Given a stationary Markov process and $n \in \mathbb{N}$, one can extend the notion of kernel to higher iterates: Define $P^m: S \times \mathcal{B}(S) \to \mathbb{R}^+$

$$P^m(x, A) = \mathbb{P}(X_{n+m} \in A | X_n = x).$$

For any $n \in \mathbb{N}$, $P^n$ generates an action on the set of measures on $(S, \mathcal{B}(S))$ in the following way. Given $\mu$ a measure on $S$, define

$$\mathcal{P}\mu(A) := \int_S P(x, A)d\mu(x),$$

and

$$\mathcal{P}^n\mu(A) := \int_S P^n(x, A)d\mu(x).$$

Using the properties of transition kernels one can prove that, {$P^n$} satisfies the semi-group property

$$\mathcal{P}^n \circ \mathcal{P}^m = \mathcal{P}^{n+m}$$

making $\mathcal{P}$ the generator of a semi-group action on probability measures on $(S, \mathcal{B}(S))$.

**Definition A.2.** A stationary Markov chain is said to be geometrically ergodic if there are $C > 0$ and $\lambda \in (0, 1)$ such that

$$d_{TV}(P^n(x_1, \cdot), P^n(x_2, \cdot)) \le C\lambda^n, \quad \forall x_1, x_2 \in S.$$

For a definition of the Total Variation distance $d_{TV}$ see the beginning of Sec. C. From Definition A.2 follows that if a Markov chain is geometrically ergodic, then there is a probability measure $\eta_0$ such that, for every probability measure $\mu$ on $(S, \mathcal{B}(S))$,

$$d_{TV}(\mathcal{P}^n\mu, \eta_0) \le C\lambda^n.$$

The measure $\eta_0$ satisfies $\mathcal{P}(\eta_0) = \eta_0$ and is also called a stationary distribution or stationary measure.

## A.2 | Sufficient conditions for geometric ergodicity

In this subsection we give a sufficient condition that ensures geometric ergodicity of a stationary Markov chain. Weaker conditions working in more general setups are available and involve *petite sets* [21] or *Lyapunov functions* [31].

**Theorem A.1** [37]. Let {$X_n$}$_{n \in \mathbb{N}_0}$ be a stationary Markov chain on $(S, \mathcal{B}(S))$ with transition kernel $P: S \times \mathcal{B}(S) \to \mathbb{R}^+$. Assume there is $\nu$ a probability measure, $\varepsilon > 0$ and $n_0 \in \mathbb{N}$ such that

$$P^{n_0}(x, \cdot) \ge \varepsilon\nu(\cdot), \quad \forall x \in S.$$

Then the Markov chain is geometrically ergodic.
Given a stationary Markov process and $n \in \mathbb{N}$, one can extend the notion of kernel to higher iterates: Define $P^m: S \times \mathcal{B}(S) \to \mathbb{R}^+$

$$P^m(x, A) = \mathbb{P}(X_{n+m} \in A | X_n = x).$$

For any $n \in \mathbb{N}$, $P^n$ generates an action on the set of measures on $(S, \mathcal{B}(S))$ in the following way. Given $\mu$ a measure on $S$, define

$$\mathcal{P}\mu(A) := \int_S P(x, A)d\mu(x),$$

and

$$\mathcal{P}^n\mu(A) := \int_S P^n(x, A)d\mu(x).$$

Using the properties of transition kernels one can prove that, {$P^n$} satisfies the semi-group property

$$\mathcal{P}^n \circ \mathcal{P}^m = \mathcal{P}^{n+m}$$

making $\mathcal{P}$ the generator of a semi-group action on probability measures on $(S, \mathcal{B}(S))$.

**Definition A.2.** A stationary Markov chain is said to be geometrically ergodic if there are $C > 0$ and $\lambda \in (0, 1)$ such that

$$d_{TV}(P^n(x_1, \cdot), P^n(x_2, \cdot)) \le C\lambda^n, \quad \forall x_1, x_2 \in S.$$

For a definition of the Total Variation distance $d_{TV}$ see the beginning of Sec. C. From Definition A.2 follows that if a Markov chain is geometrically ergodic, then there is a probability measure $\eta_0$ such that, for every probability measure $\mu$ on $(S, \mathcal{B}(S))$,

$$d_{TV}(\mathcal{P}^n\mu, \eta_0) \le C\lambda^n.$$

The measure $\eta_0$ satisfies $\mathcal{P}(\eta_0) = \eta_0$ and is also called a stationary distribution or stationary measure.

## A.2 | Sufficient conditions for geometric ergodicity

In this subsection we give a sufficient condition that ensures geometric ergodicity of a stationary Markov chain. Weaker conditions working in more general setups are available and involve *petite sets* [21] or *Lyapunov functions* [31].

**Theorem A.1** [37]. Let {$X_n$}$_{n \in \mathbb{N}_0}$ be a stationary Markov chain on $(S, \mathcal{B}(S))$ with transition kernel $P: S \times \mathcal{B}(S) \to \mathbb{R}^+$. Assume there is $\nu$ a probability measure, $\varepsilon > 0$ and $n_0 \in \mathbb{N}$ such that

$$P^{n_0}(x, \cdot) \ge \varepsilon\nu(\cdot), \quad \forall x \in S.$$

Then the Markov chain is geometrically ergodic.
### A.3 | Randomly forced systems and Markov chains

In this section we discuss the difference, in terms of mathematical definitions, between random and deterministic forcing.

By *random forcing*, we mean that given a probability space $(\Omega, \nu)$ and $f : \Omega \times X \to X$, at the $n$-th iteration we apply the map $f_{\zeta_n} := f(\zeta_n, \cdot) : X \to X$, where $\{\zeta_n\}_{n \in \mathbb{N}}$ is an i.i.d sequence of random variables defined on some probability space $(\Xi, \mathbb{P})$ with values in $\Omega$ and distributed according to $\nu$. Fixed $w \in \Xi$, the forward orbits of the system are given by

$$O(x) := \{ f_{\zeta_n(w)} \circ \dots \circ f_{\zeta_1(w)} \circ f_{\zeta_0(w)}(x) : n \in \mathbb{N}_0 \}, \quad \forall x \in X.$$

An important example of random forcing is given by additive i.i.d. noise: Consider $X = \mathbb{T}^m$, or any other set with an additive structure, a map $T: X \to X$ and $\{\zeta_n\}_{n \in \mathbb{N}_0}$ an i.i.d. sequence of random variables with values in $X$ and distributed according to $\nu$, then taking $\Omega = X$ define $f: \Omega \times X \to X$ as

$$f(\omega, x) := T(x) + \omega.$$

Composing at time $n \in \mathbb{N}$ by $f_{\zeta_n}$ corresponds to considering the recursive equation

$$\mathcal{X}_{n+1} = T(\mathcal{X}_n) + \zeta_n, \quad \forall n \in \mathbb{N}_0,$$

where $\mathcal{X}_n$ is the state of the system at time $n$. What the above means is that, calling $(\Xi, \mathbb{P})$ the underlying probability space where $\{\zeta_n\}_{n \in \mathbb{N}_0}$ are defined, $\{\mathcal{X}_n\}_{n \in \mathbb{N}_0}$ are random variables satisfying

$$\mathbb{P}(\mathcal{X}_{n+1} \in A | \mathcal{X}_n = x_n) = \mathbb{P}(\xi_n \in (A - T(x_n))),$$

and thus $\{\mathcal{X}_n\}_{n \in \mathbb{N}_0}$ is a Markov chain. In the above, $T$ denotes the intrinsic dynamics, that is, the dynamics the system would have if it did not receive any forcing, while $\xi_n$ is the random forcing noise term.

*Deterministic forcing* is also represented as application at time $n \in \mathbb{N}$ of the map $f(\zeta_n, \cdot) : X \to X$. However, in this case the sequence $\{\zeta_n\}_{n \in \mathbb{N}}$ is not required to be independent, but it should satisfy $\zeta_{n+1} = g(\zeta_n)$ for some transformation $g: \Omega \to \Omega$ that preserves the measure $\nu$. This corresponds also to the general definition of *random dynamical system* usually given in the literature (see [3]).

The difference between random and deterministic forcing is not a stark one. In fact one can show that random forcing is a particular case of deterministic forcing where $g$ is an appropriate shift map: Given $f: \Omega \times X \to X$ and $\{\zeta_n\}_{n \in \mathbb{N}}$ an i.i.d sequence with values in $\Omega$ distributed as $\nu$, we can construct the probability space $(\Omega', \nu')$ with $\Omega' := \Omega^{\mathbb{N}_0}$ and $\nu' := \nu^{\otimes \mathbb{N}_0}$. Now define the sequence of identically distributed random variables $\{\zeta_k\}_{k \in \mathbb{N}_0}$ in $\Omega'$ with $\zeta_k := \{\zeta_{n+k}\}_{n \in \mathbb{N}_0}$, and $f': \Omega' \times X \to X$

$$f'(\zeta, x) := f((\zeta)_0, x),$$

where $(\zeta)_0$ denotes the first term of the sequence $\zeta \in \Omega'$. With this definition we also have

$$\zeta_{k+1} = \sigma^{k+1}(\{\zeta_n\}_{n \in \mathbb{N}_0}) = \sigma(\zeta_{k-1})$$

where $\sigma: \Omega' \to \Omega'$ is the left shift which is easy to check that keeps the measure $\nu'$ invariant.
### A.3 | Randomly forced systems and Markov chains

In this section we discuss the difference, in terms of mathematical definitions, between random and deterministic forcing.

By *random forcing*, we mean that given a probability space $(\Omega, \nu)$ and $f: \Omega \times X \to X$, at the $n$-th iteration we apply the map $f_{\zeta_n} := f(\zeta_n, \cdot) : X \to X$, where $\{\zeta_n\}_{n \in \mathbb{N}}$ is an i.i.d sequence of random variables defined on some probability space $(\Xi, \mathbb{P})$ with values in $\Omega$ and distributed according to $\nu$. Fixed $w \in \Xi$, the forward orbits of the system are given by

$$O(x) := \{ f_{\zeta_n(w)} \circ \dots \circ f_{\zeta_1(w)} \circ f_{\zeta_0(w)}(x) : n \in \mathbb{N}_0 \}, \quad \forall x \in X.$$

An important example of random forcing is given by additive i.i.d. noise: Consider $X = \mathbb{T}^m$, or any other set with an additive structure, a map $T: X \to X$ and $\{\zeta_n\}_{n \in \mathbb{N}_0}$ an i.i.d. sequence of random variables with values in $X$ and distributed according to $\nu$, then taking $\Omega = X$ define $f: \Omega \times X \to X$ as

$$f(\omega, x) := T(x) + \omega.$$

Composing at time $n \in \mathbb{N}$ by $f_{\zeta_n}$ corresponds to considering the recursive equation

$$\mathcal{X}_{n+1} = T(\mathcal{X}_n) + \zeta_n, \quad \forall n \in \mathbb{N}_0,$$

where $\mathcal{X}_n$ is the state of the system at time $n$. What the above means is that, calling $(\Xi, \mathbb{P})$ the underlying probability space where $\{\zeta_n\}_{n \in \mathbb{N}_0}$ are defined, $\{\mathcal{X}_n\}_{n \in \mathbb{N}_0}$ are random variables satisfying

$$\mathbb{P}(\mathcal{X}_{n+1} \in A | \mathcal{X}_n = x_n) = \mathbb{P}(\xi_n \in (A - T(x_n))),$$

and thus $\{\mathcal{X}_n\}_{n \in \mathbb{N}_0}$ is a Markov chain. In the above, $T$ denotes the intrinsic dynamics, that is, the dynamics the system would have if it did not receive any forcing, while $\xi_n$ is the random forcing noise term.

*Deterministic forcing* is also represented as application at time $n \in \mathbb{N}$ of the map $f(\zeta_n, \cdot) : X \to X$. However, in this case the sequence $\{\zeta_n\}_{n \in \mathbb{N}}$ is not required to be independent, but it should satisfy $\zeta_{n+1} = g(\zeta_n)$ for some transformation $g: \Omega \to \Omega$ that preserves the measure $\nu$. This corresponds also to the general definition of *random dynamical system* usually given in the literature (see [3]).

The difference between random and deterministic forcing is not a stark one. In fact one can show that random forcing is a particular case of deterministic forcing where $g$ is an appropriate shift map: Given $f: \Omega \times X \to X$ and $\{\zeta_n\}_{n \in \mathbb{N}}$ an i.i.d sequence with values in $\Omega$ distributed as $\nu$, we can construct the probability space $(\Omega', \nu')$ with $\Omega' := \Omega^{\mathbb{N}_0}$ and $\nu' := \nu^{\otimes \mathbb{N}_0}$. Now define the sequence of identically distributed random variables $\{\zeta_k\}_{k \in \mathbb{N}_0}$ in $\Omega'$ with $\zeta_k := \{\zeta_{n+k}\}_{n \in \mathbb{N}_0}$, and $f': \Omega' \times X \to X$

$$f'(\zeta, x) := f((\zeta)_0, x),$$

where $(\zeta)_0$ denotes the first term of the sequence $\zeta \in \Omega'$. With this definition we also have

$$\zeta_{k+1} = \sigma^{k+1}(\{\zeta_n\}_{n \in \mathbb{N}_0}) = \sigma(\zeta_{k-1})$$

where $\sigma: \Omega' \to \Omega'$ is the left shift which is easy to check that keeps the measure $\nu'$ invariant.
## APPENDIX B: DISINTEGRATION OF MEASURE AND ROHLIN'S THEOREM

The following definitions and results are taken from [43], adapted to the level of generality needed in this paper.

**Definition B.1.** Let $(X, \mu)$ be a topological probability space, $Y$ a metric space and $\pi: X \to Y$ a measurable function. Call $\hat{\mu} := \pi_*\mu$. A system of conditional measures of $\mu$ with respect to $(X, \pi, Y)$ is a collection of measures $\{\mu_y\}_{y \in Y}$ such that

1) For $\hat{\mu}$-almost every $y \in Y$, $\mu_y$ is a probability measure on $\pi^{-1}(y)$.

2) For every measurable subset $B \subset X$, $y \mapsto \mu_y(B)$ is measurable and

$$\mu(B) = \int \mu_{\pi^{-1}(y)}(B) d\hat{\mu}(y).$$

When $Y$ in the above definition is a measurable partition of $X$ and $\pi(x)$ is the unique element of the partition to which $x$ belongs, then we also call $\{\mu_y\}_{y \in Y}$ a disintegration of $\mu$.

**Definition B.2.** In the same setup of Definition B.1, the topological conditional measure of $\mu$ with respect to $(X, \pi, y, Y)$ is the weak* limit (if it exists)

$$\mu_y := \lim_{\epsilon \to 0^+} \mu_{\pi^{-1}(B(y, \epsilon))}$$

where $B(y, \epsilon)$ is the ball centered at $y$ with radius $\epsilon$ with respect to the metric on $Y$ and

$$\mu_{\pi^{-1}(B(y, \epsilon))}(I) = \frac{\mu(\pi^{-1}(B(y, \epsilon)) \cap I)}{\mu(\pi^{-1}(B(y, \epsilon)))}.$$

**Theorem B.1 (Theorem 2.2 [43]).** Let $(X, \mu)$ be a compact metric probability space, let $Y$ be a separable Riemannian manifold. Let $\pi: X \to Y$ be measurable. Then for $\hat{\mu}$-almost every $y \in Y$, the topological conditional measure of $\mu$ with respect to $(X, \pi, y, Y)$ exists as in Definition B.2. Furthermore the collection of measures $\{\mu_y\}_{y \in Y}$ is a system of conditional measures as in Definition B.1. (If $\mu_y$ does not exist, set $\mu_y = 0$).

## APPENDIX C: WASSERSTEIN DISTANCE: SOME COMPUTATIONS

Consider a compact metric space $(Y, d)$. Then the Kantorovich-Wasserstein between $\mu_1, \mu_2 \in \mathcal{M}_1(Y)$ is defined as

$$d_W(\mu_1, \mu_2) := \sup_{\gamma \in \mathcal{C}(\mu_1, \mu_2)} \int_{Y \times Y} d(s, s')d\gamma(s, s')$$

where $\mathcal{C}(\mu_1, \mu_2)$ is the set of all couplings between $\mu_1$ and $\mu_2$. If we consider instead of the metric $d$ the discrete metric $d_{dis}$ defined as

$$d_{dis}(s, s') = \begin{cases} 1 & s = s' \\ 0 & s \neq s' \end{cases}$$
# APPENDIX B: DISINTEGRATION OF MEASURE AND ROHLIN'S THEOREM

The following definitions and results are taken from [43], adapted to the level of generality needed in this paper.

**Definition B.1.** Let $(X, \mu)$ be a topological probability space, $Y$ a metric space and $\pi: X \to Y$ a measurable function. Call $\hat{\mu} := \pi_*\mu$. A system of conditional measures of $\mu$ with respect to $(X, \pi, Y)$ is a collection of measures $\{\mu_y\}_{y \in Y}$ such that

1) For $\hat{\mu}$-almost every $y \in Y$, $\mu_y$ is a probability measure on $\pi^{-1}(y)$.

2) For every measurable subset $B \subset X$, $y \mapsto \mu_y(B)$ is measurable and

$$ \mu(B) = \int \mu_{\pi^{-1}(y)}(B) d\hat{\mu}(y). $$

When $Y$ in the above definition is a measurable partition of $X$ and $\pi(x)$ is the unique element of the partition to which $x$ belongs, then we also call $\{\mu_y\}_{y \in Y}$ a disintegration of $\mu$.

**Definition B.2.** In the same setup of Definition B.1, the topological conditional measure of $\mu$ with respect to $(X, \pi, y, Y)$ is the weak* limit (if it exists)

$$ \mu_y := \lim_{\epsilon \to 0^+} \mu_{\pi^{-1}(B(y, \epsilon))} $$

where $B(y, \epsilon)$ is the ball centered at $y$ with radius $\epsilon$ with respect to the metric on $Y$ and

$$ \mu_{\pi^{-1}(B(y, \epsilon))}(I) = \frac{\mu(\pi^{-1}(B(y, \epsilon)) \cap I)}{\mu(\pi^{-1}(B(y, \epsilon)))}. $$

**Theorem B.1 (Theorem 2.2 [43]).** Let $(X, \mu)$ be a compact metric probability space, let $Y$ be a separable Riemannian manifold. Let $\pi: X \to Y$ be measurable. Then for $\hat{\mu}$-almost every $y \in Y$, the topological conditional measure of $\mu$ with respect to $(X, \pi, y, Y)$ exists as in Definition B.2. Furthermore the collection of measures $\{\mu_y\}_{y \in Y}$ is a system of conditional measures as in Definition B.1. (If $\mu_y$ does not exist, set $\mu_y = 0$).

# APPENDIX C: WASSERSTEIN DISTANCE: SOME COMPUTATIONS

Consider a compact metric space $(Y, d)$. Then the Kantorovich-Wasserstein between $\mu_1, \mu_2 \in \mathcal{M}_1(Y)$ is defined as

$$ d_W(\mu_1, \mu_2) := \sup_{\gamma \in \mathcal{C}(\mu_1, \mu_2)} \int_{Y \times Y} d(s, s')d\gamma(s, s') $$

where $\mathcal{C}(\mu_1, \mu_2)$ is the set of all couplings between $\mu_1$ and $\mu_2$. If we consider instead of the metric $d$ the discrete metric $d_{dis}$ defined as

$$ d_{dis}(s, s') = \begin{cases} 1 & s = s' \\ 0 & s \neq s' \end{cases} $$
We have that

$$d_{TV}(\mu_1, \mu_2) := \sup_{y \in C(\mu_1, \mu_2)} \int_{Y \times Y} d_{dis}(s, s')d\gamma(s, s').$$

**Lemma C.1.** Let $(Y, d)$ be a metric space, $T: Y \to Y$ a Lipschitz transformation with Lipschitz constant $\text{Lip}(T)$. Then for any $\xi_1, \xi_2 \in \mathcal{M}_1(\mathbb{T})$

$$d_W(T_*\xi_1, T_*\xi_2) \le \text{Lip}(T)d_W(\xi_1, \xi_2).$$

*Proof.*

$$ \begin{aligned} d_W(T_*\xi_1, T_*\xi_2) &= \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \varphi(y)d(T_*\xi_1 - T^*\xi_2)(y) \\ &= \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \varphi \circ T d(\xi_1 - \xi_2)(y) \\ &\le \sup_{\varphi \in \text{Lip}^1(Y)} \text{Lip}(\varphi \circ T)d_W(\xi_1, \xi_2) \end{aligned} $$

and $\text{Lip}(\varphi \circ T) \le \text{Lip}(\varphi) \text{Lip}(T)$. $\square$

*Remark C.1.* The above lemma can be read in the following way: If $T: (Y, d) \to (Y, d)$ is Lipschitz, then $T_*: (\mathcal{M}_1(Y), d_W) \to (\mathcal{M}_1(Y), d_W)$ is Lipschitz with $\text{Lip}(T_*) = \text{Lip}(T)$.

**Lemma C.2.** Consider $(S, \nu)$ a measurable space with $\nu$ a probability measure, and $Y$ a compact metric space. Assume that $\{\mu_s\}_{s \in S}$ is a family of measures belonging to $\mathcal{M}_1(Y)$ and that $\exists \ell > 0$ s.t. $d_W(\mu_s, \mu_{s'}) \le \ell$ for every $s, s' \in S$. Then the measure $\bar{\mu} \in \mathcal{M}_1(Y)$ defined as

$$\bar{\mu}(A) := \int_S d\nu(s)\mu_s(A)$$

is such that $d_W(\bar{\mu}, \mu_s) \le \ell$ for all $s \in S$.

*Proof.* Pick $s \in S$

$$ \begin{aligned} d_W(\bar{\mu}, \mu_s) &= \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \varphi(y)d(\bar{\mu} - \mu_s)(y) \\ &= \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \int_S d\nu(s')\varphi(y)d(\mu_{s'} - \mu_s)(y) \\ &\le \int_S d\nu(s') \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \varphi(y)d(\mu_{s'} - \mu_s)(y) \\ &\le \int_S d\nu(s')d_W(\mu_s, \mu_{s'}) \\ &\le \ell. \end{aligned} $$

$\square$
We have that

$$d_{TV}(\mu_1, \mu_2) := \sup_{Y \in C(\mu_1, \mu_2)} \int_{Y \times Y} d_{dis}(s, s')d\gamma(s, s').$$

**Lemma C.1.** Let $(Y, d)$ be a metric space, $T: Y \to Y$ a Lipschitz transformation with Lipschitz constant $\text{Lip}(T)$. Then for any $\xi_1, \xi_2 \in \mathcal{M}_1(\mathbb{T})$

$$d_W(T_*\xi_1, T_*\xi_2) \le \text{Lip}(T)d_W(\xi_1, \xi_2).$$

*Proof.*

$$ \begin{aligned} d_W(T_*\xi_1, T_*\xi_2) &= \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \varphi(y)d(T_*\xi_1 - T^*\xi_2)(y) \\ &= \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \varphi \circ T d(\xi_1 - \xi_2)(y) \\ &\le \sup_{\varphi \in \text{Lip}^1(Y)} \text{Lip}(\varphi \circ T)d_W(\xi_1, \xi_2) \end{aligned} $$

and $\text{Lip}(\varphi \circ T) \le \text{Lip}(\varphi) \text{Lip}(T)$. $\square$

*Remark C.1.* The above lemma can be read in the following way: If $T: (Y, d) \to (Y, d)$ is Lipschitz, then $T_*: (\mathcal{M}_1(Y), d_W) \to (\mathcal{M}_1(Y), d_W)$ is Lipschitz with $\text{Lip}(T_*) = \text{Lip}(T)$.

**Lemma C.2.** Consider $(S, \nu)$ a measurable space with $\nu$ a probability measure, and $Y$ a compact metric space. Assume that $\{\mu_s\}_{s \in S}$ is a family of measures belonging to $\mathcal{M}_1(Y)$ and that $\exists \ell > 0$ s.t. $d_W(\mu_s, \mu_{s'}) \le \ell$ for every $s, s' \in S$. Then the measure $\bar{\mu} \in \mathcal{M}_1(Y)$ defined as

$$\bar{\mu}(A) := \int_S d\nu(s)\mu_s(A)$$

is such that $d_W(\bar{\mu}, \mu_s) \le \ell$ for all $s \in S$.

*Proof.* Pick $s \in S$

$$ \begin{aligned} d_W(\bar{\mu}, \mu_s) &= \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \varphi(y)d(\bar{\mu} - \mu_s)(y) \\ &= \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \int_S d\nu(s')\varphi(y)d(\mu_{s'} - \mu_s)(y) \\ &\le \int_S d\nu(s') \sup_{\varphi \in \text{Lip}^1(Y)} \int_Y \varphi(y)d(\mu_{s'} - \mu_s)(y) \\ &\le \int_S d\nu(s')d_W(\mu_s, \mu_{s'}) \\ &\le \ell. \end{aligned} $$
**Lemma C.3.** Let (Y, d) be a bounded metric space and call diam(Y) its diameter. Then

$$d_W(\mu_1, \mu_2) \leq \mathrm{diam}(Y)d_{TV}(\mu_1, \mu_2).$$

*Proof.*

$$d_W(\mu_1, \mu_2) = \sup_{\gamma \in C(\mu_1, \mu_2)} \int_{Y \times Y} d(s, s')d\gamma(s, s') \\ \leq \sup_{\gamma \in C(\mu_1, \mu_2)} \int_{Y \times Y} \mathrm{diam}(Y)d_{dis}(s, s')d\gamma(s, s') \\ = \mathrm{diam}(Y)d_{TV}(\mu_1, \mu_2). \quad \square$$

**Lemma C.4.** Assume {$\mu_i$}$_{i=1}^n$ and {$\mu'_i$}$_{i=1}^n$ are probability measures in $M_1(Y)$ and {$b_i$}$_{i=1}^n$, $b_i > 0$, are weights with $\sum_{i=1}^n b_i = 1$. Then

$$d_W \left( \sum_{i=1}^{n} b_i \mu_i, \sum_{i=1}^{n} b_i \mu'_i \right) \leq \max_i d_W(\mu_i, \mu'_i).$$

*Proof.*

$$d_W(\sum_{i=1}^{n} b_i \mu_i, \sum_{i=1}^{n} b_i \mu'_i) \leq \sup_{\varphi \in \mathrm{Lip}^1} \int_Y \varphi d \left( \sum_{i=1}^{n} b_i \mu_i - \sum_{i=1}^{n} b_i \mu'_i \right) \\ \leq \sum_{i=1}^{n} b_i \sup_{\varphi \in \mathrm{Lip}^1} \int_Y \varphi d(\mu_i - \mu'_i) \\ \leq \sum_{i=1}^{n} b_i d_W(\mu_i, \mu'_i) \\ \leq \max_i d_W(\mu_i, \mu'_i). \quad \square$$

## ACKNOWLEDGEMENTS

The first author is partially supported by the PRIN Grant 2017S35EHN “Regular and stochastic behaviour in dynamical systems” of MIUR, Italy and by INDAM - GNFM 2020 project “Deterministic and stochastic dynamical systems for climate studies”. The second author acknowledges the support of UniCredit Bank R&D group through the “Dynamics and Information Theory Institute” at the Scuola Normale Superiore. The third author acknowledges funding from the H2020 Marie Skłodowska-Curie Actions “Ergodic Theory of Complex Systems” project no. 843880. He is also grateful for the hospitality of the Centro di Ricerca Matematica Ennio de Giorgi and Scuola Normale Superiore where part of this work was carried out. The authors thank the anonymous referee, Lai-Sang Young and Stefano Galatolo for comments and discussions on preliminary versions.

Open Access Funding provided by Universita degli Studi di Pisa within the CRUI-CARE Agreement.

## JOURNAL INFORMATION

The *Journal of the London Mathematical Society* is wholly owned and managed by the London Mathematical Society, a not-for-profit Charity registered with the UK Charity Commission.
**Lemma C.3.** Let (Y, d) be a bounded metric space and call diam(Y) its diameter. Then

$$d_W(\mu_1, \mu_2) \leq \mathrm{diam}(Y)d_{TV}(\mu_1, \mu_2).$$

*Proof.*

$$ \begin{aligned} d_W(\mu_1, \mu_2) &= \sup_{\gamma \in C(\mu_1, \mu_2)} \int_{Y \times Y} d(s, s') d\gamma(s, s') \\ &\leq \sup_{\gamma \in C(\mu_1, \mu_2)} \int_{Y \times Y} \mathrm{diam}(Y) d_{dis}(s, s') d\gamma(s, s') \\ &= \mathrm{diam}(Y) d_{TV}(\mu_1, \mu_2). \end{aligned} \quad \square $$

**Lemma C.4.** Assume $\{\mu_i\}_{i=1}^n$ and $\{\mu'_i\}_{i=1}^n$ are probability measures in $M_1(Y)$ and $\{b_i\}_{i=1}^n$, $b_i > 0$, are weights with $\sum_{i=1}^n b_i = 1$. Then

$$d_W\left(\sum_{i=1}^{n} b_i \mu_i, \sum_{i=1}^{n} b_i \mu'_i\right) \leq \max_{i} d_W(\mu_i, \mu'_i).$$

*Proof.*

$$ \begin{aligned} d_W\left(\sum_{i=1}^{n} b_i \mu_i, \sum_{i=1}^{n} b_i \mu'_i\right) &\leq \sup_{\varphi \in \mathrm{Lip}^1} \int_Y \varphi d\left(\sum_{i=1}^{n} b_i \mu_i - \sum_{i=1}^{n} b_i \mu'_i\right) \\ &\leq \sum_{i=1}^{n} b_i \sup_{\varphi \in \mathrm{Lip}^1} \int_Y \varphi d(\mu_i - \mu'_i) \\ &\leq \sum_{i=1}^{n} b_i d_W(\mu_i, \mu'_i) \\ &\leq \max_{i} d_W(\mu_i, \mu'_i). \end{aligned} \quad \square $$

## ACKNOWLEDGEMENTS

The first author is partially supported by the PRIN Grant 2017S35EHN “Regular and stochastic behaviour in dynamical systems” of MIUR, Italy and by INDAM - GNFM 2020 project “Deterministic and stochastic dynamical systems for climate studies”. The second author acknowledges the support of UniCredit Bank R&D group through the “Dynamics and Information Theory Institute” at the Scuola Normale Superiore. The third author acknowledges funding from the H2020 Marie Skłodowska-Curie Actions “Ergodic Theory of Complex Systems” project no. 843880. He is also grateful for the hospitality of the Centro di Ricerca Matematica Ennio de Giorgi and Scuola Normale Superiore where part of this work was carried out. The authors thank the anonymous referee, Lai-Sang Young and Stefano Galatolo for comments and discussions on preliminary versions.

Open Access Funding provided by Universita degli Studi di Pisa within the CRUI-CARE Agreement.

## JOURNAL INFORMATION

The *Journal of the London Mathematical Society* is wholly owned and managed by the London Mathematical Society, a not-for-profit Charity registered with the UK Charity Commission.
All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.

## REFERENCES

1. J. F. Alves, C. Bonatti, and M. Viana, *SRB measures for partially hyperbolic systems whose central direction is mostly expanding*, Invent. Math. **140** (2000), no. 2, 351–398 (English).

2. J. F. Alves, C. L. Dias, S. Luzzatto, and V. Pinheiro, *SRB measures for partially hyperbolic systems whose central direction is weakly expanding*, J. Eur. Math. Soc. **19** (2017), no. 10, 2911–2946.

3. L. Arnold, *Random dynamical systems*, Springer, Berlin, 1998 (English).

4. V. Baladi and L.-S. Young, *On the spectra of randomly perturbed expanding maps*, Comm. Math. Phys. **156** (1993), no. 2, 355–385.

5. P. Berger and P. D. Carrasco, *Non-uniformly hyperbolic diffeomorphisms derived from the standard map*, Commun. Math. Phys. **329** (2014), no. 1, 239–262 (English).

6. K. Bjerklöv, *A note on circle maps driven by strongly expanding endomorphisms on $t$*, Dynam. Sys. **33** (2018), 361–368.

7. A. Blumenthal, J. Xue, and L.-S. Young, *Lyapunov exponents and correlation decay for random perturbations of some prototypical 2d maps*, Commun. Math. Phys. **359** (2018), no. 1, 347–373 (English).

8. C. Bose, A. Quas, and M. Tanzi, *Random composition of L-S-V maps sampled over large parameter ranges*, Nonlinearity **34** (2021), no. 6, 3641–3675 (English).

9. A. Boyarsky and P. Góra, *Laws of chaos. Invariant measures and dynamical systems in one dimension*, Birkhäuser, Boston, MA, 1997 (English).

10. K. Burns and A. Wilkinson, *On the ergodicity of partially hyperbolic systems*, Ann. Math. (2) **171** (2010), no. 1, 451–489 (English).

11. O. Butterley and P. Eslami, *Exponential mixing for skew products with discontinuities*, Trans. Am. Math. Soc. **369** (2017), no. 2, 783–803 (English).

12. P. D. Carrasco, *Random products of standard maps*, Commun. Math. Phys. **377** (2020), no. 2, 773–810 (English).

13. R. Castorrini and C. Liverani, *Quantitative statistical properties of two-dimensional partially hyperbolic systems*, https://arxiv.org/abs/2007.05602 (2020).

14. I. Chevyrev, P. K. Friz, A. Korepanov, and I. Melbourne, *Superdiffusive limits for deterministic fast-slow dynamical systems*, Probab. Theory Relat. Fields **178** (2020), no. 3-4, 735–770 (English).

15. B. Christian and V. Marcelo, *SRB measures for partially hyperbolic systems whose central direction is mostly contracting*, Isr. J. Math. **115** (2000), 157–193.

16. M. Cloez, B. Hairer, *Exponential ergodicity for Markov processes with random switching*, Bernoulli **21** (2015), 505–536.

17. J. De Simoi and C. Liverani, *Limit theorems for fast-slow partially hyperbolic systems*, Invent. Math. **213** (2018), no. 3, 811–1016 (English).

18. D. Dolgopyat, *Limit theorems for partially hyperbolic systems*, Trans. Am. Math. Soc. **356** (2004), no. 4, 1637–1689 (English).

19. D. Dolgopyat, *On differentiability of SRB states for partially hyperbolic systems*, Invent. Math. **155** (2004), no. 2, 389–449 (English).

20. D. Dolgopyat and B. Fayad, *Limit theorems for toral translations*, Hyperbolic dynamics, fluctuations and large deviations **89** (2015), 227–277.

21. D. Down, S. P. Meyn, and R. L. Tweedie, *Exponential and uniform ergodicity of Markov processes*, Ann. Probab. **23** (1995), no. 4, 1671–1691.

22. D. Dragičević, G. Froyland, C. Gonzalez-Tokman, and S. Vaienti, *A spectral approach for quenched limit theorems for random expanding dynamical systems*, Comm. Math. Phys. **360** (2018), no. 3, 1121–1187.

23. D. Dragičević, G. Froyland, C. González-Tokman, and S. Vaienti, *A spectral approach for quenched limit theorems for random hyperbolic dynamical systems*, Trans. Amer. Math. Soc. **373** (2020), no. 1, 629–664.

24. S. Galatolo, *Quantitative statistical stability, speed of convergence to equilibrium and partially hyperbolic skew products*, J. Éc. Polytech., Math. **5** (2018), 377–405 (English).
All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.

## REFERENCES

1. J. F. Alves, C. Bonatti, and M. Viana, *SRB measures for partially hyperbolic systems whose central direction is mostly expanding*, Invent. Math. **140** (2000), no. 2, 351–398 (English).

2. J. F. Alves, C. L. Dias, S. Luzzatto, and V. Pinheiro, *SRB measures for partially hyperbolic systems whose central direction is weakly expanding*, J. Eur. Math. Soc. **19** (2017), no. 10, 2911–2946.

3. L. Arnold, *Random dynamical systems*, Springer, Berlin, 1998 (English).

4. V. Baladi and L.-S. Young, *On the spectra of randomly perturbed expanding maps*, Comm. Math. Phys. **156** (1993), no. 2, 355–385.

5. P. Berger and P. D. Carrasco, *Non-uniformly hyperbolic diffeomorphisms derived from the standard map*, Commun. Math. Phys. **329** (2014), no. 1, 239–262 (English).

6. K. Bjerklöv, *A note on circle maps driven by strongly expanding endomorphisms on $t$*, Dynam. Sys. **33** (2018), 361–368.

7. A. Blumenthal, J. Xue, and L.-S. Young, *Lyapunov exponents and correlation decay for random perturbations of some prototypical 2d maps*, Commun. Math. Phys. **359** (2018), no. 1, 347–373 (English).

8. C. Bose, A. Quas, and M. Tanzi, *Random composition of L-S-V maps sampled over large parameter ranges*, Nonlinearity **34** (2021), no. 6, 3641–3675 (English).

9. A. Boyarsky and P. Góra, *Laws of chaos. Invariant measures and dynamical systems in one dimension*, Birkhäuser, Boston, MA, 1997 (English).

10. K. Burns and A. Wilkinson, *On the ergodicity of partially hyperbolic systems*, Ann. Math. (2) **171** (2010), no. 1, 451–489 (English).

11. O. Butterley and P. Eslami, *Exponential mixing for skew products with discontinuities*, Trans. Am. Math. Soc. **369** (2017), no. 2, 783–803 (English).

12. P. D. Carrasco, *Random products of standard maps*, Commun. Math. Phys. **377** (2020), no. 2, 773–810 (English).

13. R. Castorrini and C. Liverani, *Quantitative statistical properties of two-dimensional partially hyperbolic systems*, https://arxiv.org/abs/2007.05602 (2020).

14. I. Chevyrev, P. K. Friz, A. Korepanov, and I. Melbourne, *Superdiffusive limits for deterministic fast-slow dynamical systems*, Probab. Theory Relat. Fields **178** (2020), no. 3-4, 735–770 (English).

15. B. Christian and V. Marcelo, *SRB measures for partially hyperbolic systems whose central direction is mostly contracting*, Isr. J. Math. **115** (2000), 157–193.

16. M. Cloez, B. Hairer, *Exponential ergodicity for Markov processes with random switching*, Bernoulli **21** (2015), 505–536.

17. J. De Simoi and C. Liverani, *Limit theorems for fast-slow partially hyperbolic systems*, Invent. Math. **213** (2018), no. 3, 811–1016 (English).

18. D. Dolgopyat, *Limit theorems for partially hyperbolic systems*, Trans. Am. Math. Soc. **356** (2004), no. 4, 1637–1689 (English).

19. D. Dolgopyat, *On differentiability of SRB states for partially hyperbolic systems*, Invent. Math. **155** (2004), no. 2, 389–449 (English).

20. D. Dolgopyat and B. Fayad, *Limit theorems for toral translations*, Hyperbolic dynamics, fluctuations and large deviations **89** (2015), 227–277.

21. D. Down, S. P. Meyn, and R. L. Tweedie, *Exponential and uniform ergodicity of Markov processes*, Ann. Probab. **23** (1995), no. 4, 1671–1691.

22. D. Dragičević, G. Froyland, C. Gonzalez-Tokman, and S. Vaienti, *A spectral approach for quenched limit theorems for random expanding dynamical systems*, Comm. Math. Phys. **360** (2018), no. 3, 1121–1187.

23. D. Dragičević, G. Froyland, C. González-Tokman, and S. Vaienti, *A spectral approach for quenched limit theorems for random hyperbolic dynamical systems*, Trans. Amer. Math. Soc. **373** (2020), no. 1, 629–664.

24. S. Galatolo, *Quantitative statistical stability, speed of convergence to equilibrium and partially hyperbolic skew products*, J. Éc. Polytech., Math. **5** (2018), 377–405 (English).
25. S. Galatolo and R. Lucena, *Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps*, Discrete Contin. Dyn. Syst. **40** (2020), no. 3, 1309–1360 (English).

26. S. Galatolo, J. Rousseau, and B. Saussol, *Skew products, quantitative recurrence, shrinking targets and decay of correlations*. , Ergodic Theory Dyn. Syst. **35** (2015), no. 6, 1814–1845 (English).

27. P. Giulietti, A. Hammerlindl, and D. Ravotti, *Quantitative global-local mixing for accessible skew products*, Ann. Henri Poincaré **23** (2022), no. 3, 923–971 (English).

28. S. Gouëzel, *Statistical properties of a skew product with a curve of neutral points*, Ergodic Theory Dynam. Systems **27** (2007), no. 1, 123–151.

29. Y. Hafouta, *Limit theorems for some skew products with mixing base maps*, Ergodic Theory Dynam. Systems **41** (2019), 241–271.

30. M. Hairer and J. C. Mattingly, *Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations*. , Ann. Probab. **36** (2008), no. 6, 2050–2091 (English).

31. M. Hairer and J. C. Mattingly, *Yet another look at Harris' ergodic theorem for Markov chains*, Seminar on Stochastic Analysis, Random Fields and Applications VI (Basel) (R. Dalang, M. Dozzi, F. Russo, eds.), Springer, Basel, 2011, pp. 109–117.

32. B. Hasselblatt and Y. Pesin, *Partially hyperbolic dynamical systems*, Handbook of Dynamical Systems, vol. 1B, Elsevier, Amsterdam, 2006, pp. 1–55 (English).

33. D. Kelly and I. Melbourne, *Smooth approximation of stochastic differential equations*, Ann. Probab. **44** (2016), no. 1, 479–520 (English).

34. B. Kloeckner, *Extensions with shrinking fibers*, Ergodic Theory Dynam. Systems **41** (2021), 1795–1834.

35. A. Korepanov, Z. Kosloff, and I. Melbourne, *Explicit coupling argument for non-uniformly hyperbolic transformations*, Proc. R. Soc. Edinb., Sect. A, Math. **149** (2020), 101–130.

36. C. Liverani, *Decay of correlations*, Ann. Math. (2) **142** (1995), no. 2, 239–301 (English).

37. S. P. Meyn and R. L. Tweedie, *Markov chains and stochastic stability*, Springer-Verlag, London, 1993.

38. M. Nicol, A. Török, and S. Vaienti, *Central limit theorems for sequential and random intermittent dynamical systems*, Ergodic Theory Dynam. Systems **38** (2018), no. 3, 1127–1153.

39. W. Ott, M. Stenlund, and L.-S. Young, *Memory loss for time-dependent dynamical systems*, Math. Res. Lett. **16** (2009), no. 3, 463–475 (English).

40. M. Sambarino, *A (short) survey on dominated splittings, Mathematical Congress of the Americas: First Mathematical Congress of the Americas*, American Mathematical Society (AMS), Providence, RI, 2016, pp. 149–183 (English).

41. S. I. Santos and C. Walkden, *Distributional and local limit laws for a class of iterated maps that contract on average*, Stoch. Dyn. **13** (2013), no. 02, 1250019.

42. A. N. Shiryaev and R. P. Boas, *Probability*, 2nd ed., Springer-Verlag, Berlin, Heidelberg, 1995.

43. D. Simmons, *Conditional measures and conditional expectation; Rohlin's disintegration theorem*, Discrete Contin. Dyn. Syst. - A **32** (2012), 2565.

44. M. Stenlund, *Non-stationary compositions of Anosov diffeomorphisms*, Nonlinearity **24** (2011), no. 10, 2991.

45. D. W. Stroock, *An introduction to Markov processes*, 2nd ed., Graduate Texts in Mathematics, vol. 230, Springer, Berlin Heidelberg, 2014.

46. M. Tanzi and L.-S. Young, *Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems*, Discrete Contin. Dyn. Syst.-A **40** (2020), no. 10, 6015.

47. M. Tanzi and L.-S. Young, *Existence of physical measures in some excitation-inhibition networks*, Nonlinearity **35** (2022), no. 2, 889–915 (English).

48. M. Tsujii, *Physical measures for partially hyperbolic surface endomorphisms*, Acta Math. **194** (2005), no. 1, 37–132 (English).

49. M. Viana, *Stochastic dynamics of deterministic systems*, IMPA, Rio de Janeiro, 1997.

50. C. Villani, *Optimal transport: Old and new*, vol. 338, Springer, Berlin, 2009 (English).

51. C. Walkden and T. Withers, *Invariant graphs of a family of non-uniformly expanding skew products over Markov maps*, Nonlinearity **31** (2018), no. 6, 2726.

52. L.-S. Young, *What are SRB measures, and which dynamical systems have them?*, J. Stat. Phys. **108** (2002), no. 5-6, 733–754 (English).

53. L.-S. Young, *Comparing chaotic and random dynamical systems*, J. Math. Phys. **60** (2019), no. 5, 052701.
25. S. Galatolo and R. Lucena, *Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps*, Discrete Contin. Dyn. Syst. **40** (2020), no. 3, 1309–1360 (English).

26. S. Galatolo, J. Rousseau, and B. Saussol, *Skew products, quantitative recurrence, shrinking targets and decay of correlations*. , Ergodic Theory Dyn. Syst. **35** (2015), no. 6, 1814–1845 (English).

27. P. Giulietti, A. Hammerlindl, and D. Ravotti, *Quantitative global-local mixing for accessible skew products*, Ann. Henri Poincaré **23** (2022), no. 3, 923–971 (English).

28. S. Gouëzel, *Statistical properties of a skew product with a curve of neutral points*, Ergodic Theory Dynam. Systems **27** (2007), no. 1, 123–151.

29. Y. Hafouta, *Limit theorems for some skew products with mixing base maps*, Ergodic Theory Dynam. Systems **41** (2019), 241–271.

30. M. Hairer and J. C. Mattingly, *Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations*. , Ann. Probab. **36** (2008), no. 6, 2050–2091 (English).

31. M. Hairer and J. C. Mattingly, *Yet another look at Harris' ergodic theorem for Markov chains*, Seminar on Stochastic Analysis, Random Fields and Applications VI (Basel) (R. Dalang, M. Dozzi, F. Russo, eds.), Springer, Basel, 2011, pp. 109–117.

32. B. Hasselblatt and Y. Pesin, *Partially hyperbolic dynamical systems*, Handbook of Dynamical Systems, vol. 1B, Elsevier, Amsterdam, 2006, pp. 1–55 (English).

33. D. Kelly and I. Melbourne, *Smooth approximation of stochastic differential equations*, Ann. Probab. **44** (2016), no. 1, 479–520 (English).

34. B. Kloeckner, *Extensions with shrinking fibers*, Ergodic Theory Dynam. Systems **41** (2021), 1795–1834.

35. A. Korepanov, Z. Kosloff, and I. Melbourne, *Explicit coupling argument for non-uniformly hyperbolic transformations*, Proc. R. Soc. Edinb., Sect. A, Math. **149** (2020), 101–130.

36. C. Liverani, *Decay of correlations*, Ann. Math. (2) **142** (1995), no. 2, 239–301 (English).

37. S. P. Meyn and R. L. Tweedie, *Markov chains and stochastic stability*, Springer-Verlag, London, 1993.

38. M. Nicol, A. Török, and S. Vaienti, *Central limit theorems for sequential and random intermittent dynamical systems*, Ergodic Theory Dynam. Systems **38** (2018), no. 3, 1127–1153.

39. W. Ott, M. Stenlund, and L.-S. Young, *Memory loss for time-dependent dynamical systems*, Math. Res. Lett. **16** (2009), no. 3, 463–475 (English).

40. M. Sambarino, *A (short) survey on dominated splittings, Mathematical Congress of the Americas: First Mathematical Congress of the Americas*, American Mathematical Society (AMS), Providence, RI, 2016, pp. 149–183 (English).

41. S. I. Santos and C. Walkden, *Distributional and local limit laws for a class of iterated maps that contract on average*, Stoch. Dyn. **13** (2013), no. 02, 1250019.

42. A. N. Shiryaev and R. P. Boas, *Probability*, 2nd ed., Springer-Verlag, Berlin, Heidelberg, 1995.

43. D. Simmons, *Conditional measures and conditional expectation; Rohlin's disintegration theorem*, Discrete Contin. Dyn. Syst. - A **32** (2012), 2565.

44. M. Stenlund, *Non-stationary compositions of Anosov diffeomorphisms*, Nonlinearity **24** (2011), no. 10, 2991.

45. D. W. Stroock, *An introduction to Markov processes*, 2nd ed., Graduate Texts in Mathematics, vol. 230, Springer, Berlin Heidelberg, 2014.

46. M. Tanzi and L.-S. Young, *Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems*, Discrete Contin. Dyn. Syst.-A **40** (2020), no. 10, 6015.

47. M. Tanzi and L.-S. Young, *Existence of physical measures in some excitation-inhibition networks*, Nonlinearity **35** (2022), no. 2, 889–915 (English).

48. M. Tsujii, *Physical measures for partially hyperbolic surface endomorphisms*, Acta Math. **194** (2005), no. 1, 37–132 (English).

49. M. Viana, *Stochastic dynamics of deterministic systems*, IMPA, Rio de Janeiro, 1997.

50. C. Villani, *Optimal transport: Old and new*, vol. 338, Springer, Berlin, 2009 (English).

51. C. Walkden and T. Withers, *Invariant graphs of a family of non-uniformly expanding skew products over Markov maps*, Nonlinearity **31** (2018), no. 6, 2726.

52. L.-S. Young, *What are SRB measures, and which dynamical systems have them?*, J. Stat. Phys. **108** (2002), no. 5-6, 733–754 (English).

53. L.-S. Young, *Comparing chaotic and random dynamical systems*, J. Math. Phys. **60** (2019), no. 5, 052701.