta)}{d\bar{M}_{\bar{x}}(H_{\bar{x}}^{-1}\eta)}. $$

For all $N$, we approximate the integrable function $\tilde{g}^N = \sup\{\tilde{g}, e^{-N}\}$ and we find, in the same way as in the proof of proposition 3.5, a number $\beta > 0$ and a suitable sequence $n_i, i \in \mathbb{N}$, of upper density at least $\beta$ such that

$$ - \int \log \tilde{g}^N(\bar{x}, \eta) d\tilde{M}(\bar{x}, \eta) \le \liminf_{i \to \infty} -\frac{1}{n_i} \log \bar{M}_{\bar{f}^{L n_i} \bar{x}}(B(\bar{F}_{\bar{x}}^{(L n_i)}(H_{\bar{x}}^{-1}\eta), e^{(\gamma^- - 2\epsilon)L n_i})) + \epsilon $$

Observe that

$$ \bar{M}_{\bar{f}^{L n_i} \bar{x}}(B(\bar{F}_{\bar{x}}^{(L n_i)}(H_{\bar{x}}^{-1}\eta), e^{(\gamma^- - 2\epsilon)L n_i})) = \tilde{M}_{\tilde{f}^{L n_i} \bar{x}}(B(H_{\bar{x}}^{-1} \tilde{F}_{\bar{x}}^{(L n_i)}(\eta), e^{(\gamma^- - 2\epsilon)L n_i})) $$

and that, by our hypothesis,

$$ B(H_{\bar{f}^{L n_i} \bar{x}}^{-1} \tilde{F}_{\bar{x}}^{(L n_i)}(\eta), e^{(\gamma^- - 2\epsilon)L n_i}) \supset B(\tilde{F}_{\bar{x}}^{(L n_i)}(\eta), C^{-1} e^{\frac{1}{\beta}(\gamma^- - 2\epsilon)L n_i}). $$

Relation (3.7) follows since we can find infinitely many $n_i$ such that

$$ - \log \tilde{M}_{\tilde{f}^{L n_i} \bar{x}}(B(\tilde{F}_{\bar{x}}^{(L n_i)}(\eta), C^{-1} e^{\frac{1}{\beta}(\gamma^- - 2\epsilon)L n_i})) \le -\frac{d}{\beta}(\gamma^- - 3\epsilon)L n_i. $$

□
3.3.4. *Comments.* Theorem 3.1 extends theorem 2.2 in the different directions seen in section 2. For dependent product of matrices, the case of Markovian dependence was considered in [V79], [G79] and [R80]. The form of dependence in theorem 3.1 for linear cocycles was formulated in [L86], in order to get a common reason for theorem 2.2 and theorem 2.8 (see [L86], section IV). The non-linear version pertains to independent products of diffeomorphisms and was proven by H. Crauel (see theorem 2.10). The complete nonlinear version, the catchy name and many applications are due to A. Avila and M. Viana ([AV10]). Theorem 2.18 is also a consequence of theorem 3.1, see [AVW22] for a related result and the discussion below in section. The proofs given here are simpler for propositions 2.5 and 2.11 because we have, a priori, $\|A^{-1}\| \ge 1$ and $|(T^{-1})'| \ge 1$ so that the analog to relation (3.4) is automatic, whereas here we use (3.6) to obtain (3.4).

# 4. SOME EXAMPLES

## 4.1. Hyperbolic systems.

4.1.1. *Invariant foliations.* Recall the definition 2.12 of a partially hyperbolic $C^2$-diffeomorphism of a compact manifold $X$ and the proposition 2.13 that defines and shows the existence of the partial stable (respectively unstable) lamination $\mathcal{W}^\rho$ (resp. $\mathcal{W}^{\rho'}$), with local holonomies $h_{x,y}^{\rho}, y \in W_{\text{loc}}^{\rho}(x)$ (resp. $h_{x,y}^{\rho'}, y \in W_{\text{loc}}^{\rho'}(x)$). Recall also that we call the system hyperbolic if it is partially hyperbolic with dim $E^c = 0$. Then, there is $\rho$ (resp. $\rho'$) such that $\mathcal{W}^\rho$ (resp. $\mathcal{W}^{\rho'}$) is the stable $\mathcal{W}^s$ (resp. unstable $\mathcal{W}^u$) foliation.

If the system $(X, f)$ is hyperbolic, then the future of Markov partitions provide examples of increasing generating $\sigma$-algebras $\mathcal{B}$ to which the hypotheses of theorem 3.1 may apply. In the case of partially hyperbolic systems, it is possible to construct measurable partitions of $X$ with a "premarkov property" so that the associated $\sigma$-algebra can play the role of $\mathcal{B}$. Namely:

**Proposition 4.1.** Let $(X, f)$ be a partially hyperbolic $C^2$-diffeomorphism. Given $\rho < 1$, let $\mathcal{W}^\rho$ be the lamination obtained in proposition 2.13. Let $m$ be a $f$-invariant ergodic probability measure and $\varepsilon > 0$ small enough. There exists a measurable partition $Q$ of $X$ such that

(1) for $m$-a.e. $x \in X$, the element $Q(x)$ containing $x$ is an open set of $W_{x,\text{loc}}^\rho$ of inner diameter at least $\varepsilon$,

(2) for $m$-a.e. $x \in X$ $f(Q(x)) \subset Q(fx)$.

Let $Q$ be a partition given by proposition 4.1, $x \mapsto m_x^Q$ a family of conditional measures for $Q$. Then, for $m$-a.e. $x$, $Q(x)$ is $m_x^Q$-essentially partitioned by the images $fQ(y)$, $y \in X$ that contain some open subset of $Q(x)$. The $\sigma$-algebra $\mathcal{C}$ associated to $Q$ is thus non-decreasing. Moreover, if $f^n y \in Q(f^n x)$ for all $n \in \mathbb{Z}$, then $d_{W^\rho}(x, y) = 0$ and thus the $\sigma$-algebra $\mathcal{C}$ is generating for $f^{-1}$. One obtains non-increasing generating $\sigma$-algebras $\mathcal{B}$ as in theorem 3.1 in the same way, by applying proposition 4.1 to $f^{-1}$ and to an unstable lamination $\mathcal{W}^{\rho'}$.
A more precise premarkov partition is constructed in [AV10], propositions 4.4 and 4.5, where similar properties are obtained for all $x \in X$ and not only for $m$-almost every $x$. Proposition 4.1 with $m$ the Lebesgue measure goes back to Sinai ([S68]). See [LS82] for the general case of Pesin invariant manifolds and any invariant probability measure.

For hyperbolic systems, $W^s$ and $W^u$ form locally a system of coordinates. More precisely,

**Proposition 4.2.** Let $(X, f)$ be a transitif hyperbolic $C^2$-diffeomorphism. There is $\delta > 0$ such that, if $d(x, y) < \delta$, then $W_{x,loc}^s$ and $W_{y,loc}^u$ intersect at exactly one point, and this point $[x, y]$ depends continuously on $x, y$.

For hyperbolic systems, the notion of $(s, u)$-product probability measure (see definition 2.14) is too strong: there is only one $(s, u)$-product probability measure. We define the notion of $(s, u)$-product structure.

**Definition 4.3.** Assume that $(X, f)$ is hyperbolic. A measure $\nu$ is said to have a $(s, u)$-product structure if, in a common local chart for both foliations,

* the holonomies $h^s$ preserve the sets of measure 0 for the conditional $\nu_y^u$ of the measure $\nu$ with respect to the partition in local $W^u$-leaves or, equivalently,

* the holonomies $h^u$ preserve the sets of measure 0 for the conditional $\nu_y^s$ of the measure $\nu$ with respect to the partition in local $W^s$-leaves.

The equivalence of the two definitions is a direct application of Fubini Theorem: in a local chart with two laminations, the measure $\nu$ has the same negligible sets as the product of both transversal projections of any fiber measure. The measure of maximal entropy is a $(s, u)$-product measure and so, of course, has a $(s, u)$-product structure, but there are many others: for any Hölder continuous function $F$ on $X$, the measure that realizes the maximum of $\mu \mapsto h_\mu(f) + \int F d\mu$ over invariant probability measures $\mu$, where $h_\mu(f)$ is the entropy of the transformation $f$ for the measure $\mu$, has a $(s, u)$-product structure (see [B75]). Any invariant probability measure absolutely continuous with respect to the Lebesgue measure also has a $(s, u)$-product structure ([A67]). More generally, an invariant measure is called SRB (for Ya. G. Sinai, D. Ruelle and R. Bowen) if the conditionals on local unstable manifolds are absolutely continuous with respect to the Riemannian volume. Since the stable holonomies preserve the Riemannian volume class, a hyperbolic SRB measure has a $(s, u)$-product structure as well.

4.1.2. *Holonomy invariance.* Let $(X, \mathcal{A}, f, m, \mathcal{E}, F)$ be a smooth cocycle with $f$ a $C^2$ partially hyperbolic diffeomorphism of the compact manifold $X$. Take $\rho < 0$ with nontrivial $W^\rho$ and assume that the holonomies $h_{x,y}^\rho$ are defined for all $y \in W_x^\rho$, all $x$. We say that $F$ is compatible with the holonomies $h^\rho$ if, for all $y \in W_x^\rho$, $x \in X$, there exists an application $\tilde{h}_{x,y}^\rho : \mathcal{E}_x \to \mathcal{E}_y$ such that

* $F_y \circ \tilde{h}_{x,y}^\rho = \tilde{h}_{f(x),f(y)}^\rho \circ F_x,$

* there is $C, \beta$ such that for $d_{\mathcal{E}_x}(\xi, \eta) \le \delta$, $d_{\mathcal{E}_y}(\tilde{h}_{x,y}^\rho \xi, \tilde{h}_{x,y}^\rho \eta) \le C(d_{\mathcal{E}_x}(\xi, \eta))^\beta,$
• $(x, y, \xi) \mapsto \tilde{h}_{x,y}^{\rho}(\xi)$ is continuous and

• we have $\tilde{h}_{y,z}^{\rho} \circ \tilde{h}_{x,y}^{\rho} = \tilde{h}_{x,z}^{\rho}$, $\tilde{h}_{x,x}^{\rho} = \mathbb{1}$.

These conditions ensure that there is a $F$-invariant Hölder continuous foliation $\tilde{W}^{\rho}$ of $\mathcal{E}$ and that the leaves $\tilde{W}^{\rho}(x, \xi)$ homeomorphically project to $W^{\rho}(x)$, for all $(x, \xi) \in \mathcal{E}$. We have the following topological version of proposition 3.7:

**Proposition 4.4.** ([AV10], corollary 4.3) With the above notations, assume that $F$ is compatible with the holonomies $h^{\rho}$. Let $M$ be a $F$-invariant measure with disintegrations $M_x, x \in X$.

If $\int \min\{\gamma^{-}, 0\} dM = 0$, then there is a $W^{\rho}$-saturated subset $E$ of $X$ with full $m$-measure and a version of the disintegration $M_x$ such that, if $x, y \in E$ and are in the same $W^{\rho}$-leaf,

$$ (4.1) \qquad (\tilde{h}_{x,y}^{\rho})_*(M_x) = M_y. $$

*Proof. Step 1.* Let $P$ be a measurable partition with the premarkov property of proposition 4.1 and let $\mathcal{C}$ be the associated non-decreasing generating $\sigma$-algebra. Choose a $\mathcal{C}$-measurable section $p: M \to M$ such that for $m$-a.e. $x \in X$, $p(x) \in \mathcal{C}(x)$. Using the holonomies $\tilde{h}_{x,p(x)}^{\rho}$, we can identify the fibers so that $\mathcal{E}_{p(x)} = \mathcal{E}_x$.

*Step 2.* Define, for $x \in X$, $H: \mathcal{E} \to \mathcal{E}$ by $H(x, \xi) = (x, H_x(\xi)) := (x, \tilde{h}_{x,p(x)}^{\rho}\xi)$. Then, $G := H^{-1} \circ F \circ H$ is a deformation of $F$ such that $x \mapsto G_x$ is $\mathcal{C}$-measurable. Then, by proposition 3.7 (applied to $f^{-1}$), the mapping $x \mapsto \tilde{M}_x = (H_x)_*M_x$ is $\mathcal{C}$-measurable.

*Step 3.* It follows that there is a set of full measure $E_0$ such that if $y, z$ are in $E_0$ and in the same atom of $\mathcal{C}$, $p(y) = p(z)$ by definition, and $\tilde{M}_y = \tilde{M}_z$ by Step 2. Thus, for such $y, z$,

$$ M_y = (\tilde{h}_{p(y),y}^{\rho})_* \tilde{M}_y = (\tilde{h}_{p(z),y}^{\rho})_* \tilde{M}_z = (\tilde{h}_{z,y}^{\rho})_* (\tilde{h}_{p(z),z}^{\rho})_* \tilde{M}_z = (\tilde{h}_{z,y}^{\rho})_* M_z. $$

*Step 4.* Let $E_1 := \cap_{n \ge 0} f^{-n} E_0$. The set $E_1$ has full $m$-measure and, if $y, z$ are in $E_1$ and in the same $W^{\rho}$-leaf, there is an $n \ge 0$ such that $f^n y, f^n z$ are in $E_0$ and in the same atom of $\mathcal{C}$. By Step 3, $(\tilde{h}_{f^n x, f^n y}^{\rho})_* M_{f^n x} = M_{f^n y}$. (4.1) follows on $E_1$ by (3.1) and the equivariance of the $\tilde{h}^{\rho}$.

*Step 5.* Set $M_x := (\tilde{h}_{y,x}^{\rho})_* M_y$, for any $y \in W^{\rho}(x) \cap E_1$. By Step 4, the family $x \mapsto M_x$ is defined $m$-a.e. and is a family of disintegrations of $M$ for $\pi$ that satisfies (4.1). $\square$

<figure><img src="image_3.png" /></figure>

4.1.3. *Applications.* Assume that $(X, \mathcal{A}, f, m, \mathcal{E}, F)$ is a smooth cocycle with $f$ a $\mathcal{C}^2$ transitive hyperbolic diffeomorphism of the compact manifold $X$. Let $W^s, W^u$ be the stable and unstable foliations of $(X, f)$. They satisfy the product property of proposition 4.2. Assume also that $F$ is compatible with the holonomies $h_{\cdot, \cdot}^s$ and $h_{\cdot, \cdot}^u$. Let $M$ be an ergodic $F$-invariant measure such that $\Pi_* M = m$. Assume that both exponents $\gamma^+$ and $\gamma^-$ vanish. Then, by proposition 4.4, there are sets $Z^u \subset X$ and $Z^s \subset X$ of full measure such that the disintegrations $M_x, x \in X$ are $m$-almost everywhere $\tilde{h}^u$-equivariant along unstable leaves on $Z^u$ and $\tilde{h}^s$-equivariant along $\tilde{h}^u$-equivariant along unstable leaves on $Z^s$.
stable manifolds on $Z^s$. Assume moreover that $m$ has a $(s, u)$-product structure (see definition 4.3). Then, using proposition 4.2, one sees that there is a version of the disintegration $M_x$ that is continuous in $x$ and holonomy-equivariant. This is the starting point for many applications. For example, we can now discuss the generalization of Furstenberg theorem 2.2 to general Hölder continuous cocycles over a hyperbolic system:

**Theorem 4.5.** Let $(Y, f)$ be a topologically mixing hyperbolic system and $A : Y \to SL(2, \mathbb{R})$ an irreducible Hölder continuous cocycle. Let $m$ be an $f$-invariant measure with a $(s, u)$-product structure and $\chi \ge 0$ the $m$-a.e. value of

$$ \lim_{n \to \infty} \frac{1}{n} \log \|A(f^n y) \circ \dots \circ A(y)\|. $$

If $\chi = 0$, there exists a continuous $B : Y \to SL(2, \mathbb{R})$ such that $(B(y)) \circ A(y) \circ (B(fy))^{-1}$ is either the identity for all $y$, or a nilpotent matrix for all $y$ or a rotation for all $y$.

*Proof.* We indicate the main steps of the proof, referring to [V08] for details. We consider the system $(Y, f, m)$, the bundle $\mathcal{E} := Y \times \mathbb{P}^1$ and the Hölder continuous bundle of diffeomorphisms given by $\mathcal{F}_y([\xi]) := [A(y)\xi]$.

We have $|D_\xi F_y| = \frac{\|\xi\|^2}{\|A\xi\|^2}$ and thus, for any $\mathcal{F}$-invariant measure $M$ that projects on $m$, the exponent $\gamma^- = -2\chi = 0$. We may assume $M$ ergodic. Let $M_y$ be a family of disintegrations of $m$ for $\Pi : \mathcal{E} \to Y$. By (3.1), $A(y)_*M_y = M_{fy}$ and therefore the type of $M_y$ is $f$-invariant. By ergodicity, this type is $m$-a.e. constant and either a finite sum of equal Dirac measures or continuous.

In order to apply proposition 4.4, we have to find holonomies $\tilde{h}^s, \tilde{h}^u$ compatible with $\mathcal{F}$. In other words, we have to lift the foliations $\mathcal{W}^s, \mathcal{W}^u$ to Hölder continuous stable and unstable foliations for $\mathcal{F}$ on $\mathcal{E}$. We cannot apply proposition 2.13 because the system $(\mathcal{E}, \mathcal{F})$ is not partially hyperbolic: the decomposition with $\tilde{E}^s = E^s \otimes \{0\}, \tilde{E}^c = \{0\} \otimes T\mathbb{P}^1, \tilde{E}^u = E^u \otimes \{0\}$ is continuous, but the expansion/contraction in the central direction is not uniform, and not, a priori, dominated by the other expansion/contraction coefficients. But the exponent in the central direction is 0 and we can use Pesin theory to find $M$-a.e. stable and unstable foliations for $\mathcal{F}$ on $\mathcal{E}$. In terms of local holonomies, we find holonomies $\tilde{h}^s, \tilde{h}^u$ compatible with $\mathcal{F}$, but the Hölder continuity property becomes, for $m$-a.e. $y \in Y$, and all $z \in W_{y, toc}^\rho$,

there is $C(y), \beta$ such that for $\xi, \eta \in \mathbb{P}^1$, $d(\xi, \eta) \le \delta$, $d(\tilde{h}_{y,z}^\rho \xi, \tilde{h}_{y,z}^\rho \eta) \le C(y)(d(\xi, \eta))^\beta$.

Moreover, the function $C$ are positive $m$-a.e. and satisfy $C(f^\pm y) \le e^\epsilon C(y)$. The continuity of the mappings $(x, y) \mapsto \tilde{h}_{x,y}^s, \tilde{h}_{x,y}^u$ are also localized: there are compact sets $K_\ell \subset Y$ such that $\lim_{\ell \to \infty} m(K_\ell) = 1$, $C$ is positive on $K_\ell$ and $(x, y) \mapsto \tilde{h}_{x,y}^s, \tilde{h}_{x,y}^u$ are continuous on $K_\ell$. The arguments of the proofs of propositions 3.7 and 4.4 still work and we obtain that there are versions of the conditional measures $M_y$ on $\mathbb{P}^1$ that are holonomy equivariant $m$-a.e. on stable manifolds and versions of the conditional measures $M_y$ on $\mathbb{P}^1$ that are holonomy equivariant $m$-a.e. on unstable manifolds. Using the $(s, u)$-product structure of the measure $m$, we see that there is a version of $y \mapsto M_y$ that is continuous and holonomy equivariant on $[K_\ell, K_\ell]$
and therefore constant on $\cup_{\ell}[K_{\ell}, K_{\ell}]$. Let $\nu$ be the measure on $\mathbb{P}^1$ such that for $y \in \cup_{\ell}[K_{\ell}, K_{\ell}]$, $M_y = \nu$. We have, for $y \in \cup_{\ell}[K_{\ell}, K_{\ell}]$,

$$A(y)_{*}\nu = \nu.$$

In the general $SL(d, \mathbb{R})$, $d > 2$, case, one could conclude by looking at periodic points in $[K_{\ell}, K_{\ell}]$ and using e.g. [AKL18]. Here in dimension 2, we can use a trick from [ADZ23].

Observe that the action of $SL(2, \mathbb{R})$ on $\mathbb{P}^1$, the space of directions in $\mathbb{R}^2$, is conjugated, by $\theta \mapsto \cot \theta$ to the action on $\mathbb{R} \cup \{\infty\}$ by Moebius transformations, namely

$$\text{for } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad A \cdot x = \frac{ax + b}{cx + d}.$$

For $y \in [K_{\ell}, K_{\ell}]$, the measures $\tilde{M}_y := \cot_* M_y$ are equivariant under the holonomies $\tilde{H}_{x,y}^u := \cot \circ \tilde{h}_{x,y}^u \circ \cot^{-1}$, $\tilde{H}_{x,y}^s := \cot \circ \tilde{h}_{x,y}^s \circ \cot^{-1}$ and depend continuously on $y$. We use the conformal barycenter as defined by [DE86], namely

**Proposition 4.6.** ([DE86], Proposition 1) For each probability measure $\nu$ on $\mathbb{R} \cup \{\infty\}$ containing no atom of mass at least $1/2$, there is an unique point $b(\nu)$ in the hyperbolic upper half-space $\mathbb{H}^2$, called the conformal barycenter of $\nu$, so that the map $\nu \mapsto b(\nu)$ is equivariant under the Möbius transformations, that is, $b(A_*\nu) = A \cdot b(\nu)$ for each $A \in SL(2, \mathbb{R})$.

By invariance (3.1) and ergodicity, the measure $\cot_* \nu$ either is the average of two Dirac measures or is one Dirac measure, or satisfies the hypothesis of proposition 4.6. We conclude in each case:

(1) Assume that the measure $\cot_* \nu$ is the average of two Dirac measures $\delta_{\alpha}$ and $\delta_{\beta}$. Let $B \in SL(2, \mathbb{R})$ be such that, $B \cdot \alpha = 0$, $B \cdot \beta = \infty$. Then the matrix $BA(y)B^{-1}$ is diagonal for all $y$. Denote $a(y)$ its upperleft element. We choose $a(y) > 0$. The mapping $y \mapsto a(y)$ is Hölder continuous and satisfies, for all $f$-invariant ergodic measure $\mu$ such that $\text{support}(\mu) \subset [K_{\ell}, K_{\ell}]$, $\int \log a(y) d\mu = 0$. By Livschitz theorem, there is a Hölder continuous function $b(y)$ such that $\log a(y) = \log b(fy) - \log b(y)$. Setting $B(y) := \text{Diag}(b^{-1}(y), b(y)) \circ B$, we find that, for $m$-a.e. $y$,

$$B(y) \circ A(y) \circ B^{-1}(fy) = \text{Id}.$$

(2) Assume that the measure $\cot_* \nu$ is a single Dirac measure $\delta_{\alpha}$. Let $B \in SL(2, \mathbb{R})$ be such that, $B \cdot \alpha = 0$. Then the matrix $BA(y)B^{-1}$ is upper-triangular for all $y$. As above, we find $B(y) := \text{Diag}(b^{-1}(y), b(y)) \circ B$ such that for $m$-a.e. $y$, $B(y) \circ A(y) \circ B^{-1}(fy)$ is upper triangular with 1s on the diagonal.

(3) Otherwise, we can apply proposition 4.6 and there is $b \in \mathbb{H}^2$ which is invariant by the Möbius action of $m$-a.e. $A(y)$. Let $B \in SL(2, \mathbb{R})$ be such that, $B \cdot b = i$. Then the matrix $BA(y)B^{-1}$ is a rotation matrix for all $y$.

□
With a lot of care, one can use these ideas in many circumstances. For instance, one can generalize Furstenberg theorem 2.2 to a Hölder continuous cocycle $A : X \to SL(d, \mathbb{R})$ over a volume preserving diffeormorphism without 0 exponent.

**4.2. Random diffeomorphisms.** In this section, we consider a (compact) manifold $Y$, a finite Riemannian volume $\lambda$ on $Y$ and a probability measure $\mu$ on the group $D_{\lambda}^2(Y)$ of $C^2$-diffeomorphisms of $Y$ preserving $\lambda$. We form the random product

$$g^{(n)}(y) := g_{n-1} \circ \dots \circ g_0(y),$$

where the $g_i \in D_{\lambda}^2(Y)$ are i.i.d. with distribution $\mu$.

We assume that $\int_Y \log \|D_y g\| < + \infty$, $\int_Y \log \|(D_y g^{-1})^{-1}\| < + \infty$. The exponents $\gamma^{\pm}(y)$ are given, for $(\mu^{\otimes Z} \times \lambda)$-a.e. $(x, y) \in (D_{\lambda}^2(Y))^{\otimes Z} \times Y$, by

$$\gamma^{+}(y) := \lim_{n \to \infty} \frac{1}{n} \log \|D_y g^{(n)}\|, \quad \gamma^{-}(y) := - \lim_{n \to \infty} \frac{1}{n} \log \|(D_y g^{(n)})^{-1}\|.$$

(It is easy to see that the limits exist $(\mu^{\otimes Z} \times \lambda)$-a.e. and are $\mu^{\otimes Z}$-a.e. constant). Observe that $\gamma^{+}(y) \ge \gamma^{-}(y)$ for $\lambda$-a.e. $y$ and that, since the diffeomorphisms preserve $\lambda$, $\int_Y \gamma^{-}(y) d\lambda(y) \le 0$.

We assume that the action of $\mu$ is ergodic: there is no non-trivial measurable subset of $Y$ which is invariant under $\mu$-a.e. $g$. Then, the exponents $\gamma^{\pm}(y)$ are constants $\gamma^{\pm} \lambda$-a.e.. If $W$ is a measurable linear subbundle $W \subset TY, W \ne \{0\}$ which is invariant by the action of $Dg$ for $\mu$-a.e. $g$, we can define in the same way the exponents $\gamma_W^{\pm}$ restricted to $W$.

**Definition 4.7.** Let $\mu$ be a probability measure on $D_{\lambda}^1(Y)$, with first moment. Set $\mu^{(n)}$ for the distribution of $g^{(n)}$. We say that $\mu$ is uniformly expanding if there exist $n_0 \in \mathbb{N}_{\ge 0}, C > 0$ such that, for all $y \in Y$, all $v \in T_y Y$,

$$\int_{D_{\lambda}^1(Y)} \log \|D_y g(v)\| d\mu^{(n_0)}(g) > C.$$

**Fact ([C06], see [CD25] for a proof)** A measure $\mu$ on $D_{\lambda}^1$ with first moment and acting ergodically on $Y$ is uniformly expanding if, and only if, there exists no measurable subfield $W \subset TY, W \ne \{0\}$, which is invariant by the action of $Dg$ for $\mu$-a.e. $g$ and such that the exponent $\gamma_W^{+}$ vanishes.

Uniform expansion is the key property to establish *Stiffness* of the action of $\mu$, namely that all stationary measures are invariant. This has been established by A. Brown and F. Rodriguez Hertz ([BRH17]) when $Y$ is a real surface, S. Cantat and R. Dujardin ([CD25]) for automorphisms groups of projective complex surfaces, A. Brown, A. Eskin, S. Filip and F. Rodriguez Hertz ([BEFRH]) for higher dimension real manifolds (in this case, one has to consider uniform expansion on all exterior products $\Lambda^k TY$, see [ES23]). The proofs use the factorization method of A. Eskin and M. Mirzakhani ([EM18]).

The Invariance Principle is used to understand the case when $\mu$ is not uniformly expanding. We form $X := (D_{\lambda}^2(Y))^{\otimes Z} \times Y$, $m = (\mu^{\otimes Z} \times \lambda)$, and $f(\omega, y) = (\sigma\omega, g_0y)$, where $\sigma$ is the shift transformation on $(D_{\lambda}^2(Y))^{\otimes Z}$. We form the bundle $\pi : \mathcal{E} \to X$, where $\mathcal{E} := ((D_{\lambda}^2(Y))^{\otimes Z} \times \mathbb{P}(W))$, and the bundle $C^1$-diffeomorphism $F$ defined, for $[v] \in \mathbb{P}(W_y)$ by $F(\omega, y, [v]) := [D_y g_0(v)]$.
**Proposition 4.8.** Assume that $\mu$ has bounded support $^5$ on $D_{\lambda}^3(Y)$ and that $\gamma_W^{\pm} = 0$ $\lambda$-a.e.. Then any F-invariant probability measure that projects on $m$ is of the form $M = (\mu^{\otimes Z} \times \Lambda)$, where $\Lambda$ is a measure on $\mathbb{P}(W)$ with disintegration $\Lambda = \int_Y \Lambda_y d\lambda(y)$ in such a way that, for $\mu$-a.e. $g$,
$$ (4.2) \qquad (D_y g)_* \Lambda_y = \Lambda_{gy}. $$

*Proof:* We may assume $M$ ergodic. Observe that the exponents $\Gamma^{\pm}(\mathcal{E}, F, m) = 0$ and thus $\Gamma^{-}(\mathcal{E}, F^{-1}, m) = -\Gamma^{+}(\mathcal{E}, F, m) = 0$. We may apply the Invariance Principle on the one hand, to $F$ and the $\sigma$-algebra $\mathcal{B}$ generated by $\{g_n, n \ge 0\}$ and $y$, and on the other hand to $F^{-1}$ and the $\sigma$-algebra $\mathcal{B}'$ generated by $\{g_n, n < 0\}$ and $y$. We obtain that the disintegrations $M_{\omega,y}$ depend only on $y$. We set, for $\lambda$-a.e. $y$, $\Lambda_y := M_y$.

We have $M = (\mu^{\otimes Z} \times \int_Y \Lambda_y d\lambda(y))$. The relation (3.1) yields that, for $\mu \times \lambda$-a.e. $(g, y)$, $(D_y g)_* \Lambda_y = \Lambda_{gy}$. $\square$

Remains to understand the property (4.2). This might be simplified by the co-cycle reduction theorem (related to Zimmer's Amenable Reduction Theorem)

**Theorem 4.9 ([ACO97]).** Let $\mu$ be a probability measure on $D_{\lambda}^2(Y)$, $W \subset TY$ a $\mu$-a.e. Dg-invariant field of subspaces. Then, there exists a measurable mapping $Q: (Y \times \mathbb{P}^{\dim W-1}) \to \mathbb{P}W$, with $Q(y, \xi) = (y, Q_y(\xi))$ and $Q_y$ is a projective mapping from $\mathbb{P}^{\dim W-1}$ to $\mathbb{P}W_y$ such that any family $\nu_y, y \in Y$ of measures on $\mathbb{P}^{\dim W-1}$ that satisfies

$$ ([D_y g] \circ Q_y)_* \nu_y = (Q_{gy})_* \nu_{gy} \quad \text{for } (\mu \times \lambda)\text{-a.e. } (g, y) $$

is $\lambda$-a.e. constant.

It follows that, if (4.2) is satisfied, there is a measure $\nu$ on $\mathbb{P}^{\dim W-1}$ such that $\nu_y = (Q_y)_* \nu$ for $\lambda$-a.e. $y$ and the projective mappings $Q_{gy}^{-1} \circ [D_y g] \circ Q_y$ ($\lambda$-almost) all belong to the group of projective mappings preserving $\nu$. By studying the possible subgroups of projective transformations preserving a probability measure on $\mathbb{P}^{\dim W-1}$, one can dynamically characterize the non-uniformly expanding case ([BRH17], [CD25], [BEFRH]). One can also prove ([BM20]), see also [OP22]) that a generic (in some sense) measure $\mu$ on $D_{\lambda}^2(Y)$ is uniformly expanding and gives rise to positive exponents. This extends [BGMV03], [V08] and many other partial results.

**4.3. Dynamically coherent partially hyperbolic systems.** Recall the definition 2.15 of a dynamically coherent partially hyperbolic dynamical system. Recall that the Hölder continuous lamination $W^c$ with leaves the connected components of

$$ W_x^c := W_x^{cs} \cap W_x^{cu} $$

integrates the direction $E_x^c$ and that we denote $h_{\cdot, \cdot}^c$ the holonomy mappings between local transversals to $W^c$. The mappings $h_{\cdot, \cdot}^c$ preserve the local $W^{cs}, W^{cu}$ leaves. For the sake of presentation, we assume in this section that the holonomies $h_{x,y}^c$ are

<footnote>$^5$We apply theorem 3.1 to the derivative action on $\mathbb{P}(W)$. Given the form of the statement of theorem 3.1 we gave, we have to assume bounded support on $D_{\lambda}^3(Y)$.</footnote>
defined for all $x, y$ in the same $W^c$ leaf, that the holonomies $h^s$ and $h^c$ (respectively $h^u$ and $h^c$) commute. In suitable local charts, transversally to the local center leaves, we have two transverse foliations $\bar{W}^s$ and $\bar{W}^u$, corresponding to the traces of $W^{cs}$ and $W^{cu}$. For such systems, we have the notion of partial $(s, c, u)$-product (definition 2.16) and following theorem 2.18, it seems to be a quite common property of invariant probability measures. Nevertheless, for completeness, we define the notion of partial $(s, c, u)$-product structure.

**Definition 4.10.** Assume that $(Y, \varphi)$ is partially hyperbolic and dynamically coherent. A measure $\nu$ is said to have a partial $(s, c, u)$-product structure if

* the conditionals $\nu_y^{cs}$ of the measure $\nu$ with respect to the partition in local $W^{cs}$-leaves have a $(s, c)$-product structure and

* the conditionals $\nu_y^{cu}$ of the measure $\nu$ with respect to the partition in local $W^{cu}$-leaves have a $(u, c)$-product structure.

If a measure has a partial $(s, c, u)$-product structure, then the projected measures on the leaves of $\bar{W}^s$ and $\bar{W}^u$ are related (by the local central holonomies) to the conditional measures on the original $W^s$ and $W^u$.

**Definition 4.11.** We say that the probability $\nu$ has a $(s, c, u)$-product structure if, locally, all holonomies $h^*$, where $* = s, u, c, cs, cu$, preserve the conditional measures on the local leaves of $W^{*'}$, where $*' = \{s, u, c\} \setminus *$.

In other words, a measure $\nu$ has a $(s, c, u)$-product structure if it is locally equivalent to a product in the coordinates given by $W^c$, $\bar{W}^s$ and $\bar{W}^u$. We have

**Proposition 4.12.** If the measure $\nu$ has a partial $(s, c, u)$-product structure and, locally, the transverse measures to $W^c$ has a $(s, u)$-product structure with respect to $\bar{W}^s$ and $\bar{W}^u$, then the measure $\nu$ has a $(s, c, u)$-product structure.

This follows from lemma 6.5. Using the absolute continuity of the foliations $W^s$ and $W^u$ ([A67]), lemma 6.5 also gives that an invariant probability measure absolutely continuous with respect to the Lebesgue measure has a $(s, c, u)$-product structure as soon as the conditionals on $W^{cs}$ and $W^{cu}$ themselves have a product structure with respect to $h^s$ and $h^c$ (respectively $h^u$ and $h^c$).

Examples of dynamically coherent partially hyperbolic systems may have compact center leaves, have one-dimensional center leaves which are invariant (defined as *discretized geodesic flows*, or have the property that the action on the central leaves is quasi-isometric (for instance, this is the case for non hyperbolic ergodic automorphisms of the torus)). Invariance principle in these cases says that an invariant ergodic probability measure such that the central exponents vanish has strong invariance property with respect to the central foliation.

In the case of dynamically coherent partially hyperbolic systems with compact center leaves, the partition defined by the foliation $W^c$ is measurable and the associated conditional measures $m_y^c$ are defined $m$-a.e. ([AVW22] Proposition 3.7). In the general case of non-compact center leaves, the partition defined by the foliation $W^c$ may not be measurable. We will still denote $m_y^c$ the family of measures on
the central leaves, defined up to multiplication by a constant, on the leaves that
are proportional to the local disintegrations (see section 6.1 and [AVW15] section
3.2).

Theorem 2.18 shows that for compact extensions of hyperbolic systems, ergodic invariant measures with 0 center Lyapunov exponents are partial $(s, c, u)$-products. In other words, that the classes of measures $m_y$ up to a constant factor are $h^{u}$-invariant for $m^{cu}$-a.e.y and are $h^{s}$-invariant for $m^{cs}$-a.e.y. In this section, we survey some recent examples of applications of Invariance Principles to dynamically coherent partially hyperbolic systems with some further features.

4.3.1. *Discretized Anosov flows*. Let $g_t, t \in \mathbb{R}$, be a one parameter group of smooth diffeomorphisms of a compact manifold $Y$. It is called an Anosov flow if each $g_t$ is partially hyperbolic with the central direction being the one-dimensional tangent to the orbits of the flow.

**Definition 4.13.** Let $Y$ be a compact manifold, $g$ a smooth diffeomorphism of $Y$. The system $(Y, g)$ is called a discretized Anosov flow if it is partially hyperbolic with dim $E^c = 1$, dynamically coherent and the central leaves $W^c(y)$ are invariant by $g$.

The basic example of an Anosov flow is the geodesic flow on a manifold of negative sectional curvature and the basic examples of a discretized Anosov flow are the time-one mapping of an Anosov flow and its small enough perturbations.

Recall that a partially hyperbolic system is called accessible if for any two points $y, z$ there exists a path $x_0 = y, \dots, x_j, \dots x_k = z$ such that for each $j, 0 \le j < k$, $x_j$ and $x_{j+1}$ are in the same $W^*$ leaf, with $* = s, u$.

Let $(Y, g)$ be a discretized Anosov flow and assume that the system $(Y, g)$ is accessible and that $g$ has no fixed point. Non-compact leaves are homeomorphic to $\mathbb{R}$. Since $g$ has no fixed point, we can orientate each non compact leaf so that $y < g(y)$. This orientation is preserved by stable and unstable holonomies and is globally defined on non compact leaves. It extend by accessibility to the compact leaves as well. The space $Z$ of the segments $[y, g(y))$ is a compact space. The mapping $\psi : Z \to Z$ defined by $\psi([y, g(y))) = [g(y), g^2(y))$ is an expansive homeomorphism with expanding and contracting canonical coordinates. It follows from [R83] that the system $(Z, \psi)$ is a Smale space in the sense of D. Ruelle [R04], Chapter 7.

**Theorem 4.14.** ([AVW15]) Let $(Y, g)$ be a discretized Anosov flow and assume that the system $(Y, g)$ is accessible and that $g$ has no fixed point. Let $m$ be a $g$-invariant ergodic measure and assume that the projection of $m$ to $Z$ has a $(s, u)$-product structure. Then

• either $g$ is the time-one map of a $m$-preserving flow,

• or there exist an integer $k$ and a subset $E \subset Y$ of full $m$-measure that cuts each $W^c$-leaf in exactly $k$ orbits of $g$.

*Proof.* Let $m$ be a $g$-invariant measure. The conditional measures $m_y$ of $m$ associated to $W^c$ are defined up to a multiplicative constant. We define, for $m$-a.e.
$y \in Y$, the measure $m_y$ on $W^c(y)$ by normalizing the conditional measure so that
$m_y([y, g(y)]) = 1$.<fnref n="6" />

Consider the central exponent $\gamma$:

$$ \gamma := \lim_{n \to \infty} \frac{1}{n} \log \|D_y g^n v\| \quad \text{for } v \neq 0 \in E_y^c. $$

The case $\gamma \neq 0$ has been studied by Ruelle and Wilkinson ([RW01]: if $\gamma < 0$, say, points in $[y, g(y))$ are attracted to some points and the measures $m_y$ are discrete. By ergodicity, the measures $m_y$ have to be an average of a constant number $k$ of Dirac measures. The set $E$ of the second alternative is made of their supports.

We are also in the second alternative if $\gamma = 0$ and almost all $m_y$ are discrete. By ergodicity, almost all $m_y$ are of the same type, so it remains to consider the case when $\gamma = 0$ and, for $m$-a.e. $y$, $m_y$ is continuous. By theorem 2.18, the mappings $y \mapsto m_y$ are $m$-a.e. equivariant by the local holonomies $h^u$ and $h^s$. Since $m$ has a $(s, u)$-product structure on $Z$, we see that $m$ has a $(s, c, u)$-product structure (recall proposition 4.12).

We can parametrize each universal cover $\widetilde{W}_y^c$ by $t_y : \widetilde{W}_y^c \to \mathbb{R}$, $t_y(z) := m_y([y, z))$ (or $t_y(z) := -m_y([z, y)))$. By uniqueness of the conditional measure, for $m$-a.e. $y$, if $z, w \in \widetilde{W}_y^c$, then $t_z(w) - t_y(w) = t_y(z)$.

Observe that, when we parametrize the (universal cover of the) center leaf of $y$ by $t_y$ for all $y$, the central holonomy $h_{y,z}^c : \widetilde{W}_y^c \to \widetilde{W}_z^c$ is the translation $R_t$ by $t = t_y(z)$. Using accessibility and the $(s, u)$-product structure on $Z$, we conclude that, for $m$-a.e. $y \in Y$, the measure $(t_y)_*m_y$ on $\mathbb{R}$ is invariant under the translation $R_t$, for $(t_y)_*m_{y}$-a.e. $t$. The only continuous measures on $\mathbb{R}$ with this property are proportional to the Lebesgue measure. With our normalization, for $m$-a.e. $y \in Y$, the measure $(t_y)_*m_y$ is Lebesgue. The one-parameter flow $\psi_t, t \in \mathbb{R}$, defined by the translations along the parameter $t_y$ of $W_y^c$ is $\lambda$-a.e. well-defined and preserves $m$. The transformation $g$ is indeed the time-one of a (measurable, a.e. defined) $m$-preserving one-parameter flow. $\square$

See [AVW15], [AVW22], section 9, for further regularity of the flow in the case when $(Y, g)$ is a volume preserving discretized Anosov flow in dimension 3. Orientability of $W^c$ is also proven in [AVW22], section 9, even in the case of fixed points for $g$.

### 4.3.2. Quasi-isometric central leaves.

**Definition 4.15.** Let $(Y, f)$ be a dynamically coherent partially hyperbolic smooth system. We say that $(y, f)$ has quasi-isometric central leaves if there exists numbers $K_0, C_0$, such that, if $x, y$ are on the same $W^c$ leaf, we have, for all $n \in Z$,

$$ (4.3) \qquad K_0^{-1} d^c(x, y) - C_0 \leq d^c(f^n x, f^n y) \leq K_0 d^c(x, y) + C_0, $$

where $d^c$ is the Riemannian distance on the $W^c$ leaves.

<footnote n="6">⁶In particular, if $W_y^c$ is a circle, $[y, g(y))$ is a $k$-cover of $W_y^c$ and $m_y(W_y^c) = 1/k$.</footnote>
Systems with compact center leaves and discretized Anosov flows have quasi-isometric central leaves. Let $m$ be an ergodic $f$-invariant probability measure. We are interested in showing $h^u$ and $h^s$ invariance of the conditional measures $m_y$.

We will consider the $m$-a.e. defined and constant exponents

$$ \gamma_c^+ := \lim_{n \to \infty} \frac{1}{n} \log \|D_y g^n|_{E_y^c}\| \quad \gamma_c^- := - \lim_{n \to \infty} \frac{1}{n} \log \|(D_y g^n)^{-1}|_{E_y^c}\|. $$

If the mapping $f$ has quasi-isometric central leaves, then $\gamma_c^\pm = 0$. The following result is obtained in a recent preprint by S. Crovisier and M. Poletti.

**Theorem 4.16.** ([CP22]) Let $(Y, f)$ be a dynamically coherent partially hyperbolic smooth system, with quasi-isometric central leaves. Let $m$ be an ergodic $f$-invariant probability measure on $Y$. Then, $m$ has a partial $(s, c, u)$-product structure.

*Proof.* We assume that $\gamma_c^+ \le 0$ and show that the conditionals $m_y^{uc}$ on local $W^{uc}$ leaves are $(u, c)$-product for $m$-a.e. $y$. The proof is the same for the $(s, c)$-product property of the conditionals $m_y^{sc}$ when $\gamma_c^- \ge 0$.

The first observation is a non-trivial result in [CP22], that the problem makes sense: local central unstable leaves are nicely foliated by local unstable leaves and local centre leaves. More precisely:

**Proposition 4.17.** ([CP22] Theorem C) Let $(Y, f)$ be a dynamically coherent partially hyperbolic $C^1$ system, with quasi-isometric central leaves. Let $m$ be a $f$-invariant probability measure on $Y$. Then, there exists a full measure set $Z \subset Y$ such that for $y, z \in Z$ with $W_z^{cu} = W_y^{cu}$, the following holds:

For any $w \in W_y^c$, the leaves $W_w^u, W_z^c$ intersect at a unique point, denoted $h_{y,z}^u(w)$. The map $h_{y,z}^u : W_y^c \to W_z^c$ is a homeomorphism.

Let $\pi : \tilde{Y} \to Y$ be the continuous bundle over $Y$ defined by $\tilde{Y} = \{(y, w), w \in W_y^c\}$. We set $p : \tilde{Y} \to Y$ for the projection $p((y, w)) = w$. The fiber above $y$ is endowed with the metric $d_y$ coming from the Riemannian metric on $W_y^c$. Let $F$ be the bundle diffeomorphism of $\tilde{Y}$ defined by $F(y, w) := (fy, fw)$. We have $\pi \circ F = f \circ \pi, p \circ F = f \circ p$. For $t > 0$, write $\tilde{Y}_t := \{(y, w), w \in W_y^c; d(y, w) \le t\}$.

Assume that $f$ has quasi-isometric center leaves with constants $K_0, C_0$. Then, for $t > K_0C_0$,

$$ (4.4) \qquad \tilde{Y}_{K_0^{-1}t-C_0} \subset \cap_{n \in \mathbb{Z}} F^n \tilde{Y}_t \subset \cup_{n \in \mathbb{Z}} F^n \tilde{Y}_t \subset \tilde{Y}_{K_0t+C_0}. $$

Set $\mathcal{E}_t := \cap_{n \in \mathbb{Z}} F^n \tilde{Y}_t$. The set $\mathcal{E}_t$ is closed and $F$-invariant. By (4.4), for $t > K_0C_0$, the set $\mathcal{E}_t$ is compact and has non-empty interior in each fiber. Moreover, for $K_0/C_0 < t < t'$, $\mathcal{E}_t \subset \mathcal{E}_{t'}$.

Recall that the conditional measures $m_y$ on $W^c$ leaves are defined up to multiplication by a constant on the leaf. Such equivalence classes are invariant by $f$. The union of the supports of the measures $m_y$ is an invariant set $Z$ of full measure. For $y \in Z$, we endow the fiber above $y$ with the measure $M_y$ proportional to $m_y$, normalized by $M_y(\mathcal{E}_{K_0/C_0+t_0}) = 1$, for some $t_0 > 0$. The measure $M := \int M_y dm(y)$
is F-invariant and $M(\mathcal{E}_{K_0/C_0+t_0}) = 1$. Observe that we can choose $t_0$ in such a way that the boundary of $\mathcal{E}_{K_0/C_0+t_0}$ is negligible.

Set $\mathcal{E} := \mathcal{E}_{K_0/C_0+t_0}$. We will consider the system $(\mathcal{E}, F, M)$.

**Lemma 4.18.** ($\mathcal{E}, M$) is a compact manifold with negligible boundary. The mapping $F: \mathcal{E} \to \mathcal{E}$ preserves the boundary, is $C^{1+\alpha}$ and partially hyperbolic with decomposition $\tilde{E}^s \oplus \tilde{E}^c \oplus \tilde{E}^u$, where

$$ \tilde{E}^s := E^s \otimes \{0\}, \quad \tilde{E}^c := (E^c \otimes \{0\}) \oplus (\{0\} \otimes TW^c), \quad \tilde{E}^u := E^u \otimes \{0\}. $$

*Proof.* Only the regularity of $F$ is not immediate from the construction. But the decomposition is Hölder continuous and the derivative $DF$ is diagonal in this decomposition, with Hölder continuous components. $\square$

The proof of theorem 4.16 follows the scheme of the proof of theorem 2.18: Let $\zeta^u$ be a measurable partition of $Y$ subordinated to the unstable foliation of $(Y, f)$. We choose $\zeta^u$ small enough that, for $(y, z) \in Z$, $(y, w) \in \tilde{Y}_t$, we have $(z, h_{y,z}^u(w)) \in \tilde{W}_{loc}^u(y, w)$. We set, for $(y, w) \in \mathcal{E}$,

$$ \xi^{cu}(y, w) := \pi^{-1}(\zeta^u(\pi(y))) \text{ and } \xi^u(y, w) := \xi^{cu}(y, w) \cap \tilde{W}_{loc}^u(y, w). $$

The partitions $\xi^u$ and $\xi^{cu}$ are measurable partitions with the following properties:

(1) the atoms of $\xi^u$ (respectively $\xi^{cu}$) are subsets of $\tilde{W}^u$ leaves (respectively $\tilde{W}^{cu}$-leaves),

(2) $\xi^u$ refines $\xi^{cu}$,

(3) for M-a.e. $(y, w), \cap_{n \ge 0} F^{-n} \xi^u(F^n(y, w)) = \{(y, w)\}$ and

(4) there is a finite entropy partition $C$ such that $F^{-1}\xi^u = C \lor \xi^u$.

Observe that, since the partition $\xi^u$ is subordinated to $\tilde{W}^u$, the entropy $H(F^{-1}\xi^u|\xi^u)$ is the entropy of the system $(\mathcal{E}, F, M)$. Indeed, this equality follows from the $C^{1+\alpha}$ version of proposition 2.20 ([B22]). Then property (4) follows from a general result in entropy theory (see proposition 6.2).

**Proposition 4.19.** We have $H(C|\xi^u) \le H(C|\xi^{cu})$ with equality if and only if, for M-a.e. $(y, w) \in \mathcal{E}$, the conditionals $M_{y,w}^{cu}$ are $(u, c)$-products.

*Proof.* The proof is the same as the proof of proposition 2.19. The only modification is that the partition $C$ has finite entropy instead of being finite. This implies that, for M-a.e. $(y, w)$, the entropy of the partition $C$ for the measure $M_{y,w}^{cu}$ is finite. The inequality still follows by summing over the atoms of $C$. In the case of equality, we also have equality for individual atoms at M-a.e. $(y, w)$, since the series converges. We have $H(F^{-n}\xi^u|\xi^u) = nH(F^{-1}\xi^u|\xi^u) \le nH(F^{-1}\xi^{cu}|\xi^{cu}) = H(F^{-n}\xi^{cu}|\xi^{cu})$ for all $n > 0$. By property (4), $F^{-n}\xi^u = \bigvee_{j=0}^{n-1} F^{-j}C \lor \xi^u$. The partition $\bigvee_{j=0}^{n-1} F^{-j}C$ has finite entropy and in case of equality, we still have that, for M-a.e. $(y, w)$, the measures $M_{y,w}^u$ and $M_{y,w}^{cu}$ coincide on $\bigvee_{j=0}^{n-1} F^{-j}C$. The $(u, c)$-product property follows from property (3). $\square$
Observe that we have equality $H(F^{-1}\xi^u|\xi^u) = H(F^{-1}\xi^{cu}|\xi^{cu})$ because $H(F^{-1}\xi^u|\xi^u)$ is the entropy of the system $(\mathcal{E}, F, M)$ and $H(F^{-1}\xi^{cu}|\xi^{cu})$ is at most the entropy since the partition $\xi^{cu}$ is increasing. So, by proposition 4.19, the conditionals $M_{y,w}^{cu}$ are $(u, c)$-products. In particular, the local unstable manifolds $\bar{W}_{loc}^u$ preserve the conditional of $M$ along the fibbers $\pi^{-1}\{y\}$. In other words, the local holonomies $h_{y,z}^u$ preserve the measure $m_y$ up to a constant, on a set containing $\{w \in W_y^c, d_{W^c}(y, w) < K_0^{-1}t_0 - C_0\}$. Since this property holds for arbitrary large $t_0$, this shows that the conditionals $m_y^{uc}$ on local $W^{uc}$ leaves are $(u, c)$-product for $m$-a.e. $y$. $\square$

## 5. MISCELLANEOUS

In most developments, the Invariance Principle is a tool that is used in conjunction with other ideas. The examples of section 2 are somehow the most direct applications of the IP. In this section, we mention possible further extensions.

An important feature is remark 3.3.2. It often implies that the set of objects controlled by an IP is stable by perturbations. Details are delicate and depend on the problem, but this stability is one extra feature of many of the following results.

### 5.1. Positivity of the exponent of a linear cocycle.
The natural application is to decide when the Lyapunov exponents of a linear cocycle are not all the same. Theorem 4.5 is the basic example and there are many refinements and variants (see e.g. [BGMV03], [V08], [AV10], [ASV19], [AVW22], [CMY] and [ADZ23]). Often, in high (or not so high) regularity, one finds that the set of linear cocycles in $SL(2, \mathbb{R})$, say, with positive exponent is generic in some open set, a sharp contrast with the genericity of 0 exponent in low regularity (see [B02], [BV05]).

### 5.2. Stochastic flows.
A common source of random diffeomorphisms is the flow of random diffeomorphisms associated to a Stochastic Partial Differential Equation. In general, the reference space need not be locally compact, but is still a separable metric space, for instance, some Sobolev space of functions on a domain. See [BBPS] for the formulation and the applications of the Invariance Principle in that setting.

The flow of random diffeomorphisms associated to a Stochastic Ordinary Differential Equation connects the theory of random action of (pseudo-)groups of diffeomorphisms with the properties of diffusions on the leaves of a foliation (see [DK07]). Indeed, the discrete Markov process defined by the successive hits of a transversal by the trajectory of the leafwise diffusion can be obtained as the Markov process associated to a random diffeomorphism of the transversal. In many cases, it is possible to define a Lyapunov exponent for the random product and vanishing of this exponent is associated to the foliation admitting an invariant transversal measure.

### 5.3. $(s, u)$-product structure.
If the system $(Y, f, m)$ is partially hyperbolic, it might be important to assume that the invariant measure $m$ has a $(s, u)$-product
structure. Then, the IP might provide a partial $(s, c, u)$ product structure (see e.g. section 2.4), and proposition 4.12 would yield a $(s, c, u)$-product structure. This is the case for measures of maximal entropy and equilibrium measures for a Hölder continuous potential (see e.g. [RHRHTU], [UVY], [CP22]), or a Lebesgue measure when it is invariant, or physical measures (see e.g. [VY13], [CP22]), or else u-Gibbs measures ([UVYY]).

5.4. **Distinct exponents.** Consider an independent product of matrices in SL$(d, \mathbb{R})$, $d > 2$, with distribution $\mu$. By the subadditive ergodic theorem, there are exponents

$$ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_d, \quad \text{with} \quad \sum_j \lambda_j = 0. $$

The analog of theorem 2.2 gives conditions to ensure that $\lambda_1 > \lambda_d$ and all the discussion above concerns possible extensions and variants. It might be of interest to decide whether exponents are all distinct. In the independent case, the known criteria involve the support of the measure $\mu$. Y. Guivarc'h and A. Raugi ([GR89]) showed that this is the case as soon as there one matrix in the semi-group generated by the support of $\mu$ with moduli of eigenvalues all distinct. I. Gol'dsheid and G. Margulis ([GM89]) showed that it suffices that the semi-group generated by the support of $\mu$ is Zariski dense in SL$(d, \mathbb{R})$. There isn't a generalization of the criteria of [GR89] and [GM89] as neat as theorem 3.1, but it is still possible to obtain nice results, in particular

* The proof of Zorich-Kontsevich conjecture by A. Avila and M. Viana ([AV07]): on each transitive stratum of the Teichmüller flow, for the Masur-Veech invariant probability measure, the Zorich-Kontsevich cocycle has all exponents distinct. See for background and consequences, in particular of $\lambda_1 > \lambda_2$.

* The analog of section 5.1 for linear cocycles or for the derivative cocycle of a smooth diffeomorphisms: generically in some regularity, either all exponents are the same, or all exponents are distinct and the Oseledets splitting is dominated. See e.g. [BV05] [AB12], [RH13], [BPLV], [Mit22] and the references therein.

# 6. APPENDIX

In this Appendix, we recall notions and properties from measure theory that are used in the text.

## 6.1. Lebesgue space, measurable partition, disintegration.
A complete probability space $(X, \mathcal{A}, m)$ is called a Lebesgue space if there exist a measurable set $E$ with full measure and a countable family of measurable sets $\{C_j\}_{j \in \mathbb{N}}$ such that any point $x \in E$ is the intersection of $E$ and the sets $C_j$ that contain $x$. A Polish metric space endowed with a Borel probability measure and the completed Borel $\sigma$-algebra is a Lebesgue space.

Let $(X, \mathcal{A}, m)$ be a Lebesgue space. A partition $\xi$ of $X$ is called a measurable partition if there exist a measurable set $F$ with full measure and a countable family
of measurable sets {$D_j$}$_{j \in \mathbb{N}}$ such that for any element $P \in \xi$, $P \cap F$ is the intersection of $F$ and the sets $D_j$ that contain $P$. Let $\xi$ be a measurable partition and denote $(\hat{X}, \hat{A}, \hat{m})$ the quotient space. By definition, $(\hat{X}, \hat{A}, \hat{m})$ is a Lebesgue space. The key property of measurable partitions is that they admit disintegrations.

**Theorem 6.1.** (Rokhlin [R49]) Let $\xi$ be a measurable partition of the Lebesgue space $(X, \mathcal{A}, m)$, $(\hat{X}, \hat{\mathcal{A}}, \hat{m})$ the quotient space. There exists a family $P \mapsto m_{\xi}^{P}$ of probability measures on $X$ such that:

(1) for $\hat{m}$-a.e. $P \in \xi$, $m_{\xi}^{P}(P) = 1$,

(2) for any $\mathcal{A}$ measurable non-negative function $\psi$ on $X$, the function $P \mapsto \int_{P} \psi(x) dm_{\xi}^{P}(x)$ is $\hat{\mathcal{A}}$ measurable and

(3) $\int_X \psi dm = \int_{\hat{X}} \left( \int_P \psi(x) dm_{\xi}^{P}(x) \right) d\hat{m}(P)$.

The family $m_{\xi}^{P}$ is $\hat{m}$-essentially unique and is called the disintegration of $m$ with respect to the partition $\xi$.

In this paper, we met different kinds of measurable partitions. If we have a measurable fibration $\pi : X \to \hat{X}$, then the partition in fibers $\pi^{-1}\hat{x}, \hat{x} \in \hat{X}$ form a measurable partition if the quotient space is a Lebesgue space. In the case of a continuous foliation $\mathcal{W}$ of a Polish space with compact center leaves, the partition defined by the leaves is measurable and the associated conditional measures $m_{\hat{x}}$ are defined $\hat{m}$-a.e. ([AVW22] Proposition 3.7). In the general case of non-compact leaves, the partition defined by the leaves may not be measurable and we can only define local conditional measures on flow boxes for the foliation. It turns out that these conditional measures are proportional to one another on the overlap of different flow-boxes. We will still denote $m_{\hat{x}}$ the family of Radon measures on the leaves, defined up to multiplication by a constant, that are proportional to the local disintegrations (see [AVW15]) and we call them conditional measures on the leaves of $\mathcal{W}$.

Observe that, if $(X, \mathcal{A}, m)$ is a Lebesgue space and $\mathcal{B} \subset \mathcal{A}$ is a sub-$\sigma$-algebra, there exists a measurable partition $\xi$ such that the disintegration $P \mapsto \int \psi dm_{\xi}^{P}$ gives the conditional expectation of the function $\psi$ with respect to $\mathcal{B}$, but, in general, the elements of $\xi$ are not the atoms of $\mathcal{B}$ and, in particular for foliations with non-compact leaves, the elements of $\xi$ are non-trivial union of leaves and the disintegration is different from the conditional measures of the previous paragraph.

**6.2. Entropy.** Let $(X, \mathcal{A}, m)$ be a Lebesgue space, $f$ an invertible bimeasurable transformation preserving $m$, $\xi$ a measurable partition. The partition $\xi$ is said to be increasing if the partition $f^{-1}\xi$ is finer than the partition $\xi$. The entropy $h_m(f)$ is given by (see [R67])

$$ (6.1) \qquad h_m(f) := \sup_{\xi, f^{-1}\xi \prec \xi} H(f^{-1}\xi|\xi). $$

Assume $h_m(f) < +\infty$. An increasing measurable partition is called perfect if $H(f^{-1}\xi|\xi) = h_m(f)$. If $\mathcal{C} = \{C_j, j \in \mathbb{N}\}$ is a countable measurable partition with
finite entropy that generates $\mathcal{A}$, then the partition $\xi := \bigvee_{j \ge 0} f^j C$ is perfect. Conversely

**Proposition 6.2.** ([R67]) Let $(X, \mathcal{A}, m, f)$ be a finite entropy dynamical system and $\xi$ a perfect increasing partition. Then, there exist a partition $C = \{C_j, j \in \mathbb{N}\}$ with finite entropy such that $\xi := \bigvee_{j \ge 0} f^j C$.

### 6.3. Averaging in Besicovich spaces.

**Definition 6.3.** A metric space $X$ is called a Besicovich space if there exists a constant $C$ with the following property: given any function $x \mapsto r(x) > 0$ on $X$ such that $X \subset \cup_x B(x, r(x))$, there is a subset $Y \subset X$ such that $X \subset \cup_{y \in Y} B(y, r(y))$, and no point $x \in X$ is covered by more than $C$ balls $B(y, r(y))$, $y \in Y$.

Subsets of Euclidean spaces are Besicovich, as well as metric spaces that are bilipschitz equivalents to subsets of Euclidean spaces. A Borel probability measure $m$ on a Besicovich space has the Lebesgue density property and the averages on balls have the weak (1,1) property. For us, this means that we have the following

**Proposition 6.4.** Let $X$ be a Besicovich space, $m, m'$ Borel probability measures on $X$, $g$ the $m$-a.e. defined Radon-Nikodym derivative $g = \frac{dm'}{dm}$. For $\delta > 0$, set

$$g_{\delta}(x) := \frac{\int_{B(x, \delta)} g(x) dm}{m(B(x, \delta))} = \frac{m'(B(x, \delta))}{m(B(x, \delta))},$$

Then, for $m$-a.e. $x$, $\lim_{\delta \to 0} g_{\delta}(x) = g(x)$.

Moreover, if we set, for $N$ large,

$$g^N(x) := \sup\{g(x), e^{-N}\}, \quad g_{\delta}^N(x) := \frac{\int_{B(x, \delta)} g^N(y) dm(y)}{m(B(x, \delta))} \quad \text{and} \quad g_{\delta}^* := \sup_{\delta' \le \delta} g_{\delta'}^N,$$

the functions $-\log g^N, -\log g_{\delta}^*$ are $m$-integrable, we have $-\log g \ge -\log g^N \ge -\log g_{\delta}^*$ and

$$\lim_{\delta \to 0} \int \log \frac{g_{\delta}^*}{g^N} dm = 0.$$

See e.g. [Les21], lemmas 8,9 and 10 for a proof of these classical facts.

### 6.4. Quasi-independence.
Two random variables $Y$ and $Z$ are said to be quasi-independent if the distribution of $(Y, Z)$ has the same negligible sets as the product of the distributions of $Y$ and $Z$. They are said to be quasi-independent conditionally with respect to a third random variable $X$ if, for a.e. value $x$, $Y$ and $Z$ are quasi-independent for the conditional distributions $P_x(dy)$ and $P_x(dz)$.

**Lemma 6.5.** Let $X, Y, Z$ be random variables defined on a Lebesgue space $(\Omega, P)$. Assume that $X$ and $Y$ are quasi-independent conditionally with respect to $Z$, that $X$ and $Z$ are quasi-independent conditionally with respect to $Y$ and that $Y$ and $Z$ are quasi-independent. Then, $X, Y$ and $Z$ are quasi-independent in the sense that the distribution of $(X, Y, Z)$ has the same negligible sets as the product of the distributions of $X, Y$ and $Z$.
*Proof.* Let {$A_n$}$_{n\ge 0}$, {$B_m$}$_{m\ge 0}$, {$C_p$}$_{p\ge 0}$ be finer and finer sequences of finite partitions generating $\bar{X}$, $Y$ and $Z$. Our hypothesis is that the following limits exist and are non-zero $P$-a.e. as $n, m$ and $p$ go to infinity:

$$ \lim_{n,m,p \to \infty} \frac{P(A_n \cap B_m) P(C_p \cap B_m)}{P(A_n \cap B_m \cap C_p) P(B_m)}, \quad \lim_{n,m,p \to \infty} \frac{P(A_n \cap C_p) P(C_p \cap B_m)}{P(A_n \cap B_m \cap C_p) P(C_p)} $$

and $\lim_{m,p \to \infty} \frac{P(B_m) P(C_p)}{P(B_m \cap C_p)}$. After product and simplifications, it reads as the following limit exist and is non-zero $P$-a.e. as $n, m$ and $p$ go to infinity:

$$ \lim_{n,m,p \to \infty} \frac{P(A_n \cup B_m) P(A_n \cap C_p) P(B_m) P(C_p) P(A_n)}{P(A_n \cup B_m \cup C_p) P(A_n)} $$

Both ratios converge toward densities, which are positive $P$-a.e. since their product does not vanish $P$-a.e.<fnref n="7" /> Thus, $\lim_{n,m,p \to \infty} \frac{P(B_m) P(C_p) P(A_n)}{P(A_n \cup B_m \cup C_p)}$ is non-zero $P$-a.e. □

**6.5. Pliss Lemma.** We recall the following lemma, due to V. Pliss ([P72])

**Lemma 6.6.** Given $A \ge c_2 > c_1 > 0$, let $\beta := (c_2 - c_1)/(A - c_1)$. Then, given any real numbers $a_1, \dots, a_N$ such that

$$ \sum_{j=1}^{N} a_j \ge c_2 N \quad \text{and} \quad a_j \le A \quad \text{for every} \quad 1 \le j \le N, $$
there are $\ell > \beta N$ and $1 < n_1 < \dots < n_\ell \le N$ so that

$$ \sum_{j=n+1}^{n_i} a_j \ge c_1(n_i - n) \quad \text{for every} \quad 0 \le n < n_i \quad \text{and} \quad i = 1, \dots, \ell. $$

See e.g. [ABV00], lemma 3.1, for a proof. We will use a dynamical version of Pliss Lemma. Let $(X, \mathcal{A}, m; f)$ be an ergodic transformation of a Lebesgue space, $\varphi \in L^\infty(X, m)$, and choose $\varepsilon > 0$. An integer $n$ is called $\varepsilon$-good for $x$ if for all $j, 0 \le j < n$, we have

$$ \sum_{k=j+1}^{n} \varphi(f^k x) \ge (n-j) \left( \int \varphi dm - \varepsilon \right). $$

**Corollary 6.7.** Let $(X, \mathcal{A}, m; f)$ be an ergodic transformation of a Lebesgue space, $\varphi \in L^\infty(X, m)$, and choose $\varepsilon > 0$. There exist $\beta > 0$ such that, for $m$-a.e. $x \in X$, the asymptotic upper density of the integers that are $\varepsilon$-good for $x$ is at least $\beta$.

*Proof.* By replacing $\varphi$ by $\varphi + \text{Constante}$, we may assume that $\int \varphi dm - 2\varepsilon > 0$. Then the ergodic theorem shows that for $m$-a.e. $x$, and $N$ large enough, the sequence $\varphi(f^j x)$ satisfies the hypotheses of lemma 6.6 with $A = \|\varphi\|_\infty$, $c_2 = \int \varphi dm - \varepsilon/2$, $c_1 = \int \varphi dm - \varepsilon$. So, for $N$ large enough, there are at least $\beta N$ $\varepsilon$-good times for $x$, where $\beta = \frac{\varepsilon/2}{\|\varphi\|_\infty - \int \varphi dm + \varepsilon/2} > 0$. □

<footnote n="7">⁷Observe that the lemma with independence instead of quasi-independence is also true. The computation is the same, and now the product of the densities is 1. But by Jensen inequality, the integral of the logarithms of both densities is non-positive. So both densities are 1 and the three variables are independent.</footnote>
6.6. **Oseledets theorem.** In this subsection, we state the multiplicative ergodic theorem and the Pesin stable manifold theorem. There is the classical Oseledets theorem 6.8 and its two sided version (theorem 6.9).

We consider a probability space $(X, m)$ and an ergodic measure preserving transformation $T$ on $X$.

**Theorem 6.8.** [O68] For $g \in \text{GL}(d, \mathbb{R})$, set $|g| = \log \|g\|$, where $\|g\|$ is the norm of $g$ as an operator in the Euclidean space. Let $g : X \to \text{GL}(d, \mathbb{R})$ be measurable and such that $\int_X |g(x)^{\pm 1}| dm(x) < +\infty$. Set

$$X_n(x) := g(T^{n-1}x) \dots g(x).$$

There exists numbers $\lambda_1 > \dots > \lambda_\ell$, with multiplicities $d_1, \dots, d_\ell$, $\sum_{i=1}^\ell d_i = d$, and, for m-a.e. $x \in X$, a partial flag

$$f'(x) = \{0\} = U_{\ell+1} \subset U_\ell(x) \subset \dots \subset U_2(x) \subset U_1 = \mathbb{R}^d,$$

with $\dim U_i(x) = \sum_{j \ge i} d_j$, such that $\sum_{i=1}^\ell d_i \lambda_i = \int \log |\det g(x)| dm(x)$ and

$$(6.2) \qquad v \ne 0 \in U_i(x) \iff \lim_{n \to +\infty} \frac{1}{n} \log \|X_n(x)v\| \le \lambda_i.$$

The flag $f'(x)$ in theorem 6.8 is called the stable flag.

If the transformation $T$ is invertible $\bar{m}$-a.e., we can apply theorem 6.8 to the cocycle $(X, m, T^{-1}; g^{-1} \circ T^{-1})$. We obtain a flag $f(x)$ that is called the unstable flag. The stable flag and the unstable flags at $x$ are in general position and give rise to the Oseledets splitting:

**Theorem 6.9.** [O68] Assume moreover that the transformation $T$ is invertible m-a.e.. Then, there exists at m-a.e. $x \in X$ a decomposition

$$\mathbb{R}^d = F_1(x) \oplus \dots \oplus F_\ell(x),$$

with $\dim F_i(x) = d_i$ m-a.e. such that

$$(6.3) \qquad v \ne 0 \in F_i(x) \iff \lim_{n \to \pm\infty} \frac{1}{n} \log \|X_n(x)v\| = \lambda_i,$$

where, for $n < 0$, $X_n(x) := (g(T^{-n}x))^{-1} \dots (g(T^{-1}x))^{-1}$. Moreover, for all $i, 1 < i < \ell$,

$$(6.4) \qquad \lim_{n \to \pm\infty} \frac{1}{n} \log |\langle \oplus_{j \le i} F_j(T^n x), \oplus_{j > i} F_j(T^n x) \rangle| = 0.$$

## REFERENCES

[A67] D. V. Anosov. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Math. Inst., Vol. 90 (1967).

[AB12] A. Avila, J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Trans. Am. Math. Soc. Vol. 364 (6) (2012) 2883–2907.

[ABV00] J. F. Alves, C. Bonatti and M. Viana. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398.

[ACO97] L. Arnold, N.D. Cong and V. I. Oseledets, The essential range of a nonabelian cocycle is not a cohomology invariant, Israel J. Math. Vol. 116 (2000), 71–76.
[ADZ23] A. Avila, D. Damanik and Z. Zhang, Schrödinger operators with potentials generated by hyperbolic transformations: I-positivity of the Lyapunov exponent, *Invent. Math.* Vol. 231 (2) (2023) 851–927.

[AKL18] A. Avila, A. Kocsard and X. Liu. Livšic theorem for diffeomorphism cocycles. *Geom. Funct. Anal.* Vol. 28 (2018), 943–964.

[ASV19] A. Avila, J. Santamaría, and M. Viana. Holonomy invariance: rough regularity and applications to Lyapunov exponents. *Astérisque* Vol. 358 (2013), 13–74.

[AV07] A. Avila and M. Viana. Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. *Acta Math.* Vol. 198 (2007), 1–56.

[AV10] A. Avila and M. Viana. Extremal Lyapunov exponents: an invariance principle and applications, *Inv. math.* Vol. 181 (2010), 115–178.

[AVW15] A. Avila, M. Viana and A Wilkinson. Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows. *Journal of European Math Soc*, Vol.17 (2015) 1435–1462.

[AVW22] A. Avila, M. Viana and A Wilkinson. Absolute continuity, Lyapunov exponents and rigidity II: systems with compact center leaves. *Ergod. Th. & Dynam. Sys.*, Vol. 42 (2022), 437–490.

[B02] J. Bochi. Genericity of zero Lyapunov exponent; *Ergod. Th. & Dynam. Sys.* Vol. 22 (2002), 1667–1696.

[B22] A. Brown. Smoothness of stable holonomies inside center-stable manifolds. *Ergod. Th. & Dynam. Sys.* Vol. 42:12 (2022), 3593–3618.

[B75] R. Bowen, *Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes Math., vol. 470, Springer, 1975.

[B89] P.H. Baxendale, Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. *Probab. Theory Relat. Fields* Vol. 81(4), (1989), 521–554.

[BBPS] J. Bedrossian, A. Blumenthal and S. Pushon-Smith. Lagrangian chaos and scalar advect ion in stochastic fluid mechanics, *J. Eur. Math. Soc.* Vol 24 (2022) 1853–1990.

[BEFRH] A. Brown, A. Eskin, S. Filip and F. Rodriguez Hertz, Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method. Arxiv math. 2502.14042 (278 pages).

[BGHP20] C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. *Geom. Topol.*, Vol.24 (2020), 1751–1790.

[BGM01] C. Bonatti and X. Gómez-Mont, Sur le comportement statistique des feuilles de certains feuilletages holomorphes, *Enseignement Mathématique* Vol. 38 (2001), 15–41.

[BGMV03] C. Bonatti, X. Gómez-Mont, and M. Viana. Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices. *Ann. Inst. H. Poincaré Anal. Non Linéaire*, Vol. 20 (2003), 579–624.

[BM77] R. Bowen and B. Marcus, Unique ergodicity of the stable foliation, *Israel J. Math*. Vol.26 (1977), 43–67.

[BM20] P. G. Barrientos and D. Malicet. Extremal exponents of random products of conservative diffeomorphisms. *Math. Z.* Vol. 296 (2020), no. 3-4, 1185–1207.

[BPLV] L. Backes, M. Poletti, P. Varandas, and Y. Lima. Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems, *Ergod. Th. & Dynam. Sys.* Vol. 40 (2020), 2947–2969, With an appendix by Y. Lima.

[BRH17] A. Brown and F. Rodriguez Hertz Measure rigidity for random dynamics on surfaces and related skew products. *J. Amer. Math. Soc.* Vol. 30 no. 4, (2017) 1055–1132.

[BV05] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps. *Ann. Math.* Vol. 161 (2005), 1423–1485.

[BW10] K. Burns and A. Wilkinson. On the ergodicity of partially hyperbolic systems. *Annals of Math.* Vol.171 (2010) 451–489.

[C06] P.N. Chung. Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces (2020) (arXiv:2006.03166).

[C87] A. Carverhill, Furstenberg's Theorem for Nonlinear Stochastic Systems. *Prob. Th. Rel. Fields*, Vol.74 (1987), 529–534.

[C90] H. Crauel: Extremal exponents of random dynamical systems do not vanish. *J. Dyn. Differ. Equ.* Vol. 2(3), (1990)245–291.

[CD25] S. Cantat and R. Dujardin. Hyperbolicity for large automorphisms groups of projective surfaces. *J. École Polytechnique* Vol.12 (2025)
[CKM87] R. Carmona, A. Klein and F. Martinelli, Anderson localization for Bernoulli and other singular potentials, *Comm. Math. Phys.* Vol. 108 (1987), 41–66.

[CMY] L. Chao, K. Marin, and J. Yang. Lyapunov exponents of partially hyperbolic volume preserving maps with 2-dimensional center bundle. *Ann. Inst. H. Poincaré Anal. Non Linéaire* Vol. 35 (2018), 1687–1706.

[CP22] S. Crovisier and M. Poletti. Invariance principle and non-compact center foliation. *preprint*, ArXiv: 2210.14988v2

[D26] R. Dujardin. *Théorie ergodique*.

[DE86] A.Douady and C.J. Earle. Conformally natural extension of homeomorphisms of the circle. *Acta Math.* Vol. 157 (1986), 23–48.

[Dob59] Roland L. Dobrušin. A general formulation of the fundamental theorem of Shannon in the theory of information *Uspehi Mat. Nauk*, 14 (1959), 3–104,

[DK07] B. Deroin and V. Kleptsyn. Random conformal dynamical systems. *Geom. Funct. Anal.* Vol.17 (2007), 1043–1195.

[DKN07] B. Deroin, V. Kleptsyn and A. Navas. Sur la dynamique unidimensionnelle en régularité intermédiaire. *Acta Math*, Vol.199 (2007), 199–262.

[ES23] R. Elliott Smith. Uniformly expanding random walks on manifolds. *Nonlinearity* Vol. 36 (2023) 5955–5972.

[EM18] A. Eskin and M. Mirzakhani. Invariant and stationary measures for the SL(2, $\mathbb{R}$) action on moduli space. *Publ. Math. Inst. Hautes Études Sci.* Vol. 127 (2018) 95–324.

[F63] H. Furstenberg, Non-commuting random products. *T.A.M.S.* vol.(108) (1963), 377–428.

[FK60] H. Furstenberg and H. Kesten, Products of random matrices, *Ann. Math. Statist*. 31 (1960), 457–469.

[G79] Y. Guivarc'h. Marches aléatoires à pas markovien. *C.R. Acad. Sci. Paris Sér. A* 289, (1979), 211-213.

[G80] Y. Guivarc'h. Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire. *Astérisque*, Vol. 74 (1980) 47–98.

[GM89] I. Ya. Gol'dsheid and G. A. Margulis. Lyapunov exponents of a product of random matrices. *Russian Math Surveys*, Vol. 44 (1989), 11–71.

[GR89] Y. Guivarc'h and A. Raugi. Propriétés de contraction d'un semi-groupe de matrices inversibles. Coefficients de Liapunoff d'un produit de matrices aléatoires indépendantes *Israel J. Math.*, Vol. 65 (1989), 165–196.

[GT14] A. Gogolev and A. Tahzibi, Center Lyapunov exponents in partially hyperbolic dynamics. *J. Mod. Dyn.* 8 (2014), 549–576.

[GTY] S. Gan, Y. Tong and J. Yang, Ledrappier-Young formula for $C^1$ diffeomorphisms with dominated splitting part I: unstable entropy formula and invariance principle. preprint ArXiv math. 2509.16981 v2.

[HPS77] M. W. Hirsch, C. C. Pugh and M. Shub - *Invariant manifolds*, Lecture Notes in Math., vol. 583. Springer, 1977

[I73] K. Ishii, Localization of eigenstates and transport phenomena in the one dimensional disordered, 421–480. system. *Supp. Theor. Phys.* 53, (1973) 77-138.

[K93] T. Kaijser, On stochastic perturbations of iterations of circle maps. *Phys. D Nonlinear Phenomena* Vol. 68(2), (1993) 201–231.

[LJ87] Y. Le Jan. Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. *Ann. IHP (B) Probabilités et Statistiques*, vol. 23, (1987) 111–120.

[Les21] Pablo Lessa. Entropy and dimension of disintegrations of stationary measures *Trans. Amer. Math. Soc. Ser. B*, Vol. 8 (2021) 105–129.

[L84] F. Ledrappier, Quelques propriétés des exposants caractéristiques. In *École d'été de probabilités de Saint-Flour, XII-1982*, Springer Lect. Notes in Math.vol.1097. Springer, Berlin 1984, 305–396.

[L86] F. Ledrappier. Positivity of the exponent for stationary sequences of matrices. in *Lyapunov Exponents (Bremen,1984)* Springer Lect. Notes in Math.vol.1186. Springer, Berlin 1986, 56–73.

[LS82] F. Ledrappier and J.-M. Strelcyn. A proof of the estimation from below in Pesin's entropy formula. *Erg. Th. & Dynam. Systems*, Vol. 2 (1982), 203–219.

[LY85] F. Ledrappier and L.-S. Young. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, *Ann. of Math. (2)*, (3) Vol.122 (1985), 540–574
[M17] D. Malicet. Random walks on Homeo($S^1$). *Comm. Math. Phys.* 356 (2017), no. 3, 1083–1116.

[M69] G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature *Funkcional. Anal. i Priložen*. Vol. 3:4 (1969), 89–90; *Functional Anal. Appl.* Vol. 3:4 (1969), 335–336.

[M88] R. Mañé, A proof of the $C^1$ stability conjecture, *Inst. Hautes Études Sci. Publ. Math.* Vol. 66 (1988), 161–210.

[Mit22] D. Mitsutani, Simplicity of the Lyapunov spectrum for classes of Anosov flows, *Erg. Th. & Dynam. Systems*, Vol. 42 (2022), 1–27.

[MP25] K. Marin and M. Pelotti. Invariance Principle in dynamical systems, preprint, ArXiv: 2512.13862

[O68] V. I. Oseledets. A multiplicative ergodic theorem. Characteristic Lyapunov exponents of dynamical systems. *Trudy Moskov. Mat. Obsc* Vol. 19 (1968), 179–210.

[OP22] D. Obata and M. Poletti. Positive exponents for random products of conservative surface diffeomorphisms and some skew products. *J. Dynam. Differential Equations* 34 (2022), no. 3, 2405–2428.

[P72] V. Pliss. On a conjecture due to Smale. *Diff. Uravnenija*, Vol. 8 (1972), 262–268.

[P78] Ya. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, *Math. USSR. Izv.* Vol. 10 (1976), 1261–1302.

[P80] L. Pastur. Spectral Properties of Disordered Systems. in the One-Body Approximation, *Commun. Math. Phys.*, Vol 75 (1980), 179–196.

[R04] D. Ruelle. *Thermodynamic formalism*. 2nd ed. Cambridge Mathematical Library, University Press, Cambridge, UK (2004).

[R49] V.A. Rokhlin. On the fundamental ideas of measure theory. *Amer. Math. Soc. Transl.* Vol. 10 (1962) 1–54. translated from *Mat. Sb.* Vol, 25 (1949) 107–150.

[R67] V.A. Rokhlin. Lectures on the entropy theory of transformations with invariant measure. *Russian Math. Surveys* Vol. 22:5 (1967), 1–52.

[R80] G. Royer. Croissance exponentielle de produits markoviens de matrices aléatoires, *Ann. IHP, Sect. B*, Vol. 16, (1980). 49–62.

[R83] W. L. Reddy. Expansive canonical coordinates are hyperbolic. *Topology Appl.* Vol. 15 (1983), 205–210.

[RH13] J. Rodriguez Hertz. Some advances on the generic properties of the Oseledets splitting. *D.C.D.S.* Vol.33 (2013), 4323–4339.

[RHRHTU] F. Rodriguez Hertz, J. Rodriguez Hertz, A. Tahzibi and R. Ures. Maximizing measures for partially hyperbolic systems with compact center leaves. *Ergod. Th. & Dynam. Sys.*, Vol. 32 (2012), 825–839.

[RHRHU16] F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures. A non-dynamically coherent example on T3. *Ann. Inst. H. Poincaré Anal. Non Linéaire* Vol 33:4 (2016), 1023–1032.

[RW01] D. Ruelle and A. Wilkinson. Absolutely singular dynamical foliations. *Comm. Math. Phys.*, Vol. 219 (2001), 481–487.

[S68] Ya. G. Sinai, Classical dynamic systems with countably-multiple Lebesgue spectrum. II. *Izv. Akad. Nauk SSSR Ser. Mat.* Vol. 30 (1968), 15–68.

[S83] B. Simon, Kotani Theory for One Dimensional Stochastic Jacobi Matrices, *Commun. Math. Phys.*, Vol.89 (1983), 227–234.

[TY19] A. Tahzibi and J. Yang. Invariance principle and rigidity of high entropy measures. *Trans. Amer. Math. Soc.* 371 (2019), 1231–1251.

[UVY] R. Ures, M. Viana and J. Yang. Maximal entropy measures of diffeomorphisms of circle fiber bundles. *J. Lond. Math. Soc. (2)*, Vol. 103 (2021), 1016–1034.

[UVYY] R. Ures, M. Viana, F. Yang and J. Yang., Thermodynamical u-formalism I: measures of maximal u-entropy for maps that factor over Anosov, *Ergod. Th. & Dynam. Sys.*, Vol. 44 (2024) 290–333.

[V08] M. Viana. Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. *Annals of Mathematics* Vol. 167 (2008). 643–680.

[V14] M. Viana. *Lectures on Lyapunov exponents*, Cambridge Studies in Advanced Mathematics, vol. 145, Cambridge University Press, Cambridge, 2014.

[V79] A.D. Virtser. On products of random matrices and operators, *Theory Probab. Appl.*, Vol. 24 (1979), 367-377
[VY13] M. Viana and J. Yang. Physical measures and absolute continuity for one-dimensional center direction. *Ann. Inst. H. Poincaré Anal. Non Linéaire*, Vol. 30 (2013) 845-877.

[W17] A. Wilkinson. What are Lyapunov exponents, and why are they interesting? *Bulletin of the AMS* Vol. 54 (2017), 79–105.

[Y13] L.-S. Young, Mathematical theory of Lyapunov exponents. *J. Phys. A: Math. Theor*. Vol. 46 (2013) 254001 (17pp).

FRANÇOIS LEDRAPPIER, UNIVERSITÉ DE PARIS ET SORBONNE UNIVERSITÉ, CNRS, LPSM, BOÎTE COURRIER 158, 4, PLACE JUSSIEU, 75252 PARIS CEDEX 05, FRANCE,

Email address: fledrapp@nd.edu