# Random additive perturbation of a *k* term recurrence relation

Lisette Jager, Killian Verdure

► To cite this version:

Lisette Jager, Killian Verdure. Random additive perturbation of a *k* term recurrence relation. 2024. [hal-04851261](https://hal.science/hal-04851261v1)

HAL Id: hal-04851261

https://hal.science/hal-04851261v1

Preprint submitted on 20 Dec 2024

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 Attribution 4.0 International License
is i.i.d. and that the distribution $P_{\Theta_0}$ of one of the variables $\Theta_n$ is absolutely continuous with respect to the Lebesgue measure $m$, with density $H$.

When $\Theta$ is equal to 0, the recurrence relation is deterministic. Such problems are the subject of [6, 7]. In these articles, the recurrence relation is imbedded in $\mathbb{R}^k$ in such a way as to give rise to a piecewise expanding transformation of $\mathbb{R}^k$. The properties of the transformation and of its transfer operator imply results about the original process $X$.

Expanding maps and their ergodic properties have a long story. In dimension 1, [4, 9] studied the transfer operator of piecewise expanding transformations of an interval, applying the result of [5] to prove its quasicompacity and the existence of a spectral gap. This proved very fruitful, but in higher dimensions, the method does not apply directly. For example, [12] studies piecewise expanding transformations on a hypercube which is partitioned in smaller hypercubes. In [13] the subdomains are polyhedral and the maps, linear. The domains are not so regular in [10, 11], but the (nonlinear) transformations satisfy very strict and complicated conditions. In [15], the transformation and its properties are presented in a very axiomatized way, which sheds a new light on the transformations and on the subdomains. But it is extremely difficult to apply to a concrete example, although [2] proved that such a decomposition in “towers” is equivalent to good rates of decay of the correlations. Hence, in situations where the rate of decay is known, the towers must exist.

The difficulty one faces in dimension superior to 1 is intrinsic: [14] constructs an example of a piecewise expanding map in dimension 2, in which the successive images of a (very well chosen) open rectangle have a diameter that converges to 0, contradicting the naive intuition that the expanding property leads to evergrowing iterates and even to exactness.

In view of the shape and nonlinearity of the recurrence relation, the technics developed in [11] seemed perfectly adapted. In this work, the author studies expanding, piecewise defined transformations $T$ of a compact subset $\Omega$ of $\mathbb{R}^k$.

Under geometric conditions, he proves the existence of an absolutely continuous invariant measure $\mu$ (ACIM) for $T$. Moreover, if $T$ is mixing, the correlations decay at an exponential rate. That means that, for convenient functions $f, h$, there exist $\Lambda \in [0, 1[$ and a real constant $C(f, h)$ such that, for any $n$,

$$ \left| \int_{\Omega} f \circ T^n h \, d\mu - \int_{\Omega} f \, d\mu \int_{\Omega} h \, d\mu \right| \le C(f, h) \Lambda^n. $$

In other words, if $Y_0 \sim \mu$ and if $Y_n = T^n Y_0$, $\text{cov}(f(Y_n), h(Y_0)) \le C(f, h) \Lambda^n$.

Obtaining the mixing property is non trivial. The classical conditions on the transfer operator or the more recent random covering condition [1] are difficult to obtain. For example, in [12] the author even obtains exactness, but the shape of the partition is very regular. Therefore, the mixing condition has, up to now, to remain an additional condition.

To apply the results of [11] in our situation, one needs to derive a transformation $T$ of a compact subset of $\mathbb{R}^k$ from the function $\Phi_0$ leading the process (1). This was achieved, in the deterministic case, in [6, 7] for recurrence relations of all orders $k \ge 2$.

The present work is the following step. It deals with the full problem (1), with its stochastic perturbation $\Theta$. Instead of treating $\Theta$ as a perturbation, we study a family $(\varphi_\theta)_{\theta \in \mathbb{R}} = (\Phi_0 + \theta)_{\theta \in \mathbb{R}}$ of functions leading the process and we consider that $\theta$ is a realization of the random variable $\Theta$, with (for example) normal distribution. Each $\varphi_\theta$ gives rise to a transformation $T_\theta$ - to which we apply the results of [11]- and to its transfer operator $P_\theta$.

Following the lines of [1], we then study an *averaged transfer operator* $P$ which is, in some sense, the expected value of all the individual transfer operators $P_\theta$ (under the law of the perturbation $\Theta$). If the individual transformations $T_\theta$ satisfy the conditions in [11] uniformly in $\theta$, the operator $P$ then
satisfies the conditions of [5]. As a consequence, it has a spectral decomposition, a spectral gap, all properties which ensure the existence of an invariant measure in a certain sense. This corresponds to the annealed point of view and we are in the conditions of Example 2.7 of [1]. The averaged operator is defined in the same way as the Foias operator in [8], section 12. But, since the transformation lacks a continuity property (for a fixed value of the perturbation, our transformation is piecewise C² and not continuous), the conclusions valid for the Foias operator do not hold in our case.

The article is organized as follows.

Section 2 gives the precise setting and hypotheses. The functions $\varphi_{\theta}$ are all additive perturbations of the same function $\Phi_{0}$ defined on a compact hypercube $[-L, L]^{k}$ of $\mathbb{R}^{k}$. Since each $\varphi_{\theta}$ is supposed to take values in $[-L, L]$, it is piecewise defined on $[-L, L]^{k}$ and the precise definition domains depend on $\theta$. We then define the compact set $\Omega$ and the transformations $T_{\theta}$ derived from the $\varphi_{\theta}$ and to which we apply the results of [11]. The $T_{\theta}$ and the set are defined using a small parameter $\gamma$, which does not depend on $\theta$. To ensure its existence, we need $\Phi_{0}$ and its derivatives to satisfy the precise, quantitative conditions (17). We then give the main results. Theorem 1 states that the conditions of [11] are uniformly satisfied, Theorem 2 gives the decomposition of the averaged transfer operator $P$ and Theorem 3 gives the invariant measure for the $k$ dimensional process and for the original $\mathbb{R}$ valued process. The proofs are detailed in the following sections.

In Section 3, we use the same technics as in [6, 7] to construct the individual transformations $T_{\theta}$ of a compact set $\Omega$ of $\mathbb{R}^{k}$ and thus to imbed the recurrence relation into $\mathbb{R}^{k}$. Since the transformations depend on $\theta$, we have to check carefully that the conditions stated in [11] are satisfied uniformly with respect to $\theta$. This is particularly delicate when it comes to the precise definition domains of $\varphi_{\theta}$. The results of this section are summarized in Theorem 1.

In Section 4, according to [1] we define and study an averaged transfer operator $P$. The individual transfer operators $P_{\theta}$ have uniformity properties, for example Section 3 proves that they satisfy a Lasota-Yorke inequality with constants independent of $\theta$. The same inequality holds for $P$. Applying the result of [5] to the averaged $P$ gives a spectral decomposition of $P$ as the sum of projections on finite dimensional spaces and of a “small” residual operator $S$ (small in that it has a small spectral radius). The spectrum of $P$ then consists of a finite number of modulus 1 eigenvalues, all of them simple, and of the spectrum of $S$, which is contained in a ball of radius $< 1$, centered on the origin. This particularity is called the spectral gap.

Denote by $\mathbb{P}_{\Theta}$ the tensor product of the probability measures $\mathbb{P}_{\Theta_{0}}$ on $\mathbb{R}^{\otimes N}$. One of the projections is associated with the eigenvalue 1. Applying it to the (normalized) constant function on $\Omega$ gives an invariant measure $\mu$, in the sense that the measure $\mathbb{P}_{\Theta} \otimes \mu$ is invariant for the mapping

$$ F: \mathbb{R}^{N} \times \Omega \longrightarrow \mathbb{R}^{N} \times \Omega \\ (\vartheta, x) \longmapsto (S\vartheta, T_{\theta_{0}}(x)) \qquad (2) $$

where $S$ denotes the left shift. It remains to compute the marginal distributions to get an invariant measure for the original 1-dimensional process.
# 2 Hypotheses and main results

## 2.1 Notations, concepts, hypotheses

We now present the problem more precisely. Let $L > 0$, let the stochastic process $(X_n)$, bounded and with values in $[-L, L]$, be defined as follows. Let $\beta$ be a positive real number and $\Phi_0$ be a real-valued, $C^2$ application, defined on a neighbourhood $(-L - \beta, L + \beta)^k$ of $[-L, L]^k$.

We introduce a family $\{\varphi_{\theta}\}_{\theta \in \mathbb{R}}$ of applications indexed by $\theta \in \mathbb{R}$. Each $\varphi_{\theta}$ is defined piecewise on $[-L, L]^k$ as an additive perturbation of $\Phi_0$, with values in $[-L, L]$. More precisely, let $(O_j^{\theta})_{j \in \mathbb{Z}, \theta \in \mathbb{R}}$ be a family of open subsets of $(-L, L)^k$ which split $[-L, L]^k$ according to the values taken by $\Phi_0$ and denote by $S_j^{\theta}$ the parts of their boundaries which are inside $[-L, L]^k$:

$$ 
\begin{aligned}
O_j^{\theta} &= \{x \in (-L, L)^k, (2j-1)L < \theta + \Phi_0(x) < (2j+1)L\} \\
S_j^{\theta} &= \{x \in [-L, L]^k, (2j-1)L = \theta + \Phi_0(x)\}.
\end{aligned}
\qquad (3) $$

For every $\theta \in \mathbb{R}$ and every $x \in [-L, L]^k$, set

$$ 
\begin{aligned}
\varphi_{j,\theta}(x) &= \Phi_0(x) + \theta \quad [-L, L] \\
&= \Phi_0(x) + \theta - 2jL, \quad j = \lceil \frac{\Phi_0(x)+\theta+L}{2L} \rceil,
\end{aligned}
\qquad (4) $$

where $\lceil x \rceil$ denotes the integer part of $x$. We define $\varphi_{\theta}$ on $[-L, L]^k$ (up to a null set) by

$$ \varphi_{\theta}|_{O_j^{\theta}} = \varphi_{j,\theta}, \qquad (5) $$

provided $O_j^{\theta}$ is not empty.

If, at time $n \ge k$, the perturbation $\Theta_n$ takes the real value $\theta$, the next state $X_{n+1}$ is then given by

$$ X_{n+1} = \varphi_{\theta}(X_{n-k+1}, \dots, X_{n-k}). \qquad (6) $$

We imbed the recurrence relation (6) in $\mathbb{R}^k$, setting

$$ Y_n = (X_{n-k+1}, \dots, \gamma^{j-1}X_{n-k+j}, \dots, \gamma^{k-1}X_n), \qquad (7) $$

where $\gamma$ is a small positive parameter, to be chosen.

The conditions (17) listed below ensure that there exists a $\gamma$, independent on $\theta \in \mathbb{R}$ and small enough to allow us to apply Theorem 5.1 in [11]. The natural way, with $\gamma = 1$, doesn't allow the transformation $T_{\theta}$ to be expanding (condition (PE4) of Theorem 1). Anyway, condition (PE5) on a geometric constant $\eta$ (see (14)) imposes even stricter conditions.

Let us first define the applications and their domains. We give the convenient value of $\gamma$ in (17).

One sets

$$ \Omega = \prod_{j=1}^{k} [-\gamma^{j-1}L, \gamma^{j-1}L]. \qquad (8) $$

One denotes by $\Gamma$ the dilation linking $\Omega$ and $[-L, L]^k$:

$$ 
\begin{aligned}
\Gamma: \mathbb{R}^k &\to \mathbb{R}^k \\
x = (x_j) &\mapsto (x_j \gamma^{-j+1}),
\end{aligned}
\qquad (9) $$
hence $\Omega = \Gamma^{-1}([-L, L]^k)$.

One then sets

$$ 
\begin{aligned}
& \forall r \in [0, \beta[, \quad \Omega_r = \Gamma^{-1}((-L-r, L+r)^k) = \prod_{s=1}^{k} (-\gamma^{s-1}(L+r), \gamma^{s-1}(L+r)L), \\
& U_j^{\theta} = \Gamma^{-1}(O_j^{\theta}) = \{x \in \Omega_0, (2j-1)L < \theta + \Phi_0(\Gamma(x)) < (2j+1)L\} \\
& \Sigma_j^{\theta} = \Gamma^{-1}(S_j^{\theta}) = \{x \in \Omega, (2j-1)L = \theta + \Phi_0(\Gamma(x))\}
\end{aligned} 
\quad (10) 
$$

The open sets $\Omega_r$ are neighbourhoods of $\Omega$ and appear in Section 3. The open set $U_j^\theta$ is empty if and only if $O_j^\theta$ is empty. The properties of $\Omega$, of the $U_j^\theta$ and of the slanted parts of the boundaries, the $\Sigma_j^\theta$ (regularity, number) do not depend on the value of $\gamma > 0$.

On the set $\Omega_\beta \setminus \{x \in \Omega_\beta, \theta + \Phi_0(\Gamma(x)) \in (2\mathbb{N} + 1)L\}$, the transformation $T_\theta$ is defined by

$$ T_{\theta}(u) = \left( \frac{u_2}{\gamma}, \dots, \frac{u_k}{\gamma}, \gamma^{k-1}\varphi_{\theta}(u_1, \frac{u_2}{\gamma}, \dots, \frac{u_k}{\gamma^{k-1}}) \right), \quad (11) $$

with $\varphi_\theta$ given by (5).

We introduce an auxiliary function $S$ on $\Omega_\beta$:

$$ S(u) = \left( \frac{u_2}{\gamma}, \dots, \frac{u_k}{\gamma}, \gamma^{k-1}\Phi_0(u_1, \frac{u_2}{\gamma}, \dots, \frac{u_k}{\gamma^{k-1}}) \right) = \left( \frac{u_2}{\gamma}, \dots, \frac{u_k}{\gamma}, \gamma^{k-1}\Phi_0(\Gamma(u)) \right). \quad (12) $$

We then define $T_j^\theta$ on $\Omega_\beta$ by

$$ T_j^\theta(u) = S(u) + (0, \dots, 0, \gamma^{k-1}(\theta - 2jL)). \quad (13) $$

One sees that $T_\theta|_{U_j^\theta} = T_j^\theta|_{U_j^\theta}$.

From now on, we are interested in the $k$-dimensional valued process defined by its first term $Y_0$ and the relation

$$ Y_{n+1} = T_{\theta}(Y_n), $$

where the relationship between the $\mathbb{R}^k$ valued processus $Y = (Y_n)$ and the real valued $X = (X_n)$ is given by (7):

$$ Y_n = (X_{n-k+1}, \dots, \gamma^{j-1}X_{n-k+j}, \dots, \gamma^{k-1}X_n). $$

Lemma 9 below proves that $T_j^\theta$ is a global diffeomorphism on $\Omega_\beta$ onto its image $T_j^\theta(\Omega_\beta)$. Hence, $T_j^\theta$ is a diffeomorphism on $U_j^\theta$ onto $T_j^\theta(U_j^\theta)$. This holds for any bounded open subset of $\Omega_\beta$. Later on, we will need to restrict $T_j^\theta$ to more specific open sets (defined by (32)), in order to apply directly the results of [11].

A central parameter in [11] is the number of crossings, that is, the number of regular parts of the boundaries of the open sets of $\Omega$ which intersect at one point. In the present case, the boundaries consist of parts of hyperplanes (the boundaries of $\Omega$ itself) and of the slanted parts $\Sigma_j^\theta$. Whatever the value of the parameter $\gamma$, the $\Sigma_j^\theta$ are regular under the conditions (17) (see Lemma 13). There are at most $k$ hyperplanes crossing at one point, and one slanted part (see Figure 1 in the case when
k = 2). Since the $\Sigma_j^\theta$ are common to two open sets, they count double (see Lemma 2.1 [11]), which proves that the maximal number of crossings is $k + 2$ for a countable number of values $\theta$ and $k + 1$ for the rest of the values.

<figure><img src="image_2.png" /><figcaption>Figure 1: Open sets $U_j^\theta$ with boundaries and crossings.</figcaption></figure>

Let $Y$ be the maximal number of crossings. For a real number $\sigma > 1$, we define

$$ \eta_{0,Y}(\sigma) = \eta_0(\sigma, Y) = \frac{1}{\sqrt{\sigma}} + \frac{4}{\sqrt{\sigma} - 1} Y \frac{\gamma_{k-1}}{\gamma_k}, \quad (14) $$

where $\gamma_n$ is the volume of the $n$-dimensional unit ball in $\mathbb{R}^n$. Let $S_k(Y)$ be the real number for which $\eta_0(\sigma, Y) = 1$:

$$ S_k(Y) = \eta_{0,Y}^{-1}(1). \quad (15) $$

We choose the dilation coefficient $\sigma$:

$$ \sigma > S_k(Y) > 1 \quad (16) $$

and two real constants $C_1, C_2 > 1$. We now prescribe the conditions:

$$ \left\{ \begin{array}{ll} \gamma^{-2} & = C_1\sigma, \\ \\ \forall x \in (-L - \beta, L + \beta)^k, & \left(\frac{\partial \Phi_0}{\partial x_1}(x)\right)^2 \ge C_1^{k-1} C_2 \sigma^k, \\ \\ \forall x \in (-L - \beta, L + \beta)^k, \forall j \ge 2, & \left(\frac{\partial \Phi_0}{\partial x_j}(x)\right)^2 \le \frac{(C_1 - 1)(C_2 - 1)}{k-1} \sigma. \end{array} \right. \quad (17) $$

This gives the value of the parameter $\gamma$, which, in turn, defines the domain $\Omega$ and the transformations.

The above conditions prove that each transformation (11) is non singular in $L^1(\Omega, m)$, which implies the existence of a Perron-Frobenius operator $P_\theta$. To study the transfer operators $P_\theta$ or the averaged operator defined in (59), we use the central result of [5]. Thus, we need to introduce the following space $V_1$. We just recall here its definition. Its properties are studied in [11].

For every non-null Borel set $S$ of $\mathbb{R}^k$, for every $f \in L_m^1(\mathbb{R}^k, \mathbb{R})$, one sets :

$$ \mathrm{osc}(f, S) = \underset{S}{\mathrm{ess sup}} f - \underset{S}{\mathrm{ess inf}} f $$

where $\mathrm{ess sup}_S$ and $\mathrm{ess inf}_S$ are the essential supremum and infimum on $S$ with respect to the Lebesgue measure $m$.
One then defines the semi norm $|\cdot|_1$ by

$$|f|_1 = \sup_{0<\varepsilon<\varepsilon_1} \varepsilon^{-1} \int_{\mathbb{R}^k} \mathrm{Osc}(f, B(x, \varepsilon)) \, dx \quad (18)$$

and the set $V_1$ by

$$V_1(\Omega) = \{f \in L_m^1(\mathbb{R}^k, \mathbb{R}), \mathrm{supp} f \subset \Omega \text{ and } |f|_1 < +\infty\}. \quad (19)$$

Then, for $f \in V_1$, the norm $||\cdot||_{V_1}$ is defined by

$$\|f\|_{V_1} = \|f\|_{L_m^1} + |f|_1.$$

## 2.2 Main results

Let $m$ denote the Lebesgue measure on $\mathbb{R}^k$ and its subsets. For $X \subset \mathbb{R}^k$ and $\varepsilon > 0$, we denote by $B_\varepsilon(X)$ the following open set $B_\varepsilon(X) := \{x \in \mathbb{R}^k : d(x, X) < \varepsilon\}$.

**Theorem 1.** Under the conditions of the preceding paragraph, there exist positive real numbers $r_0, \varepsilon_m$ and $K$ and a sufficiently small $\varepsilon_0 > 0$ such that, for every $\theta \in \mathbb{R}$, the following conditions hold:

(PRE) There exist a finite family $(U_j^\theta)_j$ of pairwise disjoint open subsets of $\Omega$, a finite family $(V_j^\theta)_j = B_{\varepsilon_m}(U_j^\theta)$ such that $\overline{U_j^\theta} \subset V_j^\theta$ and $C^2$ applications $T_j^\theta : V_j^\theta \to \mathbb{R}^k$, satisfying all the following conditions:

(PE1) $\forall j, T_j^\theta|_{U_j^\theta} = T_j^\theta|_{\overline{U_j^\theta}}$ and $B_{r_0}(T_j^\theta(U_j^\theta)) \subset T_j^\theta(V_j^\theta)$;

(PE2) $\forall j, T_j^\theta$ is a $C^2$ diffeomorphism on $V_j^\theta$ onto its image. Moreover, the determinant of the Jacobian matrix is uniformly Lipschitz continuous, in the following sense:
$\forall j, \forall \varepsilon \in (0, \varepsilon_0), \forall z \in T_j^\theta(V_j^\theta), \forall x, y \in B(z, \varepsilon) \cap T_j^\theta(V_j^\theta),$

$$|\det(D(T_j^\theta)^{-1}(x)) - \det(D(T_j^\theta)^{-1}(y))| \le K |\det(D(T_j^\theta)^{-1}(z))| \varepsilon;$$

(PE3) $m(\Omega \setminus \bigcup_j U_j^\theta) = 0$;

(PE4) for all $u, v$ in $T_j^\theta(V_j^\theta)$ such that $d(u, v) \le \varepsilon_0$,

$$d((T_j^\theta)^{-1}(u), (T_j^\theta)^{-1}(v)) \le \frac{1}{\sqrt{\sigma}} d(u, v),$$

with $\sigma > 1$ defined by (16). The map $T_j^\theta$ is thus expanding;

(PE5) Set $G(\varepsilon, \varepsilon_0) = \sup_x G(x, \varepsilon, \varepsilon_0)$ with

$$G(x, \varepsilon, \varepsilon_0) = \sum_j \frac{m\left(\left(T_j^\theta\right)^{-1}\left[B_\varepsilon\left(\partial T_j^\theta(U_j^\theta)\right)\right] \cap B\left(x, \left[1-\frac{1}{\sqrt{\sigma}}\right]\varepsilon_0\right)\right)}{m\left(B\left(x, \left[1-\frac{1}{\sqrt{\sigma}}\right]\varepsilon_0\right)\right)}.$$

Then $\eta$ defined by

$$\eta(\varepsilon_0) = \frac{1}{\sqrt{\sigma}} + 2 \sup_{\varepsilon \le \varepsilon_0} \frac{G(\varepsilon, \varepsilon_0)}{\varepsilon} \varepsilon_0 \quad (20)$$

satisfies $\sup_{\delta \le \varepsilon_0} \eta(\delta) < 1$.
To sum up, the conditions of Theorem 5.1 in [11] are satisfied, with constants $r_0, \varepsilon_m, K, \varepsilon_0$ independent of $\theta \in \mathbb{R}$. In particular, condition (PE2) proves that the transformations $T_\theta$ are non singular. For each real number $\theta$, we may define the Perron-Frobenius operator associated with $T_\theta$ by

$$ \forall A \in \mathcal{B}(\Omega), \forall f \in L_m^1(\Omega), \quad \int_{T_\theta^{-1}(A)} f \, dm = \int_A P_\theta f \, dm. $$

One may check that $P_\theta f$ has the following expression:

$$ P_{\theta}f(y) = \sum_{j \in \mathbb{Z}, U_j^{\theta} \neq \emptyset} \mathbf{1}_{T_j^{\theta}(U_j^{\theta})}(y) f((T_j^{\theta})^{-1}(y)) |\det(D(T_j^{\theta})^{-1}(y))|. \quad (21) $$

According to Lemma 4.1 in [11], if $\varepsilon_0$ is small enough, there exist constants $\eta \in [0, 1[$ and $D \in \mathbb{R}^+$ such that, for all $f \in V_1$ and all $\theta \in \mathbb{R}$,

$$ |P_{\theta}f|_1 \leq \eta|f|_1 + D\|f\|_{L_m^1}. \quad (22) $$

Since these constants are expressed explicitely in terms of $\eta(\varepsilon_0)$, $\sigma$, $K$, they do not depend on $\theta$.

We formally define the averaged Perron-Frobenius operator associated with this family of transformations by

$$ P = \int_{\mathbb{R}} P_{\theta} \, d\mathbb{P}_{\theta}(\theta) = \int_{\mathbb{R}} P_{\theta} H(\theta) \, d\theta, $$

where $\mathbb{P}_{\theta}$ is the distribution of the perturbation $\Theta$ and $H$, its density function. The properties of this transfer operator are developed in 4, where a proper definition is given.

This allows one to state the following result

**Theorem 2.** Under the conditions (14) to (17) and with the notations of the preceding paragraph,

1. $P$ has finitely many eigenvalues $\lambda_1 = 1, \dots, \lambda_p \in \mathbb{C}$ of modulus 1. They are all simple. The corresponding eigenfunctions belong to $V_1$.

2. There exist $p$ finite-rank operators $\Pi_1, \dots, \Pi_p \in \mathcal{L}(V_1)$, as well as $S \in \mathcal{L}(V_1)$, such that

$$ P = S + \sum_{j=1}^{p} \lambda_j \Pi_j. $$

Furthermore, for all distinct $i, j \in \{1, \dots, p\}$, $\Pi_i^2 = \Pi_i$, $\Pi_i \Pi_j = 0$ and $\Pi_i S = S \Pi_i = 0$.

3. There exist $M > 0$ and $h > 0$ satisfying

$$ \forall n \geq 1, \quad \|S^n\|_{\mathcal{L}(V_1)} \leq \frac{M}{(1+h)^n}. $$

**Theorem 3.** Recall what we denote by $F$:

$$ F: \mathbb{R}^N \times \Omega \longrightarrow \mathbb{R}^N \times \Omega \\ (\vartheta, x) \longmapsto (S\vartheta, T_{\theta_0}(x)) $$

where $\Theta = (\Theta_n)_{n \in \mathbb{N}}$ is a sequence of i.i.d. perturbations and $S$ is the left shift. Denote by $\mathbb{P}_{\Theta}$ the tensor product $\mathbb{P}_{\Theta_0}^{\otimes N}$. Let $h_* = \frac{1}{m(\Omega)}\Pi_1 \mathbf{1}$, with $\Pi_1$ the rank one operator associated with $\lambda_1 = 1$.
In the sense of [1], the averaged transfer operator P admits an absolutely continuous invariant measure or stationary measure µ, which has density h* with respect to the (normalized) Lebesgue measure on Ω. This means that the measure P_Θ ⊗ µ is invariant by the function F recalled above.

Then, if $(\check{\vartheta}, Y)$ has distribution $\mathbb{P}_{\Theta} \otimes \mu$ and if the $\mathbb{R}^k$-valued processus $Y = (Y_n)$ and the real valued $X = (X_n)$ are related as described in (7), $X_n$ has for density any one of the following densities

$$t \mapsto \gamma^{j-1} \int_{\prod_{i \le k, i \ne j} [-\gamma^{i-1}L, \gamma^{i-1}L]} h_*(x_1, \dots, x_{j-1}, \gamma^{i-1}t, x_{j+1}, \dots, x_k) \, dx_1 \dots dx_{j-1} \, dx_{j+1} \dots dx_k \quad (23)$$

Let us prove the last fact. If $(\check{\vartheta}, Y)$ has distribution $\mathbb{P}_{\Theta} \otimes \mu$, so has $F(\check{\vartheta}, Y)$. If the initial term $(\check{\vartheta}_0, Y_0)$ has distribution $\mathbb{P}_{\Theta} \otimes \mu$, so have all terms $F^n(\check{\vartheta}_0, Y_0) = (S^n \check{\vartheta}_0, Y_n)$

This implies that $Y_n$ has the density $h_*$. Considering the marginal distributions we obtain that $X_{n-k+1}$ admits, as an density, the first marginal density of $h_*$, $\gamma^{j-1}X_{n-k+j}$ admits the j-th marginal density of $h_*$ for $j = 1, \dots, k$. This gives (23), exactly as in Theorem 2 of [7].

# 3 Proof of Theorem 1

## 3.1 Expanding property

In this part we prove that, under the conditions listed in the preceding section, the application S defined by (12) is expanding on $\Omega_\beta$. Since they differ from S by an additive constant, the applications $T_j^\theta$ share this property. We first express the difference $S(u) - S(u')$ thanks to differential calculus. There appears a matrix B, the eigenvalues of which measure the expansiveness. We prove that the eigenvalues of B are greater than or equal to $\sigma$, which gives the inequality of Proposition 4.

Note a difference with [7]. Here we compute explicitly the spectrum of B, whereas we only located it in the Gerschgörin disks. As a consequence, the conditions (17) required here are simpler than in [7]: they consist in estimates about the first derivatives of $\Phi_0$, instead of estimates about products of the same derivatives.

**Proposition 4.** Suppose that $\Phi_0$ and $\gamma$ satisfy the conditions (14) to (17). Then the function S defined by (12) satisfies

$$\forall (u, u') \in \Omega_{\beta}^{2}, \quad \|S(u) - S(u')\|^2 \geq \sigma \|u - u'\|^2.$$

**Proof.**

Let $u, u'$ belong to $\Omega_\beta$. Then

$$\|S(u) - S(u')\|^2 = \sum_{i=2}^{k} \gamma^{-2}(u_i - u_i')^2 + \gamma^{2k-2} ((\Phi_0 \circ \Gamma)(u) - (\Phi_0 \circ \Gamma)(u'))^2.$$

There exists $t_0 \in [0, 1]$ such that

$$\begin{aligned} (\Phi_0 \circ \Gamma)(u) - (\Phi_0 \circ \Gamma)(u') &= \left. \frac{d}{dt} (\Phi_0 \circ \Gamma)(tu + (1-t)u') \right|_{t=t_0} \\ &= \Gamma(u-u') \cdot \nabla (\Phi_0 \circ \Gamma)(w) = \sum_{i=1}^{k} \gamma^{-i+1} (u_i - u_i') \frac{\partial \Phi_0}{\partial x_i}(\Gamma w), \end{aligned}$$
where we set $w = t_0u + (1 - t_0)u'$. We obtain

$$ \begin{aligned} \|S(u) - S(u')\|^2 &= \sum_{i=2}^{k} \gamma^{-2}(u_i - u_i')^2 + \gamma^{2k-2} \sum_{i=1}^{k} \sum_{j=1}^{k} \gamma^{-i-j+2}(u_i - u_i')(u_j - u_j') \frac{\partial \Phi_0}{\partial x_i}(\Gamma w) \frac{\partial \Phi_0}{\partial x_j}(\Gamma w) \\ &= {}^t(u - u') \mathbf{B}(u - u') \end{aligned} $$

where the matrix $\mathbf{B} = \mathbf{B}(\Gamma w)$ is given by

$$ \mathbf{B}(x) = \frac{1}{\gamma^2} \begin{pmatrix} 0 & I_{k-1} \end{pmatrix} + \begin{pmatrix} \gamma^{k-1} \frac{\partial \Phi_0}{\partial x_1}(x) \\ \gamma^{k-2} \frac{\partial \Phi_0}{\partial x_2}(x) \\ \vdots \\ \frac{\partial \Phi_0}{\partial x_k}(x) \end{pmatrix} \begin{pmatrix} \gamma^{k-1} \frac{\partial \Phi_0}{\partial x_1}(x) & \gamma^{k-2} \frac{\partial \Phi_0}{\partial x_2}(x) & \cdots & \frac{\partial \Phi_0}{\partial x_k}(x) \end{pmatrix}. $$

$$ \text{Set} \\ v_1(x) := \gamma^{k-1} \frac{\partial \Phi_0}{\partial x_1}(x), \quad V(x) := {}^t \left( \gamma^{k-2} \frac{\partial \Phi_0}{\partial x_2}(x) \quad \cdots \quad \gamma \frac{\partial \Phi_0}{\partial x_{k-1}}(x) \quad \frac{\partial \Phi_0}{\partial x_k}(x) \right) \in \mathbb{R}^{k-1}. \quad (24) $$

The proof of the proposition is then a consequence of the following result.

**Proposition 5.** For all $x \in (-L - \beta, L + \beta)^k$, the eigenvalues $\lambda$ of $B = B(x)$ satisfy $\lambda \ge \sigma$.

**Proof of Proposition 5.**

Unsing the notations 24, $B$ can be written as

$$ \mathbf{B} = \gamma^{-2} \begin{pmatrix} 0 & 0 \\ 0 & I_{k-1} \end{pmatrix} + \begin{pmatrix} v_1^2 & v_1 {}^t V \\ v_1 V & V {}^t V \end{pmatrix}. $$

The functions $v_1$ and $V$ are taken in $x = \Gamma w$.

When $V = 0$, $\mathbf{B}$ is diagonal with two eigenvalues: a simple one, $v_1^2$ and a multiple one, $\gamma^{-2}$. The conditions (17) imply that they are $> \sigma$, since $v_1^2(x) \ge C_2\sigma$ and $\gamma^{-2} = C_1\sigma$.

We now treat the case when $V \ne 0$.

**Lemma 6.** Let $x \in (-L - \beta, L + \beta)^k$ be such that $V := V(x) \ne 0$. Then:

1. $\gamma^{-2}$ is an eigenvalue of $\mathbf{B}$, with multiplicity $k-2$. Its eigenspace is the set of vectors $\begin{pmatrix} 0 \\ X \end{pmatrix} \in \mathbb{R}^k$ with $X$ in the orthogonal of $\{V\}$ (in $\mathbb{R}^{k-1}$).

2. The real numbers

$$ \lambda_{-} := \frac{\gamma^{-2} + v_{1}^{2} + \|V\|^{2} - \sqrt{(\gamma^{-2} + v_{1}^{2} + \|V\|^{2})^{2} - 4v_{1}^{2}\gamma^{-2}}}{2}, $$

$$ \lambda_{+} := \frac{\gamma^{-2} + v_{1}^{2} + \|V\|^{2} + \sqrt{(\gamma^{-2} + v_{1}^{2} + \|V\|^{2})^{2} - 4v_{1}^{2}\gamma^{-2}}}{2}, $$

are simple eigenvalues, associated with the eigenvectors $\begin{pmatrix} t_{\pm} \\ V \end{pmatrix}$, where $t_{\pm} = (1 - \lambda_{\pm}^{-1}\gamma^{-2})v_1$.
**Proof of the lemma.**

Step 1. For every vector $(t/X) \in \mathbb{R}^k$, one checks that

$$ \mathbf{B} \begin{pmatrix} t \\ X \end{pmatrix} = \gamma^{-2} \begin{pmatrix} t \\ X \end{pmatrix} \iff (v_1 t + {}^t V X) \begin{pmatrix} v_1 \\ V \end{pmatrix} = \gamma^{-2} \begin{pmatrix} t \\ 0 \end{pmatrix}. $$

This implies that $v_1 t + {}^t V X = 0$ since $V \neq 0$. Hence $t = 0$ by (3.1), which gives $v_1 t + {}^t V X = {}^t V X = 0$ and $X \in \{V\}^{\perp}$. Reciprocally, every vector $(0/X)$ with $X \perp V$ satisfies (3.1). This proves that $\gamma^{-2}$ is an eigenvalue of $\mathbf{B}$, with multiplicity $k-2$, associated with the eigenspace

$$ E(\gamma^{-2}) = \left\{ \begin{pmatrix} 0 \\ X \end{pmatrix} : X \in \mathbb{R}^{k-1}, X \perp V \right\}. $$

Step 2. We now investigate the other eigenvalues. We look for eigenvectors taking the shape $(t/V)$ avec $t \in \mathbb{R}^*$. For $\lambda \in \mathbb{R}$, $t \in \mathbb{R}^*$, one may write

$$ \mathbf{B} \begin{pmatrix} t \\ V \end{pmatrix} = \lambda \begin{pmatrix} t \\ V \end{pmatrix} \iff \begin{cases} \lambda t = v_1^2 t + \|V\|^2 v_1 \\ \lambda = \gamma^{-2} + v_1 t + \|V\|^2 \end{cases}. $$

Remark that this system ensures that $\lambda \neq 0$. Otherwise $v_1^2 t + \|V\|^2 v_1$, $v_1 t + \|V\|^2$ and finally $\gamma^{-2}$ would be equal to 0, which is impossible.

We now solve the system. One sees that

$$ \begin{cases} \lambda t = v_1^2 t + \|V\|^2 v_1 \\ \lambda = \gamma^{-2} + v_1 t + \|V\|^2 \end{cases} \iff \begin{cases} t = (1 - \lambda^{-1}\gamma^{-2})v_1 \\ P(\lambda) = 0 \end{cases} $$

$$ \text{with } P(X) = X^2 - (\gamma^{-2} + v_1^2 + \|V\|^2)X + \gamma^{-2}v_1^2. $$

The discriminant of $P$ is $\Delta = (\gamma^{-2} + v_1^2 + \|V\|^2)^2 - 4v_1^2\gamma^{-2} > 0$. This implies that the remaining eigenvalues are the roots of $P$, that is :

$$ \lambda_{-} := \frac{\gamma^{-2} + v_{1}^{2} + \|V\|^{2} - \sqrt{\Delta}}{2} < \lambda_{+} := \frac{\gamma^{-2} + v_{1}^{2} + \|V\|^{2} + \sqrt{\Delta}}{2}. $$

And the two remaining eigenvectors take the shape $(t_{\pm}/V)$ with $t_{\pm} = (1 - \lambda_{\pm}^{-1}\gamma^{-2})v_1$. This achieves the proof of Lemma 6 $\square$

It remains to prove that $\min\{|\lambda| : \lambda \in \text{Sp}(\mathbf{B})\} \geq \sigma$ in the case when $V \neq 0$. In this case, the spectrum of the matrix $\mathbf{B}$ consists of the eigenvalues $\gamma^{-2}$ and $\lambda_{+} > \lambda_{-} > 0$. We already know that $\gamma^{-2}$ is greater than $\sigma$ and we now prove that $\lambda_{-} \geq \sigma$.

Set $c(x) := \sigma^{-1}v_1^2(x) \geq C_2$ and $\Delta = (\gamma^{-2} + v_1^2 + \|V\|^2)^2 - 4v_1^2\gamma^{-2} > 0$. Then

$$ \begin{align*} \lambda_{-} \geq \sigma &\iff (C_1 + c)\sigma + \|V\|^2 - \sqrt{\Delta} \geq 2\sigma \\ &\iff (C_1 + c - 2)\sigma + \|V\|^2 \geq \sqrt{\Delta} \\ &\iff (C_1 + c - 2)^2\sigma^2 + \|V\|^4 \\ &\phantom{\iff} + 2(C_1 + c - 2)\sigma \|V\|^2 \geq ((C_1 + c)\sigma + \|V\|^2)^2 - 4cC_1\sigma^2. \end{align*} $$
Developing $((C_1 + c)\sigma + \|V\|^2)^2 = (C_1+c)^2\sigma^2 + 2(C_1+c)\sigma \|V\|^2 + \|V\|^4$ gives the following inequality:

$$ \begin{align*} & [(C_1 + c - 2)^2 - (C_1 + c)^2 + 4cC_1] \sigma^2 - 4\sigma \|V\|^2 \ge 0 \\ \iff & 4[(C_1c - C_1 - c + 1)\sigma - \|V\|^2] \sigma \ge 0 \\ \iff & \|V\|^2 \le (C_1c - C_1 - c + 1)\sigma = (C_1 - 1)(c - 1)\sigma. \tag{**} \end{align*} $$

One may check that the conditions (17) and the definition (24) of $v_1$ and $V$ imply that, for all $x \in (-L - \beta, L + \beta)^k$,

$$ v_1^2(x) \ge C_2\sigma \quad \text{and} \quad \|V(x)\|^2 \le (C_1 - 1)(C_2 - 1)\sigma. $$

Hence, the condition (**) holds for an arbitrary $x = \Gamma w$ and we obtain that $\lambda_+ \ge \sigma$.

We have proved that $\min\{|\lambda| : \lambda \in \text{Sp}(\mathbf{B})\} \ge \sigma$, which concludes the proof of Proposition 5 and hence of Proposition 4. $\square$

## 3.2 Number of open sets, uniformity

This paragraph goes on with the rather long series of results proving that the conditions listed in Proposition 1 are satisfied.

It is not difficult to see that the number of non empty open subsets $U_j^\theta$ is bounded, see (25).

We then state a uniform version of the implicit functions theorem. It is necessary because the constants $\varepsilon_m, r_0$ and $\varepsilon_0$ of Theorem 1 express a geometric uniformity. The open sets $U_j^\theta$ are determined by the function $\Phi_0$, but we need to ensure that the slightly larger open sets $V_j^\theta$ defined by (32) do not depend on $\theta$ in a way we can't control. Similarly, we use open charts (see (37)) in a proof below and we need the charts at each point to contain a ball of the same size. Another consequence of this theorem is Lemma 13 which gives the negligibility of the boundaries. This is condition (PE3) of Theorem 1. This implies that the transformation $T_\theta$ is defined almost everywhere, since it is defined on $\Omega \setminus \bigcup_j \Sigma_j^\theta$.

Let $m$ and $M$ be the minimum and the maximum of $\Phi_0$ on $[-L, L]^k$. Let us fix $\theta \in \mathbb{R}$. If the open set $O_j^\theta$ is not empty, then the index $j$ satisfies

$$ \frac{\theta + m - L}{2L} < j < \frac{\theta + M + L}{2L}. $$

The maximal number $N$ of non empty open sets $O_j^\theta$ or $U_j^\theta$ satisfies

$$ N < \frac{M - m + 2L}{2L} + 1. \tag{25} $$

Indeed, if $O_j^\theta \ne \emptyset$, then there exists $x \in [-L, L]^k$ such that $(2j - 1)L < \Phi_0(x) + \theta < (2j + 1)L$. This gives the inequality.

**Theorem 7.** Let $E, F$ and $G$ be Banach spaces. The space $E \times F$ is endowed with the max norm. Let $O$ be an open set of $E \times F$.

Let $f: O \to G$ be $C^2$ at least on $O$. Suppose that, for every $(a, b) \in O$, $\partial_2 f(a, b) \in \text{Isom}(F, G)$. Suppose that there exist nested open sets $\Omega', \Omega''$ satisfying

$$ \Omega' \subset \bar{\Omega}' \subset \Omega'' \subset \bar{\Omega}'' \subset O, $$

with $Df, D^2f$ bounded on $\bar{\Omega}''$. Suppose that $(\partial_2 f)^{-1}$ is bounded too on $\bar{\Omega}''$. Assume that the distance between $\bar{\Omega}'$ and $\bar{C}\bar{\Omega}''$ is a positive real number $\delta > 0$.

Then there exist $r > 0, \rho > 0$ such that $0 < \rho < r < \delta$ and satisfying, for every $(a, b) \in \Omega'$, the following conditions:
1. There exist an open neighbourhood $U = B_E(a, r) \times B_F(b, r) \subset \Omega''$ of $(a, b)$ in $O$, an open neighbourhood $V$ of $a$ in $E$ and a function $\varphi \in C^2(V, F)$ such that, for every $(x, y) \in \Omega''$,

$$ (x, y) \in U \text{ and } f(x, y) = f(a, b) \iff x \in V \text{ and } y = \varphi(x). $$

$$ \text{Moreover, } B_E(a, \rho) \subset V \subset B_E(a, r). $$

2. For all $x \in V$,

$$ \partial_2 f(x, \varphi(x)) \in \text{Isom}(F, G) \text{ and } D\varphi(x) = -[\partial_2 f(x, \varphi(x))]^{-1} \circ \partial_1 f(x, \varphi(x)). $$

3. If $f \in C^k(O)$, then $\varphi \in C^k(V)$.

We apply this result to the following $C^2$ function $f$

$$ f: \begin{array}{rcl} \Omega_{\beta} & \to & \mathbb{R} \\ u & \mapsto & \Phi_0(\Gamma(u)). \end{array} \qquad (26) $$

The global open set $O$ of Theorem 7 is $O = \Omega_{\beta}$ and the intermediate sets are $\Omega' = \Omega_{\beta/3}$ and $\Omega'' = \Omega_{2\beta/3}$. They are nested:

$$ \Omega_{\beta/3} \subset \overline{\Omega_{\beta/3}} \subset \Omega_{2\beta/3} \subset \overline{\Omega_{2\beta/3}} \subset \Omega_{\beta}. $$

Denote by $d > 0$ the real number:

$$ d = \min(\text{dist}(C\Omega_{2\beta/3}, \overline{\Omega_{\beta/3}}), \text{dist}(\Omega, C\Omega_{\beta/3})), \qquad (27) $$

where the distance is the max distance, considering $\mathbb{R}^k = \mathbb{R} \times \mathbb{R}^{k-1}$.

We now turn to the function $f$. Its first and second order derivatives are bounded on the compact set $\overline{\Omega_{2\beta/3}}$. Moreover, since $\frac{\partial f}{\partial u_1}(a) = \frac{\partial \Phi_0}{\partial u_1}(\Gamma(a))$, the conditions (17) imply that $\left|\frac{\partial f}{\partial u_1}(a)\right| \ge 1$. The conditions of Theorem 7 are satisfied. As a consequence, we may state the

**Proposition 8.** There exist real numbers $r \in (0, d)$, $\rho \in (0, r)$ such that, for every $a = (a_1, \check{a}) \in \Omega_{\beta/3}$, there exist a neighbourhood $U = U_a = B(a_1, r) \times B(\check{a}, r) \subset \Omega_{2\beta/3}$ of $a$, an open set $V = V_a$ containing $\check{a}$ and such that $\check{a} \in B(\check{a}, \rho) \subset V \subset B(\check{a}, r)$ and a function $g_a: V \to \mathbb{R}$, $C^2$, satisfying

$$ ((u_1, \check{u}) \in U_a, f(u_1, \check{u}) = f(a)) \iff (\check{u} \in V_a, u_1 = g_a(\check{u})). \qquad (28) $$

The differential of $g_a$ on $V_a$ is given by:

$$ Dg_a(\check{u}) = -(\partial_1 f(g_a(\check{u}), \check{u}))^{-1}(\partial_- f(g_a(\check{u}), \check{u})), \qquad (29) $$

where we denote by $\partial_1$ (resp. $\partial$-) the partial derivative with respect to the first variable (resp. the derivative with respect to the other variables).

## 3.3 Uniform enlargement of the open sets

We begin by proving that the auxiliary transformation $S$, defined by (12), is diffeomorphic. This implies that the $T_j^\theta$ are diffeomorphic too on convenient domains. This property is often useful in what follows. We then define the open sets $V_j^\theta$, which are the original open sets $U_j^\theta$ slightly and uniformly enlarged, this uniformity being a consequence of the uniform implicit Theorem 7. The remaining part of the paragraph is devoted to the definition of the real number $r_0$, which gives condition (PE1).
**Lemma 9.** The $C^2$ application $S$ is diffeomorphic on $\Omega_\beta$ onto its image.

**Proof.** Take $u, v \in \Omega_\beta$ such that $S(u) = S(v)$. The $k-1$ first coordinates of $S$ ensure that $u_2 = v_2, \dots, u_k = v_k$. Moreover, the last coordinates of $S(u)$ and $S(v)$ give

$$ \Phi_0(u_1, \gamma^{-1}u_2, \dots, \gamma^{1-k}u_k) = \Phi_0(v_1, \gamma^{-1}v_2, \dots, \gamma^{1-k}v_k). $$

Define $g: t \mapsto \Phi_0(t, \gamma^{-1}u_2, \dots, \gamma^{1-k}u_k)$ on $(-L - \beta, L + \beta)$. Thanks to the hypotheses (17) satisfied by $\Phi_0$, $g$ is strictly monotonous. Hence the equality only takes place when $u_1 = v_1$ and $\Phi_0$ is injective on $\Omega_\beta$.

The differential of $S$ is

$$ DS(u) = \begin{pmatrix} 0 & \gamma^{-1} & 0 & \dots & 0 \\ 0 & 0 & \gamma^{-1} & \dots & 0 \\ 0 & 0 & \dots & 0 & \gamma^{-1} \\ \gamma^{k-1} \frac{\partial \Phi_0}{\partial u_1}(\Gamma(u)) & \gamma^{k-2} \frac{\partial \Phi_0}{\partial u_2}(\Gamma(u)) & \dots & & \end{pmatrix} \quad (30) $$

Its determinant is never equal to zero, which, along with the injectivity, proves Lemma 9. $\square$

**Definition of the open sets $V_j^\theta$**

Recall that $d$ is defined by (27) and that the smaller radius $\rho$ given by Proposition 8 is $< d$. One imposes

$$ \varepsilon_m = \frac{\rho}{3} \quad (31) $$

One then sets, if $U_j^\theta \neq \emptyset$,

$$ V_j^\theta = B_{\varepsilon_m}(U_j^\theta) \quad (32) $$

and one extends the definition of the application of $T_{j\theta}$ to $V_j^\theta$. According to Lemma 9, $T_{j\theta}$ is diffeomorphic on $V_j^\theta$ onto its image.

We now treat the condition (PE1) of [11].

**Lemma 10.** There exists $r_0$ such that, for every $\theta \in \mathbb{R}$, for every index $j$ such that $U_j^\theta \neq \emptyset$,

$$ B_{r_0}(S(U_j^\theta)) \subset S(V_j^\theta). $$

**Proof.** We begin by proving the existence of a real number $r_0 > 0$ such that,

$$ \forall x \in \overline{\Omega_{2\beta/3}}, \quad B(S(x), r_0) \subset S(B(x, \varepsilon_m)). \quad (33) $$

According to Lemma 9, $S$ is diffeomorphic and hence homeomorphic on $\Omega_\beta$. Therefore, for every $x \in \overline{\Omega_{2\beta/3}}$, $S(B(x, \varepsilon_m))$ is an open set containing $S(x)$. Denote by

$$ R(x) = \sup\{r > 0 : B(S(x), r) \subset S(B(x, \varepsilon_m))\}. \quad (34) $$

Suppose that $\inf_{\overline{\Omega_{2\beta/3}}} R(x) = 0$. There exists a sequence $(x_n)_{n \in \mathbb{N}} \in \overline{\Omega_{2\beta/3}}^N$ such that $(R(x_n))_n$ converges to 0. Since $\overline{\Omega_{2\beta/3}}$ is a compact set, we can assume that $(x_n)_{n \in \mathbb{N}}$ converges. Let $x_\infty \in \overline{\Omega_{2\beta/3}}$ be its limit point. Let

$$ R' = \sup \left\{ r > 0 : B(S(x_\infty), r) \subset S \left( B \left( x_\infty, \frac{\varepsilon_m}{2} \right) \right) \right\}. $$
The function $S$ is continuous at $x_\infty$. Let $\varepsilon > 0$ be such that: $\varepsilon < \frac{R'}{3}$. There exists $\eta > 0$ such that $\|y - x_\infty\| < \eta$ and $y \in \overline{\Omega_{2\beta/3}}$) $\implies$ $\|S(y) - S(x_\infty)\| < \varepsilon$. Now take $y \in \overline{\Omega_{2\beta/3}}$ such that $\|y - x_\infty\| < \min(\eta, \frac{\varepsilon_m}{2})$. Then, using the triangle inequality, one sees that

$$B\left(S(y), \frac{R'}{3}\right) \subset B\left(S(x_\infty), 2\frac{R'}{3}\right).$$

By definition of $R'$, we obtain

$$B\left(S(y), \frac{R'}{3}\right) \subset B\left(S(x_\infty), 2\frac{R'}{3}\right) \subset S\left(B\left(x_\infty, \frac{\varepsilon_m}{2}\right)\right).$$

This yields

$$S\left(B\left(x_\infty, \frac{\varepsilon_m}{2}\right)\right) \subset S\left(B\left(y, \varepsilon_m\right)\right)$$

since $B(x_\infty, \frac{\varepsilon_m}{2}) \subset B(y, \varepsilon_m)$ (by the triangle inequality and remembering that $\|x_\infty - y\| < \frac{\varepsilon_m}{2}$). To sum up, for $y \in \overline{\Omega_{2\beta/3}}$ satisfying $\|y - x_\infty\| < \min(\eta, \frac{\varepsilon_m}{2})$,

$$B\left(S(y), \frac{R'}{3}\right) \subset S\left(B(y, \varepsilon_m)\right).$$

This holds in particular for $x_n$, provided $n$ is large enough. Then $R(x_n) \ge \frac{R'}{3}$ and does not vanish. This proves (33).

We now may prove the lemma. Let $y \in B_{r_0}(S(U_j^\theta))$. There exists $x \in U_j^\theta$ such that $y \in B(S(x), r_0)$. Hence, by (33), $y \in S(B(x, \varepsilon_m)) \subset S(V_j^\theta)$. Finally

$$B_{r_0}\left(S\left(U_j^\theta\right)\right) \subset S\left(V_j^\theta\right).$$

This concludes the proof of Lemma 10. $\square$

## 3.4 Lipschitz continuity of the Jacobian determinant

We go on with the regularity properties of the transformations $T_j^\theta$, which come from the properties of the auxiliary function $S$. We prove that the Jacobian determinant satisfies the following Lipschitz property:

**Proposition 11.** There exist $K > 0$ and $\tilde{\varepsilon} > 0$ such that, for all $\theta \in \mathbb{R}$, $j \in \mathbb{Z}$ satisfying $U_j^\theta \ne \emptyset$, for all $\varepsilon \in (0, \tilde{\varepsilon})$, $z \in S(\Omega_{\beta/3})$ and $x, y \in B_\varepsilon(z)$,

$$\left|\left(\det DS^{-1}(x)\right) - \left(\det DS^{-1}(y)\right)\right| \le \frac{K}{2} \left|\left(\det DS^{-1}(z)\right)\right| \|x - y\|. \quad (35)$$

**Proof.** For $z \in S(\Omega_{\beta/3})$ define

$$g_z: \begin{array}{ccc} S(\Omega_\beta) & \longrightarrow & \mathbb{R} \\ x & \longmapsto & (\det DS^{-1}(z))^{-1} (\det DS^{-1}(x)) \end{array}$$
From (30) we get that $\det DS(S^{-1}(x)) = (-1)^{k-1} \frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(x))$. Hence for all $x \in S(\Omega_\beta)$, $g_z(x) = \left(\frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(z))\right) \left(\frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(x))\right)^{-1}$ and

$$Dg_z(x) = - \left( \frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(z)) \right) \left( \frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(x)) \right)^{-2} D \left( x \mapsto \frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(x)) \right).$$

Using conditions (17) and the $C^2$ regularity of $\Phi_0$ gives that, for $x \in S(\overline{\Omega_{2\beta/3}})$,

$$ \begin{aligned} \|Dg_z(x)\|_{op} & \le \left| \frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(z)) \right| \left| \frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(x)) \right|^{-2} \left\| D \frac{\partial \Phi_0}{\partial x_1}(\Gamma S^{-1}(x)) \right\|_{op} \|\Gamma\|_{op} \|DS^{-1}(x)\|_{op} \\ & \le \sup \left| \frac{\partial \Phi_0}{\partial x_1} \right| C_1^{-k+1} C_2^{-1} \sigma^{-k} \sup \left\| \nabla \frac{\partial \Phi_0}{\partial x_1} \right\| \gamma^{-k+1} \sup \|DS^{-1}\|_{op} =: K/2 \end{aligned} $$

where the upper bounds are taken on $[-L - \frac{2\beta}{3}, L + \frac{2\beta}{3}]^k$, except for the last one, taken on $S(\overline{\Omega_{2\beta/3}})$. Here $\|\cdot\|_{op}$ denotes the operator norm associated with the Euclidean norm.

Choose $\tilde{\varepsilon} = d(S(\overline{\Omega_{\beta/3}}), CS(\Omega_{2\beta/3}))$ (this is $> 0$). If $\varepsilon < \tilde{\varepsilon}$ and $x, y \in B_\varepsilon(z)$, the segment $[x, y]$ is included into $B_\varepsilon(z) \subset S(\overline{\Omega_{2\beta/3}})$. Then, by the fundamental theorem of differential calculus,

$$|g_z(x) - g_z(y)| \le \frac{K}{2} \|x - y\|.$$

Dividing both terms by $|(\det DS^{-1}(z))|$ yields (35). $\square$

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

Since, up to an additive constant, the applications $T_j^\theta$ coincide with $S$ on the open sets $V_j^\theta$, we obtain the following consequence:

**Corollary 12.** There exist $K > 0$ and $\tilde{\varepsilon} > 0$ such that, for all $\theta \in \mathbb{R}$, $j \in \mathbb{Z}$ satisfying $U_j^\theta \ne \emptyset$, $\varepsilon \in (0, \tilde{\varepsilon})$, $z \in T_j^\theta(V_j^\theta)$ and $x, y \in B_\varepsilon(z) \cap T_j^\theta(V_j^\theta)$,

$$|\det D(T_j^\theta)^{-1}(x) - \det(DT_j^\theta)^{-1}(y)| \le K |\det(DT_j^\theta)^{-1}(z)| \varepsilon$$

The condition (PE2) in [11] holds.

## 3.5 Geometry of the boundaries

This paragraph concludes the series of proofs concerning the conditions listed in Theorem 1. Instead of studying the expression which appears in (20), we will rely on Lemma 2.1 of [11], which gives a simpler sufficient condition. Both conditions are of geometric type: the first one is concerned with volumes of balls centered on the boundary, the second one, with the maximal number of crossings at one point (giving the number of open sets that the said balls intersect). The condition stated in (20) is satisfied if the limit of a quantity (that we will define and study below) is $< 1$. Then, for a sufficiently small parameter, the quantity itself is $< 1$.

We will follow the proof of [11] and check that the “sufficiently small” character does not depend too severely on the perturbation $\theta$. We need to give an upper bound for the volumes which appear in (20). To that end, we rectify the boundaries of the open sets and are brought back to the volume
of the portion of a ball comprized between two parallel hyperplanes. We rely on Theorem 8, on the uniform open charts defined below in (37) and which we use, too, to prove that the boundaries are null sets.

### Local charts, negligibility of the boundaries

We are concerned with the boundaries of the open sets $U_j^\theta$ defined by (10). There are portions of hyperplanes, which we do not need to rectify and the $\Sigma_j^\theta$. Recall that the function $f$ is defined by $f = \Phi_0 \circ \Gamma$ (see (4), (9), (26)). The distance $d$ is defined in (27), the real numbers $0 < \rho < r$ are given by Proposition 8, of which we will use the notations $U_a, V_a$ and $g_a$.

### Definition of the charts:

For $a = (a_1, \check{a}) \in \Omega_{\beta/3}$, set

$$U_a' := B(a_1, \rho) \times B(\check{a}, \rho). \qquad (36)$$

We define the chart by

$$\forall u = (u_1, \check{u}) \in U_a', \quad \Phi_a(u) = (u_1 - g_a(\check{u}), \check{u}). \qquad (37)$$

According to Proposition 8, $B(\check{a}, \rho) \subset V_a \subset B(\check{a}, r)$ and $g_a$, defined on $V_a$, is defined on the smaller set $B(\check{a}, \rho)$.

The application $\Phi_a$ is $C^2$ on $U_a'$ and clearly injective. Moreover,

$$D\Phi_a(u) = \begin{pmatrix} 1 & -\partial_2 g_a(\check{u}) & -\partial_3 g_a(\check{u}) & \dots & -\partial_k g_a(\check{u}) \\ 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 0 & 1 \end{pmatrix}. \qquad (38)$$

The inverse function theorem then proves that $\Phi_a$ is diffeomorphic on $U_a'$ onto its image.

Denote by $H$ the hyperplane $H = \{(0, \check{v}), \check{v} \in \mathbb{R}^{k-1}\}$. Let $u \in U_a'$ satisfy $f(u) = f(a)$. Then, according to (28), $\check{u} \in V_a$ and $u_1 = g_a(\check{u})$. Therefore $\Phi_a(u) = (0, \check{u}) \in H$. To sum up:

$$\Phi_a(\{u \in U_a' : f(u) = f(a)\}) \subset H. \qquad (39)$$

This boundary is locally rectified.

The compact set $\Sigma_j^\theta$ is covered by a finite number of charts. Since each local chart $\Phi_a$ has determinant 1, the volumes are conserved. This gives the following result:

**Lemma 13.** The sets $\Sigma_j^\theta$ ($j \in \mathbb{Z}, \theta \in \mathbb{R}$) are $C^2$ and have Lebesgue measure zero.

### Number of crossings

Consider a given point $x \in \mathbb{R}^k$ and a small neigbourhood $\mathcal{O}$ of $x$. We wish to give an upper bound for the number of boundaries of the open sets $U_j^\theta$ which intersect $\mathcal{O}$. Some boundary parts are the sides of the parallelepiped $\Omega$, of which at most $k$ intersect $\mathcal{O}$, if one is near a corner. We prove here that, if $\mathcal{O}$ is sufficiently small, it intersects at most one slanted boundary $\Sigma_j^k$ (whatever the value of $\theta$). This slanted boundary counts as two, because $\mathcal{O}$ intersects two open sets of the partition and that makes two terms in the sum of volumes we must consider (see (20) and (57)). The maximal number of crossings $Y$ is then $k+2$. Except for a countable number of parameters $\theta$, this number is even $k+1$.
**Lemma 14.** Let $\theta \in \mathbb{R}$, let $j, j'$ be two different integers such that $\Sigma_j^\theta$ and $\Sigma_{j'}^\theta$ are not empty. Take one point in each of them: $x \in \Sigma_j^\theta$, $x' \in \Sigma_{j'}^\theta$.

Let $M_{\Phi_0}^1$ and $M_\Gamma$ be positive real numbers such that

$$ \forall u \in [-L, L]^k, \|D\Phi_0(u)\| \le M_{\Phi_0}^1 \text{ and } \|\Gamma\| \le M_\Gamma $$

(this should hold for operator-norms associated with all the norms on $\mathbb{R}^k$ that we need to use). Then

$$ \|x - x'\| \ge \frac{2L}{M_{\Phi_0}^1 M_\Gamma}. $$

As a consequence, if

$$ \varepsilon_0 < \frac{L}{2M_{\Phi_0}^1 M_\Gamma}, \qquad (40) $$

the ball $B_{\varepsilon_0}(x)$ can't intersect both bands $B_{\varepsilon_0}(\Sigma_j^\theta)$ and $B_{\varepsilon_0}(\Sigma_{j'}^\theta)$.

**Proof.**

The hypotheses imply that

$$ \Phi_0(\Gamma(x)) = -\theta + (2j - 1)L, \quad \Phi_0(\Gamma(x')) = -\theta + (2j' - 1)L. $$

Applying the fundamental theorem of differential calculus gives

$$ |\Phi_0(\Gamma(x)) - \Phi_0(\Gamma(x'))| \le \sup_{y \in [x, x')} \|D\Phi_0(y)\| \|\Gamma\| \|x - x'\|. $$

Hence

$$ |\Phi_0(\Gamma(x)) - \Phi_0(\Gamma(x'))| = |2(j - j')L| \le M_{\Phi_0}^1 M_\Gamma \|x - x'\|. $$

But $|j - j'|$ is greater than 1, which gives the first result. The last point is then a consequence of the triangle inequality. $\square$

**A uniform version of Lemma 2.1 of [11]**

Let $\theta \in \mathbb{R}$ be a fixed perturbation, let $\varphi_\theta$ and $T_\theta$, be associated with $\theta$ by (11) and (13). We defined $\varepsilon_m > 0$ and the neighbourhoods $V_j^\theta$ of the $U_j^\theta$ by (31) and (32):

$$ \varepsilon_m = \frac{\rho}{3}, \quad V_j^\theta = B_{\varepsilon_m}(U_j^\theta) \subset \Omega_{\beta/3}. $$

Lemma 10 gives a positive real number $r_0$ such that, for all $\theta' \in \mathbb{R}$ and all $j$ such that $U_j^{\theta'} \neq \emptyset$,

$$ B_{r_0}(S(U_j^{\theta'})) \subset S(V_j^{\theta'}). $$

Let $x \in \mathbb{R}^k$, let $\varepsilon, \varepsilon_0$ satisfy

$$ 0 < \varepsilon < \varepsilon_0 < \min \left( \varepsilon_m, r_0, \frac{L}{2M_{\Phi_0}^1 M_\Gamma}, \tilde{\varepsilon} \right) \qquad (41) $$

with the notations of Lemma 14 and Proposition 11. Recall that $T_j^\theta$ is considered as defined on $V_j^\theta$, although its expression is well defined on the much broader set $\Omega_\beta$.
To check the condition (PE5), we must find an upper bound for the volume of the following set.

$$ (T_j^\theta)^{-1}(B_\varepsilon(\partial T_j^\theta(U_j^\theta))) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x). \quad (42) $$

According to Lemma 9, $T_\theta$ is diffeomorphic on $V_j^\theta$ onto its image. Moreover, it is expanding by Proposition 4. As in [11], we may state the following inclusions:

$$ (T_j^\theta)^{-1}(B_\varepsilon(\partial T_j^\theta(U_j^\theta))) \subset B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(\partial U_j^\theta) \subset V_j^\theta \subset B_{\frac{1}{\sqrt{\sigma}}\frac{d}{3}}(\Omega). \quad (43) $$

Formula (27) which defines $d$, implies that the set (42) is empty if $x \notin \Omega_{\beta/3}$.

The boundary of an open set $U_j^\theta$ consists of plane sets (sides of a parallelepiped) and of slanted boundaries like $\Sigma_j^\theta$. If we denote by $F_s$ these different kinds of sets, we obtain

$$ (T^\theta)^{-1}(B_\varepsilon(\partial T_\theta(U_j^\theta))) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon}(x) \subset \bigcup B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(F_s) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x), $$

and we will treat below each case (plane or not) separately.

### First case: $F_s$ is included in a hyperplane

The set $B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(F_s) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)$ has maximal volume when $F_s$ contains the center $x$ of the ball. In this case, the volume is smaller than that of a cylinder based on a $k-1$ dimensional ball, which gives:

$$ m \left( B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(F_s) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x) \right) \leq 2 \underbrace{\frac{\varepsilon}{\sqrt{\sigma}}}_{\text{height}} \times \underbrace{\gamma_{k-1} \left( \left(1 - \frac{1}{\sqrt{\sigma}}\right) \varepsilon_0 \right)}_{\text{volume of the ball}}^{k-1}. \quad (44) $$

### Second case: $F_s = \Sigma_j^\theta$

We compute an upper bound for the volume of the set below (supposed to be non empty):

$$ B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(\Sigma_j^\theta) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x). \quad (45) $$

We will use some auxiliary points and we need to locate them exactly.

#### Location of auxiliary points u, a, y, z

Let $u$ belong to the intersection. It is in the first set, which gives an $a \in \Sigma_j^\theta$ (hence in $\Omega \subset \Omega_{\beta/3}$) such that $\|u-a\| < \frac{1}{\sqrt{\sigma}}\varepsilon$. Recall that, thanks to (39), $\Phi_a(\Sigma_j^\theta \cap U_a') \subset H$.

It is in the ball too, hence $\|u-x\| < (1 - \frac{1}{\sqrt{\sigma}})\varepsilon_0$.

Therefore

$$ \|a - x\| < \frac{1}{\sqrt{\sigma}}\varepsilon + \left(1 - \frac{1}{\sqrt{\sigma}}\right)\varepsilon_0 \le \varepsilon_0 < \varepsilon_m = \frac{\rho}{3} < \frac{d}{3}. $$

Then $x \in \Omega_{\beta/3}$ (thanks to $d$) and $x$ is in the chart open set $U_a'$ (see (36)) associated with $a \in \Sigma_j^\theta$ (thanks to $\rho$).

If $y$ is in $B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)$, $\|a - y\| < 2\varepsilon_0$. Hence

$$ B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x) \subset B(a, 2\varepsilon_0). $$

This is a subset of $U_a'$ and of $\Omega_{\beta/3}$ because $2\varepsilon_0 < 2\rho/3 < 2d/3$.
And if $y \in B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(\Sigma_j^\theta) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)$, there exists $z \in \Sigma_j^\theta$ such that $\|y-z\| < \frac{1}{\sqrt{\sigma}}\varepsilon$. Then $\|a-z\| < 3\varepsilon_0$ and $z$ is in $U'_a$ too. We will use these remarks in Lemma 15.

Remark that, if $O$ is an open subset of $U'_a$, $m(\Phi_a(O)) = m(O)$. This is because $\Phi_a$ preserves the volume, as was already used to obtain Lemma 13. For the same reason, the linear transformation $D\Phi_a(x)$ preserves the volumes too:

$$m(D\Phi_a(x)^{-1}\Phi_a(O)) = m(O). \qquad (46)$$

We are brought back to the computation of the following measure

$$m\left(D\Phi_a(x)^{-1}\Phi_a\left(B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(\Sigma_j^\theta) \cap U'_a\right) \cap D\Phi_a(x)^{-1}\Phi_a\left(B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)\right)\right).$$

To this aim we have to locate the open set relatively to a convenient hyperplane.

We first establish some technical results.

**Lemma 15.** Let the points $x, a$ be as above and take $u, v$ in $U'_a$. Then we can write

$$\|D\Phi_a(x)^{-1}\Phi_a(u) - D\Phi_a(x)^{-1}\Phi_a(x)\| \le \|u - x\| + M\|u - x\|^2. \qquad (47)$$

$$\|D\Phi_a(x)^{-1}\Phi_a(u) - D\Phi_a(x)^{-1}\Phi_a(v)\| \le \|u - v\| + M\|v - x\|\|u - v\| + M\|u - v\|^2$$

with a constant $M > 0$, depending only on the bounds of $D\Phi_0, D^2\Phi_0$ and $(\partial_1\Phi_0)^{-1}$ on $\overline{\Omega_{2\beta/3}}$.

**Proof.** We apply the Taylor integral formula:

$$\begin{aligned} \Phi_a(u) - \Phi_a(x) &= D\Phi_a(x) \cdot (u-x) + \int_0^1 (1-t) D^2\Phi_a(x+t(u-x))(u-x)^2 dt \\ &= D\Phi_a(x) \cdot \left[ (u-x) + \int_0^1 (1-t) D\Phi_a(x)^{-1} D^2\Phi_a(x+t(u-x))(u-x)^2 dt \right]. \end{aligned}$$

Therefore,

$$D\Phi_a(x)^{-1}\Phi_a(u) - D\Phi_a(x)^{-1}\Phi_a(x) = (u-x) + \int_0^1 (1-t) D\Phi_a(x)^{-1} D^2\Phi_a(x+t(u-x))(u-x)^2 dt, \quad (48)$$

which will give the first inequality. To get the second one, we write

$$\Phi_a(u) - \Phi_a(v) = D\Phi_a(v) \cdot (u - v) + \int_0^1 (1 - t) D^2\Phi_a(v + t(u - v))(u - v)^2 dt$$

and develop the term $D\Phi_a(v)$. This gives

$$\Phi_a(u) - \Phi_a(v) = D\Phi_a(x) \cdot (u-v) + \int_0^1 D^2\Phi_a(x+s(v-x))(v-x) ds \cdot (u-v) + \int_0^1 (1-t) D^2\Phi_a(v+t(u-v))(u-v)^2 dt.$$

Compose by $D\Phi_a(x)^{-1}$:

$$\begin{aligned} D\Phi_a(x)^{-1}\Phi_a(u) - D\Phi_a(x)^{-1}\Phi_a(v) &= (u-v) \\ &\quad + \int_0^1 D\Phi_a(x)^{-1} D^2\Phi_a(x + s(v-x))(v-x) ds \cdot (u-v) \quad (49) \\ &\quad + \int_0^1 (1-t) D\Phi_a(x)^{-1} D^2\Phi_a(v + t(u-v))(u-v)^2 dt. \end{aligned}$$
All points appearing in the integrals belong to $U'_a$, which is a convex subset of $\Omega_{2\beta/3}$. The first, second differentials and the inverses appearing in the integrals are bounded, in norm, on $\Omega_{2\beta/3}$. If we call $M$ a common upper bound, we obtain, from (48) and (49) respectively, the first and the second inequality of (47). This completes the proof of Lemma 15. $\square$

We now turn to the set $E$ defined by

$$E = D\Phi_a(x)^{-1}\Phi_a\left(B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)\right). \qquad (50)$$

According to Lemma 15, if $y \in B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)$,

$$\|D\Phi_a(x)^{-1}\Phi_a(y) - D\Phi_a(x)^{-1}\Phi_a(x)\| \le \|y - x\|(1 + M\|y - x\|).$$

Then

$$D\Phi_a(x)^{-1}\Phi_a\left(B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)\right) \subset B(D\Phi_a(x)^{-1}\Phi_a(x), R) := F, \qquad (51)$$

with

$$R = \left(1 - \frac{1}{\sqrt{\sigma}}\right) \varepsilon_0 \left(1 + M\left(1 - \frac{1}{\sqrt{\sigma}}\right) \varepsilon_0\right),$$

since $\|y - x\| \le (1 - \frac{1}{\sqrt{\sigma}})\varepsilon_0$.

Now suppose that $y$ is in $B_{\varepsilon\frac{1}{\sqrt{\sigma}}}(\Sigma_j^\theta) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)$. We want to get an upper bound for the distance between the point $D\Phi_a(x)^{-1}\Phi_a(y)$ and the set $D\Phi_a(x)^{-1}H$, where $H$ is the hyperplane already defined by $H = \{(0, \tilde{v}), \tilde{v} \in \mathbb{R}^{k-1}\}$. Let $z \in \Sigma_j^\theta$ be such that $\|y - z\| < \varepsilon\frac{1}{\sqrt{\sigma}}$. Then $z \in U'_a$ and by Lemma 15,

$$\|D\Phi_a(x)^{-1}\Phi_a(y) - D\Phi_a(x)^{-1}\Phi_a(z)\| \le \|y - z\|(1 + M\|y - z\| + M\|x - z\|).$$

Remark that

$$\|x - z\| \le \|x - y\| + \|y - z\| \le \left(1 - \frac{1}{\sqrt{\sigma}}\right) \varepsilon_0 + \varepsilon\frac{1}{\sqrt{\sigma}},$$

with $\varepsilon \le \varepsilon_0$. Hence

$$d(D\Phi_a(x)^{-1}\Phi_a(y), D\Phi_a(x)^{-1}H) \le \varepsilon\frac{1}{\sqrt{\sigma}}\left(1 + M\varepsilon_0\left(1 + \frac{1}{\sqrt{\sigma}}\right)\right).$$

The set $H' := D\Phi_a(x)^{-1}H$ is a hyperplane too. Therefore

$$D\Phi_a(x)^{-1}\Phi_a\left(B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(\Sigma_j^\theta) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)\right) \subset B\left(D\Phi_a(x)^{-1}H, \varepsilon\frac{1}{\sqrt{\sigma}}(1 + M\varepsilon_0(1 + \sigma^{-1/2}))\right) := F_1. \quad (52)$$

The inclusions (52) and (51) give:

$$D\Phi_a(x)^{-1}\Phi_a\left(B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(\Sigma_j^\theta) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)\right) \subset F \cap F_1$$

We have seen, too, that this set $D\Phi_a(x)^{-1}\Phi_a\left(B_{\frac{1}{\sqrt{\sigma}}\varepsilon}(\Sigma_j^\theta) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x)\right)$ has the same volume as

$$B_{\varepsilon\frac{1}{\sqrt{\sigma}}}(\Sigma_j^\theta) \cap B_{(1-\frac{1}{\sqrt{\sigma}})\varepsilon_0}(x),$$
which is the set we are interested in. It implies

$$m\left(B_{\varepsilon\frac{1}{\sqrt{\sigma}}}\left(\Sigma_{j}^{\theta}\right) \cap B_{\left(1-\frac{1}{\sqrt{\sigma}}\right)\varepsilon_{0}}(x)\right) \leq m(F_{1} \cap F). \quad (53)$$

Then $F_1 \cap F$ is the intersection of a ball with the space comprised between two hyperplanes parallel to $H'$ and situated at the same distance. The volume in maximal when the center is on $H'$ and it is less than

$$\begin{align}
& \underbrace{2 \varepsilon \frac{1}{\sqrt{\sigma}} (1 + M \varepsilon_0 (1 + \sigma^{-1/2}))}_{\text{height}} \times \gamma_{k-1} \underbrace{\left( (1 - \frac{1}{\sqrt{\sigma}}) \varepsilon_0 (1 + M (1 - \frac{1}{\sqrt{\sigma}}) \varepsilon_0) \right)^{k-1}}_{\text{radius}=R} \tag{54} \\
& = 2 \gamma_{k-1} \varepsilon \frac{1}{\sqrt{\sigma}} \left( \left(1 - \frac{1}{\sqrt{\sigma}}\right) \varepsilon_0 \right)^{k-1} \times (1 + M \varepsilon_0 (1 + \sigma^{-1/2})) \left( \left(1 + M \left(1 - \frac{1}{\sqrt{\sigma}}\right) \varepsilon_0\right) \right)^{k-1} \nonumber
\end{align}$$

This upper bound holds for the simpler first case, when the boundary is a side of $\Omega$ (see (44)).

According to [11], we must find an upper bound for

$$G(\varepsilon, \varepsilon_0) = \sup_{x \in \mathbb{R}^n} G(x, \varepsilon, \varepsilon_0) \quad (55)$$

with

$$G(x, \varepsilon, \varepsilon_0) = \sum_i \frac{m(T_i^{-1}(B_{\varepsilon}(\partial T^\theta U_i^\theta)) \cap B(x, (1 - \frac{1}{\sqrt{\sigma}})\varepsilon_0))}{m(B(x, (1 - \frac{1}{\sqrt{\sigma}})\varepsilon_0))} \quad (56)$$

Precisely, $\eta(\varepsilon_0)$ is defined by

$$\eta(\varepsilon_0) = \frac{1}{\sqrt{\sigma}} + 2 \sup_{\varepsilon \le \varepsilon_0} \frac{G(\varepsilon, \varepsilon_0)}{\varepsilon} \varepsilon_0. \quad (57)$$

What precedes establishes that the sum (indexed by $i$) of all the volumes we have to consider is smaller than

$$Y \times 2\gamma_{k-1}\varepsilon \frac{1}{\sqrt{\sigma}} \left( \left(1 - \frac{1}{\sqrt{\sigma}}\right) \varepsilon_0 \right)^{k-1} \times (1 + M\varepsilon_0(1 + \sigma^{-1/2})) \left( \left(1 + M\left(1 - \frac{1}{\sqrt{\sigma}}\right) \varepsilon_0 \right) \right)^{k-1},$$

with $Y = k + 1$ if no slanted boundary $\Sigma_j^\theta$ meets the corner of $\Omega$, and $Y = k + 2$ otherwise. The slanted boundary appears twice in the sum, for two successive indices $i$.

With the definitions (55) and (56) we obtain

$$G(x, \varepsilon, \varepsilon_0) \le \frac{2\gamma_{k-1}Y\varepsilon\frac{1}{\sqrt{\sigma}}}{m\left(B\left(x, \left(1-\frac{1}{\sqrt{\sigma}}\right)\varepsilon_0\right)\right)} \left(\left(1-\frac{1}{\sqrt{\sigma}}\right)\varepsilon_0\right)^{k-1} (1+M\varepsilon_0(1+\sigma^{-1/2})) \left(\left(1+M\left(1-\frac{1}{\sqrt{\sigma}}\right)\varepsilon_0\right)\right)^{k-1},$$

which can be written more simply as

$$G(x, \varepsilon, \varepsilon_0) \le \frac{2\gamma_{k-1}Y}{\gamma_k(\sqrt{\sigma}-1)\varepsilon_0} \varepsilon \times (1 + M\varepsilon_0(1 + \sigma^{-1/2})) \left(\left(1 + M\left(1 - \frac{1}{\sqrt{\sigma}}\right)\varepsilon_0\right)\right)^{k-1}.$$

This does not depend on $x$. Similarly,

$$\frac{G(\varepsilon, \varepsilon_0)}{\varepsilon} \varepsilon_0 \le \frac{2\gamma_{k-1}Y}{\gamma_k(\sqrt{\sigma}-1)} \times (1 + M\varepsilon_0(1 + \sigma^{-1/2})) \left(\left(1 + M\left(1 - \frac{1}{\sqrt{\sigma}}\right)\varepsilon_0\right)\right)^{k-1},$$
which does not depend on $\varepsilon$. We finally obtain that the function $\eta$ is smaller than an increasing function $\bar{\eta}$ of $\varepsilon_0$:

$$ \begin{aligned} \eta(\varepsilon_0) &= \frac{1}{\sqrt{\sigma}} + 2 \sup_{\varepsilon \le \varepsilon_0} \frac{G(\varepsilon, \varepsilon_0)}{\varepsilon} \varepsilon_0 \\ &\le \overline{\eta(\varepsilon_0)} := \frac{1}{\sqrt{\sigma}} + \frac{4\gamma_{k-1}Y}{\gamma_k(\sqrt{\sigma}-1)} \times (1 + M\varepsilon_0(1 + \sigma^{-1/2})) \left( \left( 1 + M\left(1 - \frac{1}{\sqrt{\sigma}}\right)\varepsilon_0 \right) \right)^{k-1}, \end{aligned} \quad (58) $$

The limit of the upper bound $\overline{\eta(\varepsilon_0)}$, as $\varepsilon_0$ goes to 0, is

$$ \eta_Y(\sigma) = \frac{1}{\sqrt{\sigma}} + \frac{4\gamma_{k-1}Y}{\gamma_k(\sqrt{\sigma}-1)}, $$

which is assumed to be < 1 (see (14)). Therefore, there exists an $\varepsilon_0$ such that (PE5) is satisfied, which proves that all the conditions of Theorem 5.1 in [11], listed here in Theorem 1, are satisfied.

# 4 Proof of Theorems 2 and 3 : Properties of the average transfer operator

Seeing as $T_\theta : \Omega \to \Omega$ is nonsingular for every $\theta \in \mathbb{R}$, we can define its (Perron-Frobenius) transfer operator $P_\theta$.

Our goal is to define an averaged form of the transfer operator:

$$ Pf = \int_{\mathbb{R}} P_{\theta}f \, d\mathbb{P}_{\Theta_0}(\theta) = \int_{\mathbb{R}} H(\theta)P_{\theta}f \, d\theta, \quad (59) $$

for $f \in L^1(\Omega, m)$, where $H$ is a density function. This definition raises the issue of measurability in the integrand, which we shall address before establishing spectral properties for $P$.

## 4.1 Measurability and $L^p$-boundedness

**Proposition 16.** Let $f \in L^1(\Omega, m)$.

1. The mapping

$$ \begin{aligned} Qf : \quad & \Omega \times \mathbb{R} \quad \longrightarrow \quad \mathbb{R} \\ & (y, \theta) \quad \longmapsto \quad \sum_{j \in \mathbb{Z}} \mathbf{1}_{T_j^\theta(U_j^\theta)}(y) (f \circ (T_j^\theta)^{-1})(y) \left| \frac{\partial \Phi_0}{\partial x_1} (\Gamma(T_j^\theta)^{-1}(y)) \right|^{-1} \end{aligned} \quad (60) $$

is measurable.

2. For all $(y, \theta) \in \Omega \times \mathbb{R}$, $Qf(y, \theta) = P_\theta f(y)$.

3. The mapping

$$ \begin{aligned} \bar{Q}f : \quad & \Omega \times \mathbb{R} \quad \longrightarrow \quad \mathbb{R} \\ & (y, \theta) \quad \longmapsto \quad H(\theta)P_\theta f(y) \end{aligned} $$

is measurable and integrable over $\Omega \times \mathbb{R}$.
Proof.

1. For fixed $j \in \mathbb{Z}$, since the set $\{(y, \theta) \in \mathbb{R}^k \times \mathbb{R} : y \in T_j^\theta(U_j^\theta)\}$ is open, the map $(y, \theta) \mapsto \mathbf{1}_{T_j^\theta(U_j^\theta)}(y)$ is measurable. So are the maps $(y, \theta) \mapsto (f \circ (T_j^\theta)^{-1})(y)$ and $(y, \theta) \mapsto \left|\frac{\partial \Phi_0}{\partial x_1}(\Gamma(T_j^\theta)^{-1}(y))\right|^{-1}$. $Qf$ is therefore a series of measurable functions. In addition, for fixed $\theta$, only finitely many of the $U_j^\theta$ are nonempty, thus for every $(y, \theta) \in \Omega \times \mathbb{R}$, all but a finite number of the series' terms vanish. This makes the series pointwise convergent and thus $Q$ measurable.

2. For fixed $\theta \in \mathbb{R}$ and $A \subset \Omega$ open,

$$ \int_A P_\theta f(y) \, dy = \int_{(T^\theta)^{-1}(A)} f(x) \, dx = \int_{(T^\theta)^{-1}(A)} \sum_{j \in \mathbb{Z}} \mathbf{1}_{U_j^\theta}(x) f(x) \, dx $$

whence we get by substituting $x = (T_j^\theta)^{-1}(y)$ in each term:

$$ \begin{aligned} \int_A P_\theta f(y) \, dy &= \int_A \sum_{j \in \mathbb{Z}} (\mathbf{1}_{U_j^\theta} \circ (T_j^\theta)^{-1})(y) (f \circ (T_j^\theta)^{-1})(y) \left| \det DT_j^\theta ((T_j^\theta)^{-1}(y)) \right| \, dy \\ &= \int_A \sum_{j \in \mathbb{Z}} \mathbf{1}_{T_j^\theta(U_j^\theta)}(y) (f \circ (T_j^\theta)^{-1})(y) \left| \frac{\partial \Phi_0}{\partial x_1} (\Gamma(T_j^\theta)^{-1}(y)) \right|^{-1} \, dy \\ &= \int_A Qf(y, \theta) \, dy \end{aligned} $$

With this equality holding for open $A \subset \Omega$, a measure extension theorem allows us to induce that $P_\theta f(y) = Qf(y, \theta)$ for every $(y, \theta) \in \Omega \times \mathbb{R}$.

3. $\bar{Q}$ is measurable as it is the product of two measurable functions, and for all $\theta \in \mathbb{R}$, $|P_\theta f| \le P_\theta |f|$ for which Fubini-Tonelli yields:

$$ \iint_{\Omega \times \mathbb{R}} H(\theta) P_{\theta} |f|(y) \, d\theta \, dy = \int_{\mathbb{R}} H(\theta) \left( \int_{\Omega} P_{\theta} |f|(y) \, dy \right) \, d\theta = \|f\|_{L^1}, $$

hence the result.

<figure><img src="image_4.png" /></figure>

This ensures that the operator described in (59) is well-defined. Notice that the expression given for $P_\theta$ in (60) allows us to establish an $L^\infty \to L^\infty$ bound:

$$ |P_{\theta}f(x)| \le N \|f\|_{L^{\infty}} C_1^{-\frac{k-1}{2}} C_2^{-\frac{1}{2}} \sigma^{-\frac{k}{2}} \quad (61) $$

where $N$ denotes the maximal number of nonempty open subsets $U_j^\theta$ for fixed $\theta \in \mathbb{R}$. Since this number has a bound that is uniform in $\theta$ (see (25)), we are able to derive a bound for the average operator on $L^\infty$ after establishing $L^1$-boundedness, which will be achieved by showing that $P$ is a Markov operator.

We recall that, for some measure space $(X, \mathcal{T}, \mu)$, a Markov operator over $X$ in the sense of Lasota & Mackey (see [8]) is a linear operator $T: L^1(X, \mu) \to L^1(X, \mu)$ such that for all nonnegative $f \in L^1$, $Tf \ge 0$ and $\|Tf\|_{L^1} = \|f\|_{L^1}$.

**Theorem 17.** *P* is a Markov operator.
### Proof

Let $f \in L^1(\Omega, m)$ such that $f \ge 0$. Thanks to properties (3.2.4), (3.2.5) from [8], the $P_\theta$ are Markov operators, hence $Pf = \int_{\mathbb{R}} H(\theta) P_\theta f \, d\theta \ge 0$, and

$$ \begin{aligned} \|Pf\|_{L^1} &= \int_{\Omega} Pf(y) \, dy = \iint_{\Omega \times \mathbb{R}} H(\theta) P_{\theta} f(y) \, d\theta \, dy \\ &= \int_{\mathbb{R}} H(\theta) \left( \int_{\Omega} P_{\theta} f(y) \, dy \right) \, d\theta = \int_{\mathbb{R}} H(\theta) \|P_{\theta} f\|_{L^1} \, d\theta \\ &= \int_{\mathbb{R}} H(\theta) \|f\|_{L^1} \, d\theta = \|f\|_{L^1}, \end{aligned} $$

establishing that $P$ also is a Markov operator.

<figure><img src="image_5.png" /></figure>

Proposition 3.1.1 of [8] then allows us to establish that:

**Corollary 18.** For all $f \in L^1(\Omega, m)$, $\|Pf\|_{L^1} \le \|f\|_{L^1}$.

**Theorem 19.** For all $p \in [1, \infty]$, $P: L^p \to L^p$ is a bounded linear operator.

### Proof

By interpolation of (61) and Corollary 18, $P$ maps $L^p$ onto itself and by setting $s = 1 - \frac{1}{p}$, one has:

$$ \|P\|_{L(L^p)} \le \|P\|_{L(L^1)}^{1-s} \|P\|_{L(L^\infty)}^s \le N^s C_1^{-\frac{(k-1)s}{2}} C_2^{-\frac{s}{2}} \sigma^{-\frac{ks}{2}}, $$

and thus $P \in \mathcal{L}(L^p)$.

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

## 4.2 Lasota-Yorke inequality and spectral properties

Our aim now is to apply the Ionescu-Tulcea & Marinescu spectral decomposition theorem on a suitable $P$-stable subspace. The space of functions of weak oscillation, as defined in [1, 11], is one such subspace:

$$ V_1(\Omega) = \left\{ f \in L^1(\mathbb{R}^k) : \supp f \subset \Omega, |f|_1 := \sup_{0<\varepsilon<\varepsilon_0} \varepsilon^{-1} \int \mathrm{osc}(f, B_\varepsilon(x)) \, dx < +\infty \right\}, \quad (62) $$

where for any non-null measurable set $A$ and all $f \in L^1$, the oscillation of $f$ over $A$ is defined in [1] by

$$ \mathrm{osc}(f, A) = \ess\sup_{x,y \in A} |f(x) - f(y)|. $$

The first ingredient required for the decomposition theorem to apply is a Lasota-Yorke inequality on $P$. One such estimate was established in [11] which can be applied to our $P_\theta$:

**Lemma 20.** Under the conditions (14) to (17) of Section 2.1, there exist $\eta \in [0, 1[$ and $D \in \mathbb{R}^+$ such that, for all $\theta \in \mathbb{R}$, $\forall f \in V_1$,

$$ |P_{\theta} f|_1 \le \eta |f|_1 + D \|f\|_{L_m^1}. \quad (63) $$

### Proof.

This is a consequence of the proof of Lemma 4.1 of [11]. For $\varepsilon \in (0, \varepsilon_0)$ of Theorem 1, Lemma 4.1
gives a Lasota-Yorke inequality for the Perron-Frobenius operator $P_{\theta}$, of the shape (63), with explicit $\eta$ and $D$ given by

$$ \eta = \left(1 + \frac{K}{\sqrt{\sigma}}\varepsilon_0\right) \left(\frac{1}{\sqrt{\sigma}} + 2 \sup \frac{G(\varepsilon, \varepsilon_0)\varepsilon_0}{\varepsilon}\right) $$

and

$$ D = \frac{K}{\sqrt{\sigma}}\varepsilon + 2 \left(1 + \frac{K}{\sqrt{\sigma}}\varepsilon\right) G(\varepsilon, \varepsilon_0). $$

With the notation (58), we get

$$ \eta \le \left(1 + \frac{K}{\sqrt{\sigma}}\varepsilon_0\right) \overline{\eta(\varepsilon_0)}, \quad D \le \frac{K}{\sqrt{\sigma}}\varepsilon_0 + \left(1 + \frac{K}{\sqrt{\sigma}}\varepsilon_0\right) \overline{\eta(\varepsilon_0)}. $$

For $\varepsilon_0$ sufficiently small, $\eta$ is smaller than 1. Neither of them depends on $\theta$ since the constants $K, \sigma$ and the function $\overline{\eta(\varepsilon_0)}$ do not depend on $\theta$. $\square$

It follows from Lemma 20 that $V_1$ is invariant under the operators $P_{\theta}$ for all $\theta \in \mathbb{R}$. In the following, we shall establish a similar estimate for $P$.

**Lemma 21.** For all $f \in V_1(\Omega)$ and $\varepsilon > 0$, the map

$$ \begin{array}{ccc} \mathbb{R}^k \times \mathbb{R} & \to & \mathbb{R} \\ (x, \theta) & \mapsto & \operatorname{osc}(P_{\theta}f, B_{\varepsilon}(x)) \end{array} $$

is measurable.

**Proof**

Let $a > 0$ and $O_{\theta} : (y, z) \in \Omega^2 \mapsto |P_{\theta}f(y) - P_{\theta}f(z)|$. We shall denote by $\chi_a$ the indicator function of the set

$$ \{(x, y, z, \theta) \in \mathbb{R}^k \times \Omega \times \Omega \times \mathbb{R} : y, z \in B_{\varepsilon}(x) \text{ and } O_{\theta}(y, z) > a\}, $$

whose measurability is easily deduced from that of $(\theta, y) \mapsto P_{\theta}f(y)$. Define

$$ M_a : (x, \theta) \in \mathbb{R}^k \times \mathbb{R} \mapsto \iint_{\Omega^2} \chi_a(x, y, z, \theta) \, dy \, dz $$

By the Fubini-Tonelli theorem, $M_a$ is measurable, thus so is the set

$$ \left\{ (x, \theta) \in \mathbb{R}^k \times \mathbb{R} : \underset{y,z \in B_{\varepsilon}(x)}{\operatorname{ess\,sup}} O_{\theta}(y,z) > a \right\} = \left\{ (x, \theta) : m\left( \left\{ (y,z) \in B_{\varepsilon}(x)^2 : O_{\theta}(y,z) > a \right\} \right) > 0 \right\} \\ = \{(x, \theta) : M_a(x, \theta) > 0\} = M_a^{-1}((0, +\infty)) $$

Hence the map $(x, \theta) \in \mathbb{R}^k \times \mathbb{R} \mapsto \underset{y,z \in B_{\varepsilon}(x)}{\operatorname{ess\,sup}} O_{\theta}(y,z)$ is measurable, which concludes the proof. $\square$

The proof for the main result in this section utilizes the following well-known estimate:

$$ \sup_{x \in X} \int_Y g(x, y) \, d\mu_Y(y) \le \int_Y \sup_{x \in X} g(x, y) \, d\mu_Y(y) $$

for all nonnegative, bounded $g \in L^1(X \times Y, \mu_X \otimes \mu_Y)$. A measure-theoretic version shall also be utilized, for which we give a short proof.
**Lemma 22.** Let $(X, \mathcal{T}_X, \mu_X)$, $(Y, \mathcal{T}_Y, \mu_Y)$ be σ-finite measure spaces, and consider a nonnegative essentially bounded $g \in L^1(X \times Y, \mu_X \otimes \mu_Y)$. The mapping $y \in Y \mapsto \text{ess sup}_{x \in A} g(x, y)$ is measurable, and

$$ \text{ess sup}_{x \in A} \int_Y g(x, y) \, d\mu_Y(y) \le \int_Y \text{ess sup}_{x \in A} g(x, y) \, d\mu_Y(y) \quad (64) $$

for all $A \in \mathcal{T}_X$ with positive measure.

### Proof

Measurability is established as in Lemma 21. Consider $f : (x, y) \mapsto \text{ess sup}_A g(\cdot, y) - g(x, y)$ and denote by $\chi$ the indicator of $f^{-1}(\mathbb{R}_-^*)$. By Fubini-Tonelli,

$$ \iint_{X \times Y} \chi = \int_Y \mu_X \left( \left\{ x \in X : g(x, y) > \text{ess sup}_A g(\cdot, y) \right\} \right) d\mu_Y(y) = 0, $$

which entails that $f^{-1}(\mathbb{R}_-^*)$ is null and thus that $f \ge 0$ almost everywhere. Therefore, for $\mu_X$-almost every $x \in X$,

$$ \int_Y g(x, y) \, d\mu_Y(y) \le \int_Y \text{ess sup}_A g(\cdot, y) \, d\mu_Y(y), $$

hence (64).

<figure><img src="image_7.png" /></figure>

We may now move on to the proof of this section's main result:

**Theorem 23.** There exist $\eta \in (0, 1)$ and $C > 0$ such that, for all $f \in V_1(\Omega)$, $Pf \in V_1$ and

$$ \|Pf\|_{V_1} \le \eta \|f\|_{V_1} + C \|f\|_{L^1} $$

### Proof

By Lemma 20, for all $f \in V_1$ and $\theta \in \mathbb{R}$,

$$ |P_{\theta}f|_1 \le \eta |f|_1 + D \|f\|_{L^1} $$

where $\eta \in (0, 1)$, $D > 0$ do not depend on $f$ or $\theta$. Furthermore, (64) and the definition of $P$ yield:

$$ \begin{aligned} |Pf|_1 &= \sup_{0<\varepsilon<\varepsilon_0} \varepsilon^{-1} \int_{\mathbb{R}^k} \text{ess sup}_{y,z \in B_\varepsilon(x)} |Pf(y) - Pf(z)| \, dx \\ &= \sup_{0<\varepsilon<\varepsilon_0} \varepsilon^{-1} \int_{\mathbb{R}^k} \text{ess sup}_{y,z \in B_\varepsilon(x)} \left| \int_{\mathbb{R}} H(\theta) (P_\theta f(y) - P_\theta f(z)) \, d\theta \right| \, dx \\ &\le \sup_{0<\varepsilon<\varepsilon_0} \varepsilon^{-1} \int_{\mathbb{R}^k} \text{ess sup}_{y,z \in B_\varepsilon(x)} \int_{\mathbb{R}} H(\theta) |P_\theta f(y) - P_\theta f(z)| \, d\theta \, dx \\ &\le \sup_{0<\varepsilon<\varepsilon_0} \varepsilon^{-1} \int_{\mathbb{R}^k} \int_{\mathbb{R}} H(\theta) \text{ess sup}_{y,z \in B_\varepsilon(x)} |P_\theta f(y) - P_\theta f(z)| \, d\theta \, dx \\ &= \sup_{0<\varepsilon<\varepsilon_0} \varepsilon^{-1} \int_{\mathbb{R}} \int_{\mathbb{R}^k} H(\theta) \text{osc}(P_\theta f, B_\varepsilon(x)) \, dx \, d\theta \\ &\le \int_{\mathbb{R}} H(\theta) \left( \sup_{0<\varepsilon<\varepsilon_0} \varepsilon^{-1} \int_{\mathbb{R}^k} \text{osc}(P_\theta f, B_\varepsilon(x)) \, dx \right) \, d\theta = \int_{\mathbb{R}} H(\theta) |P_\theta f|_1 \, d\theta, \end{aligned} $$

and it follows from (63) that

$$ |Pf|_1 \le \int_{\mathbb{R}} H(\theta) (\eta |f|_1 + D \|f\|_{L^1}) \, d\theta = \eta |f|_1 + D \|f\|_{L^1}. \quad (65) $$
Recall in addition that $\|Pf\|_{L^1} \le \|f\|_{L^1}$ for all $f \in V_1$; adding this to (65) yields

$$ \begin{aligned} \|Pf\|_{V_1} &\le \eta |f|_1 + D \|f\|_{L^1} + \|f\|_{L^1} \\ &= \eta |f|_1 + D \|f\|_{L^1} + (\eta + 1 - \eta) \|f\|_{L^1} \\ &= \eta \|f\|_{V_1} + (D + 1 - \eta) \|f\|_{L^1}, \end{aligned} $$

and the result is established for $C := D + 1 - \eta$. $\square$

**Proposition 24.** *P is continuous on $V_1$. Furthermore, for $\lambda \in \mathbb{C}^* \setminus \sigma(S)$ such that $\lambda \ne \lambda_i$ for $1 \le i \le p$, The operator $P - \lambda I$ has a continuous inverse given by*

$$ (P - \lambda I)^{-1} = (S - \lambda I)^{-1} \left( I - \sum_{i=1}^{p} \frac{\lambda_i}{\lambda_i - \lambda} \Pi_i \right). $$

and the resolvent operator of P has simple poles at $\lambda_1, \dots, \lambda_p$ and the eigenvalues $\lambda_i$ have index 1.

**Proof.**

Let $v \in V_1$ and let us solve $(P - \lambda I)u = v$. In this case, $u$ satisfies

$$ (S - \lambda I)u + \sum_{i=1}^{p} \lambda_i \Pi_i u = v. $$

Applying $\Pi_j$ and using the commutation relations gives

$$ (\lambda_j - \lambda)\Pi_j u = \Pi_j v. $$

The hypotheses on $\lambda$ allow us to invert $(S - \lambda I)$ and to write

$$ u = (S - \lambda I)^{-1} \left( v - \sum_{i=1}^{p} \frac{\lambda_i}{\lambda_i - \lambda} \Pi_i v \right). $$

This is the expression mentioned in the proposition. One may check that it gives a preimage of $v$, since (for a nonzero $\lambda$)

$$ \Pi_j(S - \lambda I)^{-1} = (S - \lambda I)^{-1} \Pi_j = -\frac{1}{\lambda} \Pi_j. $$

The assertion on the poles of the resolvent operator is straightforward. The assertion about the index of the eigenvalues $\lambda_i$ comes from Theorem VII.3.18 in [3]. $\square$

We now have every ingredient we need for Ionescu-Tulcea & Marinescu's theorem to apply, which gives points 1, 2, 3 of Theorem 2. Simplicity of the eigenvalues follows from Proposition 24.

**Lemma 25.** *The averaged transfer operator P has the eigenvalue $\lambda_1 = 1$, and $h_* := \Pi_1(\frac{1}{m(\Omega)})$ is a nonnegative fixed point.*

**Proof**

Applying Lemma (4.1) of [5] with $c = 1$, for all $f \in V_1$ gives a function $\tilde{f} \in V_1$ such that

$$ \lim_{n \to \infty} \left\| \tilde{f} - \frac{1}{n} \sum_{k=1}^{n} P^k f \right\|_{L^1} = 0. $$
Taking $\lambda_1 = 1$, the projector $\Pi_1$ from Theorem 2 on $V_1$ is then defined by setting $\tilde{f} = \Pi_1 f$. It is a positive operator, in the sense that, if $g \in V_1$ and $g \ge 0$, then $\Pi_1 g \ge 0$ too (consider an a.e. converging subsequence of $\frac{1}{n} \sum_{k=1}^n P^k g$, which is a sequence of $\ge 0$ functions).

One may check that

$$ \int_{\Omega} \frac{1}{n} \sum_{k=1}^{n} P^k f \, dm = \int_{\Omega} f \, dm, $$

since $\int_{\Omega} Pf \, dm = \int_{\Omega} f \, dm$ due to $P$ being Markov. Hence, if $f > 0$, $\Pi_1 f$ is not the zero function because $\int_{\Omega} \Pi_1 f \, dm = \int_{\Omega} f \, dm \ne 0$. Moreover, the function $\tilde{f} = \Pi_1 f$ satisfies $P\tilde{f} = \tilde{f}$. Thus $h_* = \Pi_1(\frac{1}{m(\Omega)})$ is a fixed point of $P$ and it is nonnegative by positiveness of $\Pi_1$. $\square$

## Proof of Theorem 3

Here $\theta = (\theta_n)_{n \in \mathbb{N}} = (\theta_0, \tilde{\theta}) \in \mathbb{R}^N$ is a real sequence. In (2), we defined a mapping $F$ on $\mathbb{R}^N \times \Omega$, setting $F((\theta_0, \tilde{\theta}), x) = (\tilde{\theta}, T^{\theta_0}(x))$. We seek to establish that $\mathbb{P}_{\Theta} \otimes \mu$ is invariant for $F$. It suffices to show that

$$ (\mathbb{P}_{\Theta} \otimes \mu)(A \times B) = (\mathbb{P}_{\Theta} \otimes \mu)(F^{-1}(A \times B)) $$

for all $A = A_0 \times \dots \times A_n \times \mathbb{R} \times \dots$ with $A_0, \dots, A_n \in \mathcal{B}(\mathbb{R})$, $n \in \mathbb{N}$, and all $B \in \mathcal{B}(\Omega)$, as sets of this form generate the Borel $\sigma$-algebra over $\mathbb{R}^N \times \Omega$.

We have

$$ \begin{aligned} (\mathbb{P}_{\Theta} \otimes \mu)(F^{-1}(A \times B)) &= \iint_{\mathbb{R}^N \times \Omega} \mathbf{1}_{A \times B}(F(\theta, x)) \, d(\mathbb{P}_{\Theta} \otimes \mu)(\theta, x) \\ &= \int_{\mathbb{R}^N} \mathbf{1}_A(S\theta) \left[ \int_{\Omega} \mathbf{1}_B(T^{\theta_0}(x)) \, d\mu(x) \right] \, d\mathbb{P}_{\Theta}(\theta) \\ &= \iint_{\mathbb{R} \times \mathbb{R}^N} \mathbf{1}_{\mathbb{R} \times A}(\theta_0, \tilde{\theta}) \left[ \int_{(T^{\theta_0})^{-1}(B)} \, d\mu(x) \right] \, d(\mathbb{P}_{\Theta_0} \otimes \mathbb{P}_{\Theta})(\theta_0, \tilde{\theta}), \end{aligned} $$

thanks to the independence of the variables $(\Theta_t)$, which allows us to split the integral as follows:

$$ \begin{aligned} (\mathbb{P}_{\Theta} \otimes \mu)(F^{-1}(A \times B)) &= \int_A \left[ \int_{\mathbb{R}} \int_{(T^{\theta_0})^{-1}(B)} \, d\mu(x) \, d\mathbb{P}_{\Theta_0}(\theta_0) \right] \, d\mathbb{P}_{\Theta}(\tilde{\theta}) \\ &= \int_A \left[ \int_{\mathbb{R}} \int_{(T^{\theta_0})^{-1}(B)} h_* \, dm(x) \, d\mathbb{P}_{\Theta_0}(\theta_0) \right] \, d\mathbb{P}_{\Theta}(\tilde{\theta}) \\ &= \int_A \left[ \int_{\mathbb{R}} \int_B P_{\theta_0} h_* \, dm(x) \, d\mathbb{P}_{\Theta_0}(\theta_0) \right] \, d\mathbb{P}_{\Theta}(\tilde{\theta}) \end{aligned} $$

by definition of $\mu$ and $P_{\theta_0}$. It then follows by applying Fubini's theorem that

$$ \begin{aligned} (\mathbb{P}_{\Theta} \otimes \mu)(F^{-1}(A \times B)) &= \int_A \left[ \int_B \int_{\mathbb{R}} P_{\theta_0} h_* \, d\mathbb{P}_{\Theta_0}(\theta_0) \, dm(x) \right] \, d\mathbb{P}_{\Theta}(\tilde{\theta}) \\ &= \int_A \left[ \int_B P h_* \, dm(x) \right] \, d\mathbb{P}_{\Theta}(\tilde{\theta}), \end{aligned} $$

whence, since $Ph_* = h_*$,

$$ \begin{aligned} (\mathbb{P}_{\Theta} \otimes \mu)(F^{-1}(A \times B)) &= \int_A \left[ \int_B \, d\mu(x) \right] \, d\mathbb{P}_{\Theta}(\tilde{\theta}) \\ &= \iint_{A \times B} \, d(\mathbb{P}_{\Theta} \otimes \mu)(\tilde{\theta}, x) = (\mathbb{P}_{\Theta} \otimes \mu)(A \times B), \end{aligned} $$

which establishes the invariance of $\mu$. $\square$
References

[1] Aimino, R., Nicol, M., Vaienti, S.: Annealed and quenched limit theorems for random expanding dynamical systems, Probab. Theory Relat. Fields 162, No 1-2, 233-274 (2015)

[2] Alves, J.F., Freitas, J.M., Luzzatto, S., Vaienti, S.: From rates of mixing to recurrence times via large deviations. Advances in Mathematics 228, no. 2, 1203-1236 (2011)

[3] Dunford, N., Schwartz, J. Linear operators. Part I. General theory. John Wiley and sons, Inc., New York, 1988

[4] Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Mathematische Zeitschrift 180, 119-140 (1982)

[5] Ionescu Tulcea, C.T., Marinescu, G.: Théorie ergodique pour des classes d'opérations non complètement continues. Annals of Mathematics 52, no. 2, 140-147 (1950)

[6] Jager, L., Maes, J., Ninet, A.: Exponential decay of correlations for a real-valued dynamical system embedded in $\mathbb{R}^2$. Comptes rendus - Mathématique 353, no. 11, 1041-1045 (2015)

[7] Jager, L., Maes, J., Ninet, A.: Exponential decay of correlations for a real-valued dynamical system generated by a $k$ dimensional system. Acta Appl. Math. doi 10.1007/s10440-018-0192-z (2018)

[8] Lasota, A., Mackey, M.C.: Chaos, fractals and noise : stochastic aspects of dynamics. Springer Verlag, New York (1998)

[9] Lasota, A., Yorke, J.A. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186(1973), 481-488 (1974)

[10] Liverani, C. Multidimensional expanding maps with singularities: a pedestrian approach. Ergodic Theory Dyn. Syst. 33, No 1, 168-182 (2013)

[11] Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Israel Journal of Mathematics 116, 223-248 (2000)

[12] Socala, J. Exactness of expanding mappings, Ann. Pol. Math. 49, no 1, 93-99 (1988)

[13] Tsujii, M. Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143, No 2, 349-373 (2001)

[14] Tsujii, M. Piecewise expanding maps on the plane with singular ergodic properties. Ergodic Theory Dyn. Syst. 20, No6, 1851-1857 (2000)

[15] Young, L.S.: Recurrence times and rates of mixing. Israel Journal of Mathematics 110, 153-188 (1999)