2 Statement of the main results

In what follows, we introduce the setting and state the main results of this paper. However, for the sake of simplicity, we postpone several technical definitions to the next sections. Throughout, we assume that $(X,d)$ is a complete and separable metric space, and that $T_1, \ldots T_k: X \to X$ are continuous, surjective and Ruelle expanding maps (cf. Definition 3.2). Moreover, we always assume that the semigroup $\mathcal {S}$ generated by these maps is jointly topologically mixing and finitely aperiodic (cf. Definitions 3.3 and 3.4).

Moreover, as we are interested in thermodynamic quantities, we fix Hölder continuous functions $\varphi _1, \ldots , \varphi _k: X \to \mathbb {R}$ and define, for a finite word $v=i_1 \ldots i_n$ ,

$$ \begin{align*} T_v:= T_{i_n}\circ \cdots \circ T_{i_1} \quad\mbox{and}\quad \varphi_v := \varphi_{i_1} + \varphi_{i_2} \circ T_{i_1} + \cdots + \varphi_{i_n} \circ T_{i_1 i_2 \ldots i_{n-1}}. \end{align*} $$

This then gives rise to a family of Ruelle operators $\{L_v\}$ and a further family of operators $\{\mathbb {P}_{u}^{v}\}$ , defined by

$$ \begin{align*} L_v(f)(x) := \sum_{T_v(y)=x} e^{\varphi_v(y)}f(y), \quad \mathbb{P}_{u}^{v}(f) = \frac{L_{v}(f \cdot L_{u}(\mathbf{1}))}{L_{uv}(\mathbf{1})}, \end{align*} $$

for f in a suitable function space and with $\mathbf {1}$ referring to the constant function of value $1$ . Moreover, to guarantee that $L_{v}(\mathbf {1})$ is well defined, we also assume that the functions $\varphi _i$ are summable (cf. Definition 4.1). As it will turn out below, the analysis of this family of operators allows us to ignore the problem of the non-existence of invariant densities due to purely functorial reason and was, according to the authors’ knowledge, first employed in [Reference Bessa and Stadlbauer3].

The two main features of these quotients are that $\mathbb {P}_{u}^{v}(\mathbf {1}) = \mathbf {1}$ and that the iteration rule $\mathbb {P}_{uv}^w\circ \mathbb {P}_{u}^v = \mathbb {P}_{u}^{vw}$ holds. It follows from the first that the dual operators $\{(\mathbb {P}_{u}^{v})^{\ast }\}$ act on the space of probability measures $\mathcal {M}_1(X)$ , and from the second that it is possible to adapt methods for Markov operators as in [Reference Bressaud, Fernández and Galves5, Reference Hairer and Mattingly17, Reference Kloeckner, Lopes and Stadlbauer23, Reference Stadlbauer31] to obtain geometric convergence. Our first principal result now establishes this kind of convergence. In here, $\overline {W}$ refers to the Wasserstein metric and $\overline {D}$ to the Hölder coefficient with respect to the equivalent metric $d^{\ast }$ (cf. equation (5.1)). We refer the reader to §4 for the necessary definitions and notation.

Theorem A. Suppose the Ruelle expanding semigroup $\mathcal {S}$ is jointly topologically mixing and finitely aperiodic, and that every potential $\varphi _i$ is $\alpha $ -Hölder and summable. Then there exist $k_0 \in {\mathbb N}$ and $s \in (0,1)$ such that for all finite words $u,v$ with length $|v|\ge k_0$ and $\nu _1 , \nu _2\in \mathcal {M}_1(X)$ and every Hölder continuous observable $f: X\to \mathbb R$ with $\overline {D}(f) < \infty $ ,

$$ \begin{align*} \overline{W}({\mathbb{P}_{u}^{v}}^{\ast}(\nu_1), {\mathbb{P}_{u}^{v}}^{\ast} (\nu_2)) &\leq s^{|v|} \overline{W}( \nu_1 , \nu_2),\\ \quad \overline{D}(\mathbb{P}_{u}^{v}(f)) &\leq s^{|v|} \overline{D}(f). \end{align*} $$

This theorem implies that for any infinite word $\omega =i_1 i_2 \ldots $ and measure $\nu \in \mathcal {M}_1(X)$ , the limit

$$ \begin{align*} \mu_{\omega}:= \lim_{l \to \infty} ( {\mathbb{P}_{\emptyset}^{i_1 \ldots i_l}})^{\ast}(\nu) \end{align*} $$

exists, is independent of $\nu $ and the speed of convergence is exponential. This means that, under some mild assumptions on the set of Ruelle expanding maps, any non-autonomous sequence of dynamics admits a probability measure that rules its dynamics and that this measure is a non-autonomous conformal measure in the following sense: there exists $\unicode{x3bb} _{u,\omega }> 0$ such that $L_u^{\ast }(\mu _{\omega }) = \unicode{x3bb} _{u,\omega } \mu _{u\omega }$ (see Proposition 6.1). Furthermore, for any left infinite word $\tilde \omega = \ldots i_{-2} i_{-1}$ , the limit

$$ \begin{align*} \mu_{\tilde{\omega},\omega}:= \lim_{l \to \infty} ( {\mathbb{P}_{i_{-l} \ldots i_{-1}}^{i_1 \ldots i_l}})^{\ast}(\nu) \end{align*} $$

exists, varies Hölder continuously with $\omega $ , is independent of $\nu $ , and the speed of convergence is exponential. As shown in Proposition 6.3, this measure is invariant in the non-autonomous setting, and if $\tilde {\omega }$ and $\omega $ are periodic extensions of the finite word w, that is, $\tilde {\omega } = \ldots ww$ and ${\omega } = ww \ldots $ , then $\mu _{\tilde {\omega },\omega }$ is the unique equilibrium state of $(T_w,\varphi _w)$ (cf. Proposition 6.5). In fact, the set of all measures $\{\mu _{\tilde {\omega },\omega }\}$ , where $\tilde {\omega }$ , $\omega $ run through all infinite words is the closure of these equilibrium states and can be used to define a compactification of the semigroup (Proposition 9.4).

A further application of Theorem A is related to an invariance principle as the contraction allows us to apply the general invariance principle in [Reference Cuny and Merlevède10] and gives rise to the following result (for a similar result for continued fractions with restricted entries, see [Reference Stadlbauer and Zhang32]). Here, $[\omega ]_n$ stands for the initial n-word of an infinite word $\omega $ .

Theorem B. Suppose the finitely Ruelle expanding semigroup $\mathcal {S}$ is jointly topologically mixing and finitely aperiodic, and that every potential $\varphi _i$ is $\alpha $ -Hölder and summable. Suppose $\omega \in \Sigma $ , $f\in {\mathcal H}_{\alpha }$ . Let $f_n=f -\int f\circ T_{[\omega ]_n} \,d\mu _{\omega }$ for every $n\in \mathbb N_0$ , and let $s_n^2 = \mathbb E_{\mu _{\omega }}(\sum _{k=0}^{n-1} f_k\circ T_{[\omega ]_k})^2$ for $n\ge 1$ and assume that $ \sum _n s_n^{-4}<\infty $ . Then there exists a sequence $(Z_n)$ of independent centred Gaussian random variables such that

$$ \begin{align*} &\qquad\qquad \sup_n\bigg|\sqrt{\textstyle \sum_{k=0}^{n-1} \mathbb E_{\mu_{\omega}} Z_k^2}-s_n\bigg|<\infty,\\ \sup_{0\leq k \leq n-1} & \bigg| \sum_{i=0}^k f_i\circ T_{[\omega]_i} - \sum_{i=0}^k Z_i \bigg| = o\bigg(\sqrt{s^2_n \log \log s^2_n}\bigg) \text{ almost surely}. \end{align*} $$

We then relate and apply these results to random dynamical systems, that is, we assume that the $T_i$ are chosen with respect to a given probability measure $\rho $ . So, it is sufficient to fix a measure $\rho $ either on the shift spaces $\Sigma := \{1,\ldots ,k\}^{\mathbb {N}}$ or $\Sigma _{\mathbb {Z}} := \{1,\ldots ,k\}^{\mathbb {Z}}$ and consider the almost sure behaviour, referred to as quenched, and the behaviour in average, referred to as annealed behaviour. In this setting, Proposition 6.1 provides existence and exponential decay towards the quenched random conformal measure $\mu _{\omega }$ , whereas the bilateral result in Proposition 6.3 implies the same statement for the quenched equilibrium state $\mu _{\tilde {\omega },\omega }$ .

To relate these quenched results to their annealed counterparts, we consider in here as in [Reference Baladi2] the annealed operators

$$ \begin{align*} \mathcal{A}_n := \sum_{|w|=n} \rho(\{\omega: [\omega]_n=w\}) L_w. \end{align*} $$

A fundamental problem of these operators is that, in general, $\mathcal {A}_{n+m} \neq \mathcal {A}_n \circ \mathcal {A}_m$ , which makes it impossible to apply methods from spectral theory. However, if we assume that $\rho $ is supported on a topologically mixing, one-sided subshift of finite type, it is possible to control the asymptotic behaviour of $\{\mathcal {A}_n\}$ , which is our third main result. In here, $\theta $ refers to the one-sided shift map.

Theorem C. Suppose the Ruelle expanding semigroup $\mathcal {S}$ is jointly topologically mixing and finitely aperiodic, and that every potential $\varphi _i$ is $\alpha $ -Hölder and summable. Moreover, suppose that $\rho $ is supported on a topologically mixing, one-sided subshift of finite type and that $d\rho /d\rho \circ \theta $ is Hölder continuous. Then there exist $r\in (0,1)$ , a positive function $h\in {\mathcal H}_{\alpha }$ and $\beta>0$ such that for all $f \in \mathcal {H}_{\alpha }$ and every large $n \ge 1$ ,

$$ \begin{align*} \bigg| \frac{\mathcal{A}_n(f)(x)}{\beta^n h(x)} - \int f \,d\pi \bigg| \ll r^n (\overline D(f) +\|f\|_m). \end{align*} $$

Now assume that $\rho $ is a Bernoulli measure, so that the maps $T_i$ are chosen independently. Then, by independence, it follows that $\mathcal {A}_n = (\mathcal {A}_1)^n$ . Hence, as an immediate corollary, one obtains that

$$ \begin{align*} (\mathcal{A}_1)^n (hf)(x) /\beta^n h(x) \longrightarrow \int f(x) h(x)\,d\pi(x) \end{align*} $$

exponentially fast, which is a well-known version of Ruelle’s operator theorem for independently chosen maps $T_i$ (cf. Proposition 3.1 in [Reference Baladi2]). As this is the key step for existence and uniqueness of the annealed equilibrium state (cf. Proposition 3.3 in [Reference Baladi2]), one obtains Theorem 1 in [Reference Baladi2] for independent and identically distributed Ruelle expanding maps as a corollary.

We now return to the general case of a one-sided subshift of finite type with exponential decay of correlations and now assume, in addition, that $\rho $ is $\theta $ -invariant. In this setting, we obtain an annealed version of decay of correlations.

Theorem D. Suppose that the assumptions of Theorem C hold and that $\rho $ is $\theta $ -invariant. Then there exist a probability measure $\tilde {\pi }$ , $r \in (0,1)$ and $k_1 \in \mathbb {N}$ such that

$$ \begin{align*} &\bigg| \int \sum_{|v|=n} \mathbf{1}_{[v]}(\omega) f (T_v(x)) g(x) \,d\mu_{\omega}(x) \,d\rho(\omega) - \int f \,d\tilde{\pi} \int g \,d\mu_{\omega} \,d\rho \bigg| \\ &\quad \leq r^n \int |f| \,d\mu_{\omega} \,d\rho \bigg( \overline D(g) + \int |g| \,d\mu_{\omega} \,d\rho \bigg) \end{align*} $$

for all $g \in {\mathcal H}_{\alpha }$ and $f: X \to \mathbb {R}$ integrable with respect to $d\mu _{\omega }(x) \,d\rho (\omega )$ .

The latter reveals an unexpected connection between quenched and annealed dynamics. Indeed, it is noticeable that despite the fact that quenched and annealed random dynamical systems often measure different complexities of the dynamics (see e.g. [Reference Carvalho, Rodrigues and Varandas6, Proposition 8.3] for an explicit formula in the context of free semigroup actions), in Theorem D, we obtain an annealed decay of correlations with respect to a probability $d\mu _{\omega } \,d\rho $ obtained via quenched asymptotics. These results for both quenched and annealed dynamical systems will appear as Theorems 5.1, 7.3, 7.4 and 8.3 below. Moreover, the authors would like to point out that, according to their knowledge, Theorems C and D are the first annealed results for a dependent choice of the maps $\{T_i\}$ . Finally, in §9, we discuss applications to non-autonomous conformal iterated function systems, the thermodynamic formalism of semigroup actions and a boundary construction through equilibrium states.

3 Semigroups of Ruelle expanding maps on non-compact spaces

We always assume that $(X,d)$ is a complete and separable metric space and that ${\mathcal W}$ is a finite alphabet. For every $i\in {\mathcal W}$ , let $T_i:X \to X$ be a continuous, surjective transformation and let $\mathcal {S}$ be the semigroup generated by $\{T_i\}_{i\in {\mathcal W}}$ , that is,

$$ \begin{align*} \mathcal{S} =\{ T_{i_k}\circ T_{i_{k-1}} \circ \cdots \circ T_{i_1} {:}\, k \in {\mathbb N}, \, {i_1},{i_2},\ldots, {i_k} \in {\mathcal W} \}. \end{align*} $$

For every $k\in \mathbb N$ and every finite word $v = {i_1}{i_2}\ldots {i_k}\in {\mathcal W}^k$ , set

$$ \begin{align*} T_v:= T_{i_k}\circ \cdots \circ T_{i_1}. \end{align*} $$

Then each element of $\mathcal {S}$ is equal to $T_v$ for some finite word v, but v might not be uniquely determined (e.g. if two generators $T_a, T_b$ commute, then $T_{ab}=T_{ba}$ ). Observe that, with the usual concatenation of words, we have that $T_{vw} = T_w \circ T_v$ and, in particular, that the map from $\bigcup _{k\geq 1} {\mathcal W}^k \to \mathcal S$ given by $v \mapsto T_v$ is a semigroup anti-homomorphism, referred to as the coding of $\mathcal {S}$ . This coding naturally defines a free semigroup action $\mathcal S \times X \to X$ , $(T_v,x) \mapsto T_v(x)$ determined by $\mathcal S$ .

For every finite word $v\in {\mathcal W}^k$ , denote its length by $|v|=k$ . For $x\in X$ and $A\subset X$ , let $B_r(x)=\{y\in X: d(x,y)<r\}$ and $\ B_r(A)=\{y\in X: d(x, y)<r \text { for some } x\in A\}.$ For a finite word $v=i_1\!\ldots i_k$ , define dynamical distance

$$ \begin{align*} d_v(x,y) :=\sup\{d(x,y),\,d(T_{i_1\ldots i_j}(x), T_{i_1\ldots i_j}(y)), 1\le j<|v| \} \end{align*} $$

and dynamical ball

$$ \begin{align*} B_r^v(x):=\{y\in X: d_v(x,y)<r\}. \end{align*} $$

Later, we will also consider infinite words. The transformations $T_i, i\in {\mathcal W}$ in this paper are always Ruelle expanding maps as introduced in [Reference Ruelle29]. However, here, we do not require that the base space is compact and, in particular, the set of preimages of a point might be countably infinite. Recall that this notion of expanding map is defined as follows.

Definition 3.1. T is said to be $(a,\unicode{x3bb} )$ -Ruelle expanding, for some $a>0$ and $\unicode{x3bb} \in (0, 1)$ , if for any $x, {y}, \tilde {x} \in X$ with $d(x, {y})<a$ and $T(\tilde {x})=x$ , there exists a unique $\tilde {y}\in X$ with $T(\tilde {y})={y}$ and $d(\tilde {x}, \tilde {y})<a$ , and such that this $\tilde y$ satisfies

$$ \begin{align*} d(\tilde{x}, \tilde{y}) \leq \unicode{x3bb} d(x,y). \end{align*} $$

Examples of Ruelle expanding maps include $C^1$ expanding maps on compact Riemannian manifolds, distance expanding maps on compact metric spaces and one-sided subshifts of countable type. In particular, our setting includes distance expanding maps on non-compact metric spaces. Observe that as we only consider a finite alphabet ${\mathcal W}$ , we may choose the same parameters a and $\unicode{x3bb} $ for all $T_i, i\in {\mathcal W}$ .

We extend to the semigroup $\mathcal S$ the notions of topological mixing and finite aperiodicity, which are usually defined for the iteration of a single map. They are known from graph directed Markov systems [Reference Mauldin and Urbański25] or from the big images and preimages property for shift spaces [Reference Sarig30].

Definition 3.4. An $(a,\unicode{x3bb} )$ -Ruelle expanding semigroup $\mathcal S$ is said to be n-finitely aperiodic (see Figure 1) if there exist $n\in \mathbb N$ , a finite subset $K \subset X$ and $r>0$ such that for all $x \in X$ and $w \in {\mathcal W}^n$ , one can find $\xi , \eta \in K$ satisfying:

1. (1) there is $\xi ^{\ast } \in T_w^{-1}(\xi )$ with $d_w(x, \xi ^{\ast })<a$ ;

2. (2) there is $x^{\ast }\in T_w^{-1}(x)$ with $d(x^{\ast },\eta )< a$ and $d_w(x^{\ast }, \eta )<r$ .

Figure 1 Finite aperiodicity.

The first condition is modelled after the big image condition, the second after the big preimage condition.

We now present two classes of examples of jointly topologically mixing and finitely aperiodic semigroups.

Example 3.4. We now construct a class of semigroups generated by a finite number of skew products over the same topological Markov chain and provide sufficient conditions for joint topological mixing and finite aperiodicity.

To do so, we recall the notion of a topological Markov chain with the big images and preimages property. So assume that $A=(a_{ij})_{i,j \geq 0}$ is a matrix with values in $\{0,1\}$ without rows or columns equal to $0$ . We then refer to

$$ \begin{align*} \Sigma := \{ (x_i: i \in \mathbb{N} \cup\{0\}) : x_i \in \mathbb{N} \cup\{0\}, a_{x_{i}x_{i+1}} = 1 \text{ for all } i \geq 0 \} \end{align*} $$

as a topological Markov chain with transition matrix A. Furthermore, we say that A is aperiodic if for any pair $(i,j)$ , there exists $n_0 \in \mathbb {N}$ such that the coordinate $(i,j)$ of the nth power $A^{n}$ is strictly positive for all $n> n_0$ . Moreover, we say that $\Sigma $ has the big images and preimages property if there exits a finite subset $L \subset \mathbb {N} \cup \{0\}$ such that for each $n \in \mathbb {N} \cup \{0\}$ , there exist $k,l \in L$ such that $a_{kn} =1$ and $a_{nl} =1$ . It is worth noting here that the non-triviality of rows and columns imply that $\Sigma $ is non-compact with respect to the product of the discrete topology on $\mathbb {N} \cup \{0\}$ . In combination with the big images and preimages property, this then implies that $\Sigma $ is even locally non-compact.

We now show that the left shift $\sigma : \Sigma \to \Sigma $ is a topologically mixing $1$ -aperiodic Ruelle expanding map with respect to the metric $d_{\sigma }((x_i),(y_i)) := 2^{-\min \{i : x_i \neq y_i\}}$ , which is compatible with the product topology on $\Sigma $ . First, note that $d_{\sigma }(x,y)\leq 3/4$ implies that x and y share the same first coordinate. In particular, the restriction of $\sigma $ on balls of radius $3/4$ is a homeomorphism and expands distances by $2$ . That is, $\sigma $ is $({\textstyle \frac 34,\frac 12})$ -Ruelle expanding. Moreover, it follows from aperiodicity of A and finiteness of L that there exists $m_0$ such that for any pair $(i,j)$ in L, $\sigma ^{m_0}([i])\subset [j]$ , where $[a] \subset \Sigma $ refers to those elements in $\Sigma $ , whose first coordinate is equal to a. Hence, it follows from big images and preimages that $\sigma ^{m_0+2}([a]) = \Sigma $ for any $a \in \mathbb {N} \cup \{0\}$ . This then implies that $\sigma $ is topologically mixing. To see that $\sigma $ is $1$ -aperiodic in the sense of Definition 3.4, it remains to choose for each $i \in L$ an element $x_i \in [i]$ and check that $\{x_i : i \in L\}$ satisfies the conditions of Definition 3.4.

Now fix $(X,d)$ is as in Example 3.2, $\unicode{x3bb} \in (0,1)$ , $a> 0$ and a finite set ${\mathcal W}$ . Furthermore, assume that the set of $(a,\unicode{x3bb} )$ -Ruelle expanding maps on X is non-empty and that for any $w \in {\mathcal W}$ , $\kappa _w$ associates to each $ \mathbb {N} \cup \{0\}$ a Ruelle expanding map, that is,

$$ \begin{align*} \kappa_w : \mathbb{N} \cup\{0\} \to \{ T: X \to X \,|\,T \mbox{ is } (a,\unicode{x3bb})\mbox{-Ruelle expanding} \}. \end{align*} $$

In particular, $\kappa _w$ gives rise to the skew product

$$ \begin{align*} T_w: \Sigma \times X \to \Sigma \times X, \, ((x_i),y) \mapsto (\sigma((x_i)),T_{\kappa_w(x_0)}(y) ) \end{align*} $$

and the semigroup $\mathcal {S}$ generated by $\{T_w : w \in {\mathcal W}\}$ . With respect to $d_{\mathcal {S}}((x,y),(\bar {x},\bar {y})):= d_{\sigma }(x,\bar {x}) + d(y,\bar {y})$ , one then obtains the following.

Without specifying, $\mathcal S$ is always $(a,\unicode{x3bb} )$ -Ruelle expanding in this paper. We use the notation $x\ll y, x\gg y, x\asymp y$ to indicate that there exists a positive constant C such that $x\le Cy, x\ge Cy, C^{-1}y\le x\le C y$ , respectively.

4 Quotients of Ruelle operators

In this section, we introduce a family of quotients of Ruelle operators, which will act as strict contractions on the set of probability measures. It provides an effective construction of the relevant measures, whereas a normalization of the Ruelle operators through invariant functions has no dynamical significance in the setting of semigroups or sequential dynamics due to purely functorial reasons, as noted in Remark 6.6 below.

To begin with, let $\varphi _i:X \to {\mathbb R}$ , $i\in {\mathcal W}$ be a continuous function. We also call $\varphi _i$ a potential. Define for a finite word $v = {i_1} {i_2} \ldots {i_k}\in {\mathcal W}^k$ ,

$$ \begin{align*} \varphi_v(x):= \varphi_{{i_1}}(x) + \varphi_{{i_2}}(T_{i_1}(x)) + \cdots + \varphi_{{i_k}}(T_{i_1\ldots i_{k-1}}(x)). \end{align*} $$

Then the Ruelle operator $L_v$ is defined by

$$ \begin{align*} L_v(f)(x) := \sum_{T_v(y)=x} e^{\varphi_v(y)}f(y) \end{align*} $$

for f in a suitable function space. Note that it follows from $T_v\circ T_u = T_{uv}$ that ${L_v\circ L_{u}= L_{uv}}$ for any two finite words $u, v$ . We now define the adequate function space. For $\alpha \in (0,1]$ and $f:X \to {\mathbb R}$ , the Hölder coefficient $D_{\alpha }(f)$ is

$$ \begin{align*} D_{\alpha}(f):= \sup_{x,y \in X, x \neq y} \frac{|f(x)-f(y)|}{d(x,y)^{\alpha}} \end{align*} $$

and the space of $\alpha $ -Hölder functions ${\mathcal H}_{\alpha }^*$ is

$$ \begin{align*} \mathcal{H}^{\ast}_{\alpha} := \{ f : D_{\alpha}(f) < \infty \}. \end{align*} $$

Let ${\mathcal H}_{\alpha }$ denote the subspace of bounded functions in $\mathcal {H}^{\ast }_{\alpha }$ . It is well known that ${\mathcal H}_{\alpha }$ is a Banach space with respect to the norm $\|\cdot \|:=\|\cdot \|_{\infty } + D_{\alpha }(\cdot )$ . We are now in position to specify the class of potentials considered here.

Suppose $\varphi _i$ is $\alpha $ -Hölder for every $i\in {\mathcal W}$ . We shall estimate distortion of $\varphi _v$ . Due to the $(a,\unicode{x3bb} )$ -Ruelle expanding property, for $v=i_1\ldots i_k \in {\mathcal W}^k$ and $x, y, \tilde {x}\in X$ with ${d(x, y) < a}$ and $T_v(\tilde x)=x$ , there exists a unique point $\tilde y\in T_v^{-1}(y)\cap B_a^v(\tilde x)$ . Moreover,

$$ \begin{align*} d(\tilde x, \tilde y) < \unicode{x3bb}^k d(x, y),\quad d(T_{i_1 \ldots i_j}(\tilde x), T_{i_1 \ldots i_j}(\tilde y)) < \unicode{x3bb}^{k-j} d({x},{y}),\quad 1\le j<k. \end{align*} $$

Hence, the inverse branch

(4.1) $$ \begin{align} (T_v)_{\tilde x}^{-1}: B_a(x) \to B_a^v(\tilde x),\quad y\mapsto \tilde y \end{align} $$

is well defined and contracts the distance at every intermediate step by $\unicode{x3bb} $ . It follows that for any pair $x, y$ with $d(x,y)<a$ , there is a bijection from $T_v^{-1}(x)$ to $T_v^{-1}(y)$ given by

(4.2) $$ \begin{align} \tilde x\mapsto \tilde y_{\tilde x}:=(T_v)^{-1}_{\tilde x}(y). \end{align} $$

Now Hölder continuity implies that whenever $d(x,y)<a$ ,

(4.3) $$ \begin{align} |\varphi_v(\tilde x)-\varphi_v(\tilde y_{\tilde x})| \leq \frac{\max_{i \in {\mathcal W}} D_{\alpha}(\varphi_{i})}{1-\unicode{x3bb}^{\alpha}} d({x},{y})^{\alpha} =: C_{\varphi} d({x},{y})^{\alpha}. \end{align} $$

It follows from a simple argument that $L_v$ maps $\mathcal {H}_{\alpha }$ to ${\mathcal H}_{\alpha }$ if $\varphi _v$ is also summable.

As we are interested in operators that leave invariant the constant function $\mathbf {1}$ , define for finite words $u, v$

$$ \begin{align*} \mathbb P_{u}^v(f) := \frac{L_v(f \cdot L_{u}(\mathbf{1}))}{L_{uv}(\mathbf{1})} = \frac{L_{uv}(f\circ T_u)}{L_{uv}(\mathbf{1})}. \end{align*} $$

It is clear from the definition that

$$ \begin{align*} \mathbb{P}_{u}^v(\mathbf{1}) =\mathbf{1}. \end{align*} $$

The motivation to consider these families of operators stems from the simple observation that for finite words $u,v,w$ ,

$$ \begin{align*} \mathbb{P}_{uv}^w\circ \mathbb{P}_{u}^v(f) = \frac{L_w( \mathbb{P}_{u}^v(f) \cdot L_{uv}(\mathbf{1}))}{L_{uvw}(\mathbf{1})} = \frac{L_w( L_v(f \cdot L_u(\mathbf{1})))}{L_{uvw}(\mathbf{1})} = \mathbb{P}_{u}^{vw}(f). \end{align*} $$

Hence, with

$$ \begin{align*} \mathbb{P}^{w}(f) := L_w(f)/L_w(\mathbf{1}), \end{align*} $$

for a sequence of finite words $v_1, \ldots v_k$ ,

(4.4) $$ \begin{align} \mathbb{P}^{v_1\ldots v_k} = \mathbb{P}_{v_1\ldots v_{k-1}}^{v_k} \circ \mathbb{P}_{v_1\ldots v_{k-2}}^{v_{k-1}} \circ \cdots \circ \mathbb{P}_{v_1 v_2}^{v_3} \circ \mathbb{P}_{v_1}^{v_2} \circ \mathbb{P}^{v_1}. \end{align} $$

As a first result, we obtain $\mathcal {H}_{\alpha }$ -invariance of these quenched operators.

Lemma 4.1. $\mathbb {P}_{u}^{v}$ is a bounded operator on $\mathcal {H}_{\alpha }$ . Furthermore, for $f \in \mathcal {H}_{\alpha }$ and $x,y$ with $d(x,y)<a$ ,

(4.5) $$ \begin{align} |\mathbb{P}_{u}^{v}(f)(x) - \mathbb{P}_{u}^{v}(f)(y)| \leq C_{\varphi} (2 \|f\|_{\infty} + \unicode{x3bb}^{|v|} D_{\alpha}(f) ) d(x,y)^{\alpha}. \end{align} $$

Proof. Following verbatim the proof of Lemma 2.1 in [Reference Bessa and Stadlbauer3], one obtains that for $x,y$ with $d(x,y)<a$ ,

$$ \begin{align*} |L_v( f L_u(\mathbf{1}))(x) - L_v( f L_u(\mathbf{1}))(y)| \leq C_{\varphi} L_{uv}(\mathbf{1})(x) (\|f\|_{\infty} + \unicode{x3bb}^{|v|}D_{\alpha}(f)) d(x,y)^{\alpha}. \end{align*} $$

The estimate (4.5) follows from this as in [Reference Bessa and Stadlbauer3]. It remains to show that the operators are bounded and leave invariant $\mathcal {H}_{\alpha }$ . As $\mathbb {P}_{u}^{v}$ maps positive functions to positive functions and $\mathbb {P}_{u}^{v}(\mathbf {1})= \mathbf {1}$ , we have $\|\mathbb {P}_{u}^{v}(f) \|_{\infty } \leq \|f\|_{\infty }$ . Furthermore, by considering the cases ${d(x,y)<a}$ and $d(x,y)\geq a$ separately, we obtain

$$ \begin{align*} D_{\alpha}(\mathbb{P}_{u}^{v}(f)) \leq \max\{ C_{\varphi} (2 \|f\|_{\infty} + \unicode{x3bb}^{|v|} D_{\alpha}(f)), 2 a^{-\alpha} \|f\|_{\infty} \}, \end{align*} $$

which proves that $\mathbb {P}_{u}^{v}:\mathcal {H}_{\alpha } \to \mathcal {H}_{\alpha } $ is a well-defined and bounded operator.

We observe that Lemma 4.1, which requires Hölder continuity of the potentials and no further assumption on topological irreducibility, is one of the principal ingredients to prove that the duals of the previous operators act as contractions on the space of probabilities. The other ingredient is the following result for which finite aperiodicity is essential.

Proof. First, note that for any $x,y\in X$ with $d(x,y)<a$ and any finite word v, the bijection of equation (4.2) and the estimate (4.3) imply that $L_v(\mathbf {1})(x) \asymp L_v(\mathbf {1})(y)$ .

Suppose $\mathcal S$ is n-finitely aperiodic. Let K be a finite set and $r>0$ be given by finite aperiodicity. It follows from the Ruelle expanding property and joint topological mixing that there exists $m\in \mathbb N$ such that for all $\xi , \eta \in K$ and $|w|\ge m$ , there exists $\eta ^* \in X$ with $T_w(\eta ^*)=\eta $ and $d(\eta ^*, \xi )<a$ .

We now show the lemma for any $x, y\in X$ and all finite words v with $|v|>2n+m$ . Take such a finite word v, we will select preimages of x as follows, illustrated in Figure 2.

Figure 2 Selection of preimages.

Decompose $v=upwq$ , where $u,w,p,q$ are finite words and $|p|=|q|=n, |w|=m$ . Note that

$$ \begin{align*} L_v(\mathbf 1)(x)=L_{wq}(L_{up}(\mathbf 1))(x) \le \sup_{i\in{\mathcal W}}\|L_i(\mathbf 1)\|_{\infty}^{n+m}\sup_{x'\in T^{-1}_{wq}(x)}\! L_{up}(\mathbf 1)(x'). \end{align*} $$

Fix $x'\in T^{-1}_{wq}(x)$ . For any $\tilde x\in T^{-1}_{up}(x')$ , let $\hat x=T_u(\tilde x)$ . There exist by condition (1) of finite aperiodicity, $\xi \in K$ and $\xi ^{\ast }\in T_p^{-1}(\xi )$ such that $d_p(\hat x, \xi ^{\ast })<a$ . Let $\tilde \xi ^{\ast }=(T_u)_{\tilde x}^{-1}(\xi ^{\ast })$ , the inverse branch defined in equation (4.1). Then using equation (4.3),

$$ \begin{align*} e^{\varphi_{up}(\tilde x)}=e^{\varphi_u(\tilde x)}e^{\varphi_p(\hat x)}\le e^{C_{\varphi} a^{\alpha}+\varphi_{u}(\tilde \xi^{\ast})} e^{na^{\alpha}+\varphi_p(\xi^{\ast})}=e^{C_{\varphi} a^{\alpha}+na^{\alpha}}e^{\varphi_{up}(\tilde \xi^{\ast})}. \end{align*} $$

Because $d_{up}(\tilde x, \tilde \xi ^{\ast })<a$ and $T_{up}(\tilde \xi ^{\ast })=\xi $ , one has $\tilde x=(T_{up})^{-1}_{\tilde \xi ^{\ast }}(x')$ and $\tilde \xi ^{\ast }=(T_{up})^{-1}_{\tilde x}(\xi )$ . Therefore, different $\tilde x$ is associated to different $\tilde \xi ^{\ast }$ , so that

$$ \begin{align*} L_{up}(\mathbf 1)(x')=\sum_{\tilde x\in T_{up}^{-1}(x')} e^{\varphi_{up}(\tilde x)}\ll \sum_{\tilde x\in T_{up}^{-1}(x') } e^{\varphi_{up}(\tilde \xi^{\ast})}\le \sum_{\xi\in K} L_{up}(\mathbf 1)(\xi). \end{align*} $$

Hence,

$$ \begin{align*} L_{v}(\mathbf 1)(x)\ll \sum_{\xi\in K}L_{up}(\mathbf 1)(\xi). \end{align*} $$

However, there exist by condition (2) of finite aperiodicity, a preimage $x^{\ast }\in T_q^{-1}(x)$ and $\eta \in K$ such that $d(x^{\ast },\eta )<a$ and $\eta \in B_{r}^q(x^{\ast }).$ As $d(x^{\ast }, \eta )<a$ , we know that $L_{upw}(\mathbf 1)(x^{\ast })\asymp L_{upw}(\mathbf 1)(\eta )$ . Then,

$$ \begin{align*} L_v(\mathbf 1)(x)\ge e^{\varphi_q(x^{\ast})}L_{upw}(\mathbf 1)(x^{\ast})\gg e^{\varphi_q(\eta)-nr^{\alpha}}L_{upw}(\mathbf 1)(\eta)\gg L_{upw}(\mathbf 1)(\eta). \end{align*} $$

The last estimate holds because $q\in {\mathcal W}^n$ and $\eta \in K$ both range over finite sets. Now for any $\xi \in K$ , one can find $\eta ^*\in T_w^{-1}(\eta )$ such that $d(\eta ^*, \xi )<a$ , then find such a $\eta ^*_0$ for $\xi _0$ that achieves $\max _{\xi _{\in } K}L_{up}(\mathbf 1)(\xi )$ . Then, $L_{up}(\mathbf 1)(\xi _0)\asymp L_{up}(\mathbf 1)(\eta ^{\ast }_0)$ and

$$ \begin{align*} L_{upw}(\mathbf 1)(\eta)&=\sum_{\eta^*\in T_w^{-1}(\eta)}e^{\varphi_w(\eta^{\ast})} L_{up}(\mathbf 1)(\eta^*)\ge e^{\varphi_w(\eta_0^{\ast})}L_{up}(\mathbf 1)(\eta_0^{\ast})\\ &\gg e^{\varphi_w(\eta_0^{\ast})}L_{up}(\mathbf 1)(\xi_0)\gg L_{up}(\mathbf 1)(\xi_0). \end{align*} $$

The last estimate holds because $\varphi _w$ is continuous, $\eta _0^{\ast }\in \overline {B_a(\xi _0)}$ , $\xi _0\in K$ and $w\in {\mathcal W}^m$ range over finite sets. Therefore,

$$ \begin{align*} L_v(\mathbf 1)(x)\gg \max_{\xi\in K} L_{up}(\mathbf 1)(\xi). \end{align*} $$

All the constants absorbed into $\ll $ or $\gg $ are determined by $\mathcal S, \varphi , K, m, n$ (essentially by $\mathcal S$ and $\varphi $ ), in particular independent of $v, x, y$ . It follows from the above estimates that $L_v(\mathbf {1})(x) \asymp L_v(\mathbf {1})(y)$ for any $x,y\in X$ .

Lastly, when $|v|\le 2n+m$ , take any finite word $|v'|>2n+m$ , then for any $x\in X$ ,

$$ \begin{align*} L_{v'v}(\mathbf 1)(x)=L_v(L_{v'}(\mathbf 1))(x)=\sum_{\tilde x\in T_v^{-1}(x)} e^{\varphi(\tilde x)}L_{v'}(\mathbf 1)(\tilde x) &\asymp \sum_{\tilde x\in T_v^{-1}(x)} e^{\varphi(\tilde x)}L_{v'}(\mathbf 1)(x)\\ &= L_v(\mathbf 1)(x)L_{v'}(\mathbf 1)(x) \end{align*} $$

by the already-proven case. So $L_v(\mathbf 1)(x)\asymp L_{v'v}(\mathbf 1)(x)/L_{v'}(\mathbf 1)(x)$ , and hence for any $x, y\in X$ , $L_v(\mathbf 1)(x)\asymp L_v(\mathbf 1)(y)$ .

5 Contraction in the Wasserstein distance

Let $\mathcal {M}_1(X)$ refer to the space of Borel probability measures on X. Recall that the Wasserstein distance W of $\mu , \nu \in \mathcal {M}_1(X)$ defined by

$$ \begin{align*} W(\mu, \nu) := \inf\bigg\{\!\int d(x,y) \,dP: P \in \Pi(\mu, \nu) \bigg\} \end{align*} $$

is a compatible metric with weak convergence, where $\Pi (\mu , \nu )$ refers to the couplings of $\mu $ and $\nu $ , that is, the set of probability measures on $X \times X$ with marginal distributions $\mu $ and $\nu $ . Moreover, by Kantorovich’s duality,

$$ \begin{align*} W(\mu, \nu) = \sup \bigg\{ \bigg| \int f \,d(\mu-\nu)\bigg|: \sup_{x \neq y} \frac{|f(x)-f(y)|}{d(x,y)}\leq 1 \bigg\}. \end{align*} $$

Let ${\mathbb P_u^v}^{\ast }$ denote the dual operator of $\mathbb P_u^v$ on $\mathcal M_1(X)$ . To obtain a contraction of $W({\mathbb {P}_{u}^{v}}^{\ast }(\cdot ), {\mathbb {P}_{u}^{v}}^{\ast }(\cdot ))$ , the estimates of Lemma 4.1 indicate that for a-close measures, one should consider $(d(x,y))^{\alpha }$ instead of $d(x,y)$ . However, for distant measures, the method of proof below based on an idea in [Reference Hairer and Mattingly17] (see also [Reference Bessa and Stadlbauer3, Reference Kloeckner, Lopes and Stadlbauer23, Reference Stadlbauer31, Reference Stadlbauer and Zhang32]) requires a truncated distance. We consider

(5.1) $$ \begin{align} {d}^{\ast}(x,y) := \min\{1, \Delta \,{d(x,y)^{\alpha}} \}, \quad \Delta:= \max\{4C_{\varphi},a^{-\alpha}\}. \end{align} $$

Observe that, by construction, $d(x,y)< a$ whenever ${d}^{\ast }(x,y)<1$ . To see that $d^{\ast }$ is a metric, observe that the triangle inequality follows from $x^{\alpha } + y^{\alpha } \geq (x+y)^{\alpha }$ for $x,y \geq 0$ and $0< \alpha \leq 1$ , which is an inequality that easily can be deduced from the concavity of $x \mapsto x^{\alpha }$ . The remaining assertion that $d^{\ast }(x,y) = 0$ if and only if $x=y$ is trivial.

We now introduce the space of ${d}^{\ast }$ -Lipschitz functions. To do so, recall that the Lipschitz coefficient is defined by $D_{d^{\ast }}(f) := \sup \{ |f(x) -f(y)|/d^{\ast }(x,y) : x\neq y\}$ and that f is a bounded Lipschitz continuous function with respect to $d^{\ast }$ if and only if $\|f\|:= \| f \|_{\infty } + D_{d^{\ast }}(f)< \infty $ . To identify these functions in terms of the metric d, set

$$ \begin{align*} \overline{D}(f) := \max\{\sup_{x,y \in X} |f(x)-f(y)|, D_{\alpha}^{{\tiny loc}}(f)/\Delta \}, \end{align*} $$

where

$$ \begin{align*} D_{\alpha}^{{\tiny loc}}(f):= \sup\bigg\{ \frac{|f(x)-f(y)|}{d(x,y)^{\alpha}} : x,y \in X, 0< d(x,y) < \Delta^{-{1}/{\alpha}}\bigg\}. \end{align*} $$

Now observe that it follows from the construction that $\overline {D}(f) = D_{d^{\ast }}(f)$ , $\overline D(f)\le 2\|f\|_{\infty }+ \Delta ^{-1} D_{\alpha }(f)$ and $D_{\alpha }(f)\le \Delta \overline D(f)$ . Hence, the norms $\|\cdot \|_{\infty }+D_{\alpha }^{ {\tiny loc}} (\cdot )$ and $\|\cdot \|_{\infty }+ D_{d^{\ast }}(\cdot )$ are equivalent. In particular, by Kantorovich’s duality, the Wasserstein metric $\overline {W}$ with respect to $d^{\ast }$ is characterized through local Hölder continuous functions with respect to d by

$$ \begin{align*} \overline{W}(\mu, \nu) = \sup \bigg\{ \bigg| \int f \,d(\mu- \nu) \bigg|: \overline{D}(f) \leq 1 \bigg\}. \end{align*} $$

Theorem 5.1. Suppose that $\mathcal {S}$ is jointly topologically mixing and a finitely aperiodic Ruelle expanding semigroup, and that every potential $\varphi _i$ is $\alpha $ -Hölder and summable. Then there exist $k_0 \in {\mathbb N}$ and $s \in (0,1)$ such that for all finite words $u,v$ with $|v|\ge k_0$ and $\nu _1 , \nu _2\in \mathcal {M}_1(X)$ and f with $\overline {D}(f) < \infty $ ,

$$ \begin{align*} \overline{W}({\mathbb{P}_{u}^{v}}^{\ast}(\nu_1), {\mathbb{P}_{u}^{v}}^{\ast} (\nu_2)) &\leq s^{n} \overline{W}( \nu_1 , \nu_2),\\ \quad \overline{D}(\mathbb{P}_{u}^{v}(f)) &\leq s^{n} \overline{D}(f). \end{align*} $$

Proof. As in [Reference Hairer and Mattingly17], we first prove the assertions for Dirac measures and then extend the partial result by optimal transport to arbitrary probability measures.

(1) Local contraction. Assume that ${d}^{\ast }(x,y)<1$ and that f is $d^{\ast }$ -Lipschitz continuous. Since $d(x,y)<a$ as soon as ${d}^{\ast }(x,y)<1$ , Lemma 4.1 gives that

$$ \begin{align*} |\mathbb{P}_{u}^{v}(f)(x) - \mathbb{P}_{u}^{v}(f)(y) |\leq (2C_{\varphi} \|f\|_{\infty} + \unicode{x3bb}^{|v|} D_{\alpha}^{{\tiny loc}}(f))(d(x,y))^{\alpha}. \end{align*} $$

Furthermore, as $\mathbb {P}_{u}^{v}(\mathbf {1})=\mathbf {1}$ , one may suppose without loss of generality that $\inf f=0$ , and therefore, $\|f\|_{\infty } \leq \overline {D}(f)$ . Dividing by $\Delta $ and choosing $k_0$ such that $\unicode{x3bb} ^{k_0} \leq 1/4$ , it follows that for v with $|v| \geq k_0$ ,

$$ \begin{align*} |\mathbb{P}_{u}^{v}(f)(x) - \mathbb{P}_{u}^{v}(f)(y) | \leq \bigg( \frac{\|f\|_{\infty}}{2} + \frac{D_{\alpha}^{{\tiny loc}}(f)}{4 \Delta} \bigg) d^{\ast}(x,y) \leq \frac{3 \overline{D}(f)}{4}d^{\ast}(x,y). \end{align*} $$

Hence, by Kantorovich’s duality,

$$ \begin{align*}\overline{W}({\mathbb{P}_{u}^{v}}^{\ast}(\delta_x), {\mathbb{P}_{u}^{v}}^{\ast}(\delta_y)) \leq \tfrac34 d^{\ast}(x,y) = \tfrac34 \overline{W}(\delta_x,\delta_y).\end{align*} $$

(2) Global contraction. If ${d}^{\ast }(x,y) =1$ , an upper bound for $\overline {W}$ can be obtained by construction of a coupling based on finite aperiodicity. To do so, fix an open set U of diameter smaller than $a/2$ . Suppose $\mathcal S$ is $n_1$ -finitely aperiodic and $K, r$ are given by finite aperiodicity. As $\mathcal S$ is jointly topologically mixing, one can find $n_2$ such that $T_{w}(U) \cap B_{a}(\xi ) \neq \emptyset $ for all $w\in {\mathcal W}^{n_2}$ and $\xi \in K$ and that $\unicode{x3bb} ^{n_2} <1/8$ . Choose $n_3$ large such that $C_{n_3}:=\Delta (a\unicode{x3bb} ^{n_3})^{\alpha }<1/2.$ Let $k_0= n_1+n_2+n_3$ .

Let $n\ge k_0$ . For $v\in {\mathcal W}^n$ , write $v=v_3v_2v_1$ , where $|v_1|=n_1, |v_2|=n_2$ and $|v_3|\ge n_3$ . For any $x\in X$ , we will select a preimage $x^{\#}$ in $T_{v_2v_1}^{-1}(x)$ as below, illustrated in Figure 3.

Figure 3 The map $x\mapsto x^{\#}$ .

Let $\eta \in K$ and ${x}^{\ast } \in X$ be given by condition (2) of finite aperiodicity so that $T_{v_1}({x}^{\ast })\,{=}\,x, d(x^{\ast }, \eta )<a$ and $x^{\ast }\in B_{r}^{v_1}(\eta )$ . Now the choice of $n_2$ and Ruelle expanding property allow us to find a preimage $\eta '\in T_{v_2}^{-1}(\eta )$ such that $\eta '\in B_{a/8}(U)$ . Use the Ruelle expanding property again to find a preimage $x^{\#}\in T_{v_2}^{-1}(x^{\ast })$ such that $x^{\#} \in B_{a/8}(\eta ')\subset B_{a/4}(U)$ . One has $|\varphi _{v_2}(x^{\#})-\varphi _{v_2}(\eta ')|\le C_{\varphi } a^{\alpha }$ by equation (4.3), so that

$$ \begin{align*} |\varphi_{v_2v_1}(x^{\#}) - \varphi_{v_2v_1}(\eta')| \leq C_{\varphi} a^{\alpha}+n_1 r^{\alpha} \max_{i\in {\mathcal W}} D_{\alpha}(\varphi_i), \end{align*} $$

and hence

$$ \begin{align*} e^{ \varphi_{v_2v_1}(x^{\#})} \asymp e^{\varphi_{v_2v_1}(\eta')}=e^{\varphi_{v_2}(\eta')}e^{\varphi_{v_1}(\eta)}. \end{align*} $$

Since $\eta '$ lies in a fixed bounded region $B_{a/8}(U)$ and $\varphi $ is continuous and $\eta \in K, v_1\in {\mathcal W}^{n_1}, v_2\in {\mathcal W}^{n_2}$ range over finite sets, one concludes that for all $x\in X, v_1\in {\mathcal W}^{n_1}, v_2\in {\mathcal W}^{n_2}$ ,

(5.2) $$ \begin{align} e^{ \varphi_{v_2v_1}(x^{\#})}\asymp 1. \end{align} $$

For any pair $(x,y)\in X^2$ , find as before $x^{\#}, y^{\#}\in B_{a/4}(U)$ . Then $d(x^{\#}, y^{\#})<a$ . As stated in equation (4.2), there is a bijection $\tilde x\mapsto \tilde y$ from $T_{v_3}^{-1}(x^{\#})$ to $T_{v_3}^{-1}(y^{\#})$ . Pair $(\tilde x, \tilde y)$ together by this bijection and set a subprobability measure on $X^2$ ,

$$ \begin{align*} Q_{(x,y)} := \min \bigg\{\sum_{(\tilde x, \tilde y)} \frac{e^{\varphi_{v}(\tilde x)}L_u(\mathbf{1})(\tilde x)}{L_{uv}(\mathbf{1})(x)} \; \delta_{(\tilde{x},\tilde{y})},\ \sum_{(\tilde x, \tilde y)} \frac{e^{\varphi_{v}(\tilde y)}L_u(\mathbf{1})(\tilde y)}{L_{uv}(\mathbf{1})(y)} \; \delta_{(\tilde{x},\tilde{y})}\bigg\}. \end{align*} $$

Note that $Q_{(x,y)}(X^2)=Q_{(x,y)}(\{(z_1, z_2):d(z_1, z_2)<a\unicode{x3bb} ^{|v_3|}\})$ . For any $A\subset X$ ,

$$ \begin{align*} Q_{(x,y)}(A\times X)\le \sum_{T_v(z)=x} \frac{e^{\varphi_v(z)}\mathbf 1_A\cdot L_u(\mathbf 1)(z)}{L_{uv}(\mathbf 1)(x)}=\frac{L_v(\mathbf 1_A\cdot L_u(\mathbf 1))}{L_{uv}(\mathbf 1)}(x)={\mathbb P_{u}^{v}}^{\ast}(\delta_x)(A) \end{align*} $$

and similarly $Q_{(x,y)}(X\times A)\le {\mathbb {P}_{u}^{v}}^{\ast }(\delta _y)(A).$ Hence, there exists a further subprobability measure R such that $P:= Q_{(x,y)}+R \in \Pi ({\mathbb {P}_{u}^{v}}^{\ast }(\delta _x), {\mathbb {P}_{u}^{v}}^{\ast }(\delta _y))$ (see, e.g. [Reference Hairer and Mattingly17]). Therefore, due to the choice of $n_3$ ,

$$ \begin{align*} \overline{W}({\mathbb{P}_{u}^{v}}^{\ast}(\delta_x), {\mathbb{P}_{u}^{v}}^{\ast}(\delta_y)) &\leq \int d^{\ast}(z_1, z_2) dP \\ &\leq \Delta (a \unicode{x3bb}^{|v_3|})^{\alpha} P(\{ d(z_1, z_2)< a \unicode{x3bb}^{|v_3|} \}) + P(\{ d(z_1, z_2)\geq a \unicode{x3bb}^{|v_3|} \})\\ &\leq 1- C_{n_3} P(\{ d(z_1, z_2)< a \unicode{x3bb}^{|v_3|} \}) \leq 1 - C_{n_3} Q_{(x,y)}(X^2). \end{align*} $$

To get a lower bound for $Q_{(x,y)}(X^2)$ , use equation (5.2) to see

$$ \begin{align*} Q_{(x,y)} (X^2) &\asymp \min \bigg\{\sum_{T_{v_3}(\tilde x)=x^{\#}} \frac{e^{\varphi_{v_3}(\tilde x)}L_u(\mathbf{1})(\tilde x)}{L_{uv}(\mathbf{1})(x)} , \sum_{T_{v_3}(\tilde y)=y^{\#}} \frac{e^{\varphi_{v_3}(\tilde y)}L_u(\mathbf{1})(\tilde y)}{L_{uv}(\mathbf{1})(y)} \bigg\}\\ &=\min\bigg\{\frac{L_{uv_3}(\mathbf 1)(x^{\#})}{L_{uv}(\mathbf 1)(x)},\ \frac{L_{uv_3}(\mathbf 1)(y^{\#})}{L_{uv}(\mathbf 1)(y)} \bigg\}. \end{align*} $$

Applying Lemma 4.2, we get that for any $\xi _0\in K$ ,

$$ \begin{align*} Q_{(x,y)}(X^2)\asymp \frac{1}{L_{v_2v_1}(\mathbf{1})(\xi_0)} \geq \min \{ (L_w(\mathbf{1})(\xi))^{-1} : \xi \in K, w\in {\mathcal W}^{n_1+n_2}\}>0. \end{align*} $$

Hence, there is a lower bound $N \leq Q_{(x, y)}(X^2)$ , independent of $x,y \in X$ and $v\in {\mathcal W}^n$ . Therefore, increasing $n_3$ so that $C_{n_3}N<1$ if needed,

$$ \begin{align*}\overline{W}({\mathbb{P}_{u}^{v}}^{\ast}(\delta_x), {\mathbb{P}_{u}^{v}}^{\ast}(\delta_y)) \leq 1 - C_{n_3} N = (1 - C_{n_3} N ) d^{\ast}(x,y) = (1 - C_{n_3}N ) \overline{W}( \delta_x,\delta_y). \end{align*} $$

Combining part (1) with part (2) of the proof and letting $t:= \max \{3/4, 1 - C_{n_3}N\}<1$ , we obtain that there exists $k_0$ such that for all finite words $u, v$ with $|v|\ge k_0$ and $x,y\in X$ ,

$$ \begin{align*} \overline{W}({\mathbb{P}_{u}^{v}}^{\ast}(\delta_x), {\mathbb{P}_{u}^{v}}^{\ast}(\delta_y)) \leq t \overline{W}(\delta_x,\delta_y). \end{align*} $$

Using Kantorovich’s duality, for f with $\overline {D}(f) \leq 1$ , it follows that

$$ \begin{align*} |\mathbb{P}_{u}^{v}(f)(x) - \mathbb{P}_{u}^{v}(f)(y)| = \bigg| \int f d{\mathbb{P}_{u}^{v}}^{\ast}(\delta_x) - \int f \,d{\mathbb{P}_{u}^{v}}^{\ast}(\delta_y) \bigg| \leq t. \end{align*} $$

(3) Contraction for arbitrary probability measures. The extension to arbitrary probability measures is a standard application of optimal transport and omitted as the proof is a straightforward adaption of [Reference Hairer and Mattingly17], [Reference Stadlbauer31] or [Reference Kloeckner, Lopes and Stadlbauer23]. We obtain that for any finite words $u, v$ with $|v|\ge k_0$ and any probability measures $\nu _1, \nu _2$ ,

$$ \begin{align*} \overline{W}({\mathbb{P}_{u}^{v}}^{\ast}(\nu_1), {\mathbb{P}_{u}^{v}}^{\ast}(\nu_2)) \leq t \overline{W}(\nu_1,\nu_2). \end{align*} $$

(4) Iteration. By the iteration rules given in equation (4.4), the theorem follows for $s = t^{1/2k_0}$ . $\Box $

6 Conformal measures, quenched exponential decay and continuity

From now on, we always assume that $\mathcal {S}$ is jointly topologically mixing and finitely aperiodic and every potential $\varphi _i$ is $\alpha $ -Hölder and summable, so that Theorem 5.1 holds. It has immediate consequences for the existence and regularity of two types of compact sets of probability measures, which are canonical generalizations of conformal measures and equilibrium states to the context of semigroups.

6.1 One-sided dynamics

Denote by $\Sigma =\{i_1i_2\ldots : i_1, i_2,\ldots \in {\mathcal W}\}$ the set of infinite words and by $\theta (i_1i_2\ldots )=i_2i_3\ldots $ the shift map. For an infinite word $\omega =i_1 i_2\ldots \in \Sigma $ and $k \in \mathbb {N}$ , let

$$ \begin{align*} [\omega]_k:=i_1 \ldots i_k \in{\mathcal W}^k. \end{align*} $$

The first family of measures is constructed as follows, which generalizes the notion of conformal measures.

Proposition 6.1. For any finite word u, infinite word $\omega $ and measure $\nu \in \mathcal {M}_1(X)$ , the limit

$$ \begin{align*} \mu_{u,\omega}:= \lim_{l \to \infty} {\mathbb{P}_{u}^{[\omega]_l}}^{\ast}(\nu) \end{align*} $$

exists and is independent of $\nu $ . Furthermore, with $k_0$ and s given by Theorem 5.1, the following statements hold.

1. (1) For $k\geq k_0$ and any $\omega , \tilde {\omega } \in \Sigma $ with $[\omega ]_k = [\tilde {\omega }]_k$ , $\overline {W}(\mu _{u,\omega }, \mu _{u,\tilde {\omega }}) \leq s^k$ .

2. (2) For $k\geq k_0$ and $f \in \mathcal {H}_{\alpha }$ ,

   $$ \begin{align*} \bigg\| \mathbb{P}_{u}^{[\omega]_k}(f) - \int f \,d\mu_{u,\omega} \bigg\| \leq 2 s^k \overline{D}(f). \end{align*} $$

3. (3) Let $\mu _{\omega }:= \mu _{\emptyset , \omega }$ , then

   $$ \begin{align*} \mu_{u\omega}={\mathbb P^u}^*(\mu_{u,\omega}),\quad \mu_{u,\omega} = \mu_{u\omega}\circ T_u^{-1}. \end{align*} $$
   If v is a finite word,
   $$ \begin{align*} \mu_{u, v\omega}={\mathbb P_{u}^{v}}^{\ast} (\mu_{uv, \omega}). \end{align*} $$

4. (4) Let $\unicode{x3bb} _{u, \omega } := \int L_u(\mathbf {1}) \,d\mu _{\omega } $ , then

   $$ \begin{align*} L_u^{\ast}(\mu_{\omega}) = \unicode{x3bb}_{u,\omega} \mu_{u\omega}, \end{align*} $$
   and if v is a finite word,
   $$ \begin{align*} \unicode{x3bb}_{uv,\omega} = \unicode{x3bb}_{u,v\omega} \unicode{x3bb}_{v,\omega}. \end{align*} $$

5. (5) The measures $\mu _{u, \omega }$ and $\mu _{\omega }$ are absolutely continuous to each other and

   $$ \begin{align*} h_{u,\omega}:=\frac{d\mu_{u,\omega}}{d\mu_{\omega}}=\unicode{x3bb}_{u,\omega}^{-1} L_u(\mathbf{1}). \end{align*} $$

Proof. For probability measures $\nu ,\tilde {\nu }$ on X and $l>k\ge k_0$ , Theorem 5.1 implies

$$ \begin{align*} \overline{W}({\mathbb{P}_{u}^{[\omega]_k}}^{\ast}(\nu), {\mathbb{P}_{u}^{[\omega]_l}}^{\ast}(\tilde{\nu})) = \overline{W}({\mathbb{P}_{u}^{[\omega]_k}}^{\ast}(\nu), {\mathbb{P}_{u}^{[\omega]_k}}^{\ast}\circ {\mathbb{P}_{u[\omega]_k}^{[\theta^k \omega]_{l-k}}}^{\ast} (\tilde{\nu}))\leq s^k. \end{align*} $$

Hence, $\{{\mathbb {P}_{u}^{[\omega ]_k}}^{\ast }(\nu )\}_{k\geq k_0}$ is a Cauchy sequence and $\mu _{u,\omega } := \lim _k {\mathbb {P}_{u}^{[\omega ]_k}}^{\ast }(\nu )$ exists and is independent of $\nu $ . This, in particular, implies the estimate in item (1). To show item (2), it suffices to consider $\nu =\delta _x$ . If $k \geq k_0$ , we have that

$$ \begin{align*} \bigg|\mathbb{P}_{u}^{[\omega]_k}(f)(x)- \int f \,d\mu_{u,\omega}\bigg| \leq \overline{D}(f) s^k. \end{align*} $$

The estimate in item (2) then follows from this combined with Theorem 5.1.

The second part of item (3) follows from

$$ \begin{align*} \int \mathbb P_u^v(f)\,d\mu_{uv,\omega} =\lim_{k \to \infty} \mathbb P_{uv}^{[\omega]_k}\circ \mathbb P_u^v(f)(x)=\lim_{k\to\infty}\mathbb P_{u}^{v[\omega]_k}(f)(x)=\int f \,d\mu_{u, v\omega}. \end{align*} $$

The first part of item (3) follows from this and

$$ \begin{align*} \int f \,d\mu_{u,\omega} &= \lim_{k \to \infty} \frac{L_{[\omega]_k} (f L_u(\mathbf{1}))(x)}{ L_{u[\omega]_k}(\mathbf{1})(x)} = \lim_{k \to \infty} \frac{L_{u[\omega]_k}(f\circ T_u)(x)}{L_{u[\omega]_k}(\mathbf{1})(x)}\\ & = \int f\circ T_u \,d\mu_{u\omega} = \int f\,d\mu_{u\omega}\circ T_u^{-1}. \end{align*} $$

Item (4) holds because

$$ \begin{align*} \int L_u(f)\,d\mu_{\omega} & = \lim_{k \to \infty} \frac{L_{[\omega]_k} (L_u(f))(x)}{ L_{[\omega]_k}(\mathbf{1})(x)} = \lim_{k \to \infty} \frac{L_{u[\omega]_k} (f)(x)}{ L_{u[\omega]_k}(\mathbf{1})(x)}\cdot \frac{ L_{u[\omega]_k}(\mathbf{1})(x)}{ L_{[\omega]_k}(\mathbf{1})(x)}\\ & = \int f \,d\mu_{u\omega} \int L_u(\mathbf{1}) \,d\mu_{\omega} \end{align*} $$

and

$$ \begin{align*} \unicode{x3bb}_{uv,\omega}\mu_{uv\omega}=L_{uv}^{\ast}(\mu_{\omega})=L_u^{\ast} L_v^{\ast}(\mu_{\omega})=L_u^{\ast}(\unicode{x3bb}_{v,\omega}\mu_{v\omega})=\unicode{x3bb}_{v,\omega}\unicode{x3bb}_{u,v\omega}\mu_{uv\omega}. \end{align*} $$

Item (5) follows from

$$ \begin{align*}\int f \,d\mu_{u,\omega} = \lim_{k \to \infty} \frac{L_{[\omega]_k}(\mathbf{1})(x)}{L_{u[\omega]_k}(\mathbf{1})(x)} \cdot \frac{L_{[\omega]_k}(f L_u(\mathbf{1}))(x)}{L_{[\omega]_k}(\mathbf{1})(x)} = \frac{1}{\unicode{x3bb}_{u,\omega}} \int f L_u(\mathbf{1}) \,d\mu_{\omega}.\\[-43pt] \end{align*} $$

Remark 6.2. Recall that a probability measure $\nu $ is $(T_w,\varphi _w)$ -conformal, where w is a finite word, if there exists $c> 0$ such that $L_w^{\ast }(\nu )=c\nu $ . Consider $\overline {w}:= ww\ldots \in \Sigma $ and $\mu _{\overline {w}}=\mu _{\emptyset , \overline w}$ given by Proposition 6.1. By item (4) of the same proposition, $L_w^{\ast }(\mu _{\overline {w}})=\unicode{x3bb} _{w,\overline {w}} \mu _{\overline {w}}$ , hence $\mu _{\overline {w}}$ is conformal. Moreover, item (1) and $\mu _{u\overline {w}}\circ T_u^{-1} = \mu _{u,\overline {w}}$ imply

$$ \begin{align*} \{\mu_{u,\omega} :\omega \in \Sigma\} = \overline{\bigg\{ \mu_{u\overline{w}} \circ T_u^{-1} : w\in\bigcup_{k\geq 1} {\mathcal W}^k\bigg\}}. \end{align*} $$

As $\Sigma $ is compact and $\omega \mapsto \mu _{u,\omega }$ is Lipschitz continuous by statement (1) of Proposition 6.1, $\{\mu _{u,\omega } : \omega \in \Sigma \}$ is compact. It is also worth mentioning that item (1) ensures that for any finite word u, the family $\Sigma \ni \omega \mapsto \mu _{u,\omega }$ is Hölder continuous. Finally, the fact that any two asymptotic limits are equivalent (recall item (5)) will be useful to provide an application to characterize the boundary of a semigroup action in §9.

6.2 Two-sided compositions

We shall find a second family of probabilities which generalizes the notions of invariant measures and equilibrium states. To attain that goal, despite the fact that the underlying dynamics is not invertible, we need to consider forward iterations of maps determined by two-sided sequences. Let $\Sigma ^-$ refer to the set of left-infinite words, that is, $ \Sigma ^- = \{ \ldots i_2 i_1 : i_1, i_2,\ldots \in {\mathcal W} \},$ and for $k \in \mathbb {N}$ and $\sigma =\ldots i_2 i_1 \in \Sigma ^-$ , define

$$ \begin{align*} {}_k[\sigma] := i_k \ldots i_2 i_1\in{\mathcal W}^k. \end{align*} $$

Proposition 6.3. For any $\sigma \in \Sigma ^-$ , $\omega \in \Sigma $ and $\nu \in \mathcal {M}_1(X)$ , the limit

$$ \begin{align*} \mu_{\sigma,\omega}:= \lim_{k,l \to \infty} {\mathbb{P}_{_k[\sigma]}^{[\omega]_l}}^{\ast}(\nu) \end{align*} $$

exists and is independent of $\nu $ . Furthermore, with $k_0$ and s given by Theorem 5.1, the following statements hold.

1. (1) For $k,l$ with $k\wedge l\ge k_0$ and $\sigma ,\tilde {\sigma } \in \Sigma ^-, \omega ,\tilde {\omega } \in \Sigma $ with $_k[\sigma ]={_k[\tilde {\sigma }]}, [\omega ]_l=[\tilde {\omega }]_l$ , $\overline {W}(\mu _{\sigma , \omega }, \mu _{\tilde {\sigma },\tilde {\omega }}) \leq s^{k\wedge l}$ .

2. (2) For $k,l$ with $k\wedge l\ge k_0$ and $f\in \mathcal {H}_{\alpha }$ ,

   $$ \begin{align*} \bigg\| \mathbb{P}_{_k[\sigma]}^{[\omega]_l}(f) - \int f \,d\mu_{\sigma, \omega} \bigg\| \leq 2 s^{k\wedge l} \overline{D}(f). \end{align*} $$

3. (3) For a finite word u, $\mu _{\sigma u,\omega } = \mu _{\sigma , u\omega }\circ T_u^{-1}$ .

4. (4) The measures $\mu _{\sigma , \omega }$ and $\mu _{\omega }$ are absolutely continuous to each other and $h_{\sigma , \omega }:= d\mu _{\sigma , \omega }/d\mu _{\omega }$ satisfies

   $$ \begin{align*} \| h_{_k[\sigma],\omega} - h_{\sigma, \omega} \| \ll s^k, \end{align*} $$
   where $\mu _{\omega }$ and $h_{_k[\sigma ], \omega }$ are as given in the previous proposition.

Proof. As a consequence of Proposition 6.1(2), Lemmas 4.1 and 4.2, for any finite word u, infinite word $\omega \in \Sigma $ and $l \geq k_0$ , we have that

(6.1) $$ \begin{align} \| L_{u[\omega]_l}(\mathbf{1})/ L_{[\omega]_l}(\mathbf{1}) - \unicode{x3bb}_{u,\omega} \| \leq s^l \overline{D}(L_u(\mathbf{1})) \leq C s^l \unicode{x3bb}_{u,\omega}, \end{align} $$

for some $C>0$ . Hence, for finite words $v \in \mathcal {W}^k$ , $w \in \mathcal {W}^l$ , $k \geq k_0$ and f Hölder continuous,

$$ \begin{align*} & | \mathbb{P}_{v}^{w}(f) - \mathbb{P}_{uv}^{w}(f) | \\ &\quad \leq\bigg| \frac{L_{w}(f L_{v}(\mathbf{1}))}{L_{vw}(\mathbf{1}) } - \frac{L_{w}(f L_{uv}(\mathbf{1}))}{\unicode{x3bb}_{u,\overline{vw}} L_{vw}(\mathbf{1}) } \bigg| + \bigg| \frac{L_{w}(f L_{uv}(\mathbf{1}))}{\unicode{x3bb}_{u,\overline{vw}} L_{vw}(\mathbf{1}) } - \frac{L_{w}(f L_{uv}(\mathbf{1}))}{L_{uv w}(\mathbf{1}) } \bigg| \\ &\quad \leq \frac{L_{w}(|f| L_{v}(\mathbf{1}) | 1- {L_{uv}(\mathbf{1})}/{ \unicode{x3bb}_{u,\overline{vw}}L_{v}(\mathbf{1}) }| )}{L_{vw}(\mathbf{1}) } + \frac{L_{w}(|f| L_{uv}(\mathbf{1}))}{L_{uv w}(\mathbf{1}) } \bigg| \frac{L_{uv w}(\mathbf{1})}{ \unicode{x3bb}_{u,\overline{vw}}L_{vw}(\mathbf{1}) } -1 \bigg| \\ &\quad\leq C ( \mathbb{P}_{v}^{w}(|f|) s^k + \mathbb{P}_{uv}^{w}(|f|) s^{k+l} ), \end{align*} $$

where we used the notation $\overline {u}:=(u u \ldots )$ to denote the periodic word formed by u blocks. Now assume that $\nu $ and $\tilde {\nu }$ are probability measures and f is Hölder continuous with ${\overline {D}(f) \leq 1}$ and $\inf _{x \in X} f(x) =0$ . In particular, $\|f\|_{\infty } \leq 1$ . By the above and Proposition 6.1, for $\sigma ,\tilde {\sigma } \in \Sigma ^-$ and $\omega ,\tilde {\omega } \in \Sigma $ such that $_k[\sigma ]={_k[\tilde {\sigma }]}, [\omega ]_l=[\tilde {\omega }]_l$ and $k\wedge l\geq k_0$ ,

$$ \begin{align*} & \bigg| \int \mathbb{P}_{_k[\sigma]}^{[\omega]_{l}}(f) \,d\nu - \int \mathbb{P}_{_k[\tilde{\sigma}] }^{[\tilde{\omega}]_l}(f) \,d\tilde{\nu} \bigg| \\ &\quad\leq \int \bigg| \mathbb{P}_{_k[\sigma]}^{[\omega]_{l}}(f) - \mathbb{P}_{_k[\tilde{\sigma}]}^{ [\omega]_{l}}(f) \bigg| \,d\nu + \bigg| \int \mathbb{P}_{_k[\tilde{\sigma}]}^{[\omega]_{l}}(f) \,d\nu - \int \mathbb{P}_{_k[\tilde{\sigma}]}^{[\omega]_{l}}(f) \,d\tilde{\nu} \bigg| \\ &\quad\leq C ( 2 \|\mathbb{P}_{_k[\sigma]}^{[\omega]_{l}}(f)\|_{\infty} s^k + \|\mathbb{P}_{_k[\sigma]}^{[\omega]_{l}}(f)\|_{\infty} s^{k+l} + \|\mathbb{P}_{_k[\tilde{\sigma}]}^{[\omega]_{l}}(f)\|_{\infty} s^{k+l} ) + 2 s^{l} \\ &\quad\leq 2C(s^k + s^{k+l}) + 2s^l \ll s^{k\wedge l}. \end{align*} $$

Hence, by Kantorovich’s duality and completeness of the space of probability measures, $\lim _{k,l\to \infty } {\mathbb {P}_{_k[\sigma ]}^{[\omega ]_{l}}}^{\ast } (\nu )$ exists, is independent of $\nu $ and the estimate in part (1) holds. Part (2) is an immediate consequence of part (1), and the proof of part (3) follows as in Proposition 6.1. Proposition 6.1(5) indicates that $h_{\sigma ,\omega }$ is the limit of $h_{_k[\sigma ],\omega }$ and by the first argument in Proposition 2.2 in [Reference Bessa and Stadlbauer3], it follows that $\|h_{_k[\sigma ],\omega } - h_{_l[\sigma ],\omega }\|_{\infty } \ll s^{k\wedge l}$ . Then the argument in there can be easily adapted to obtain exponential convergence with respect to $\|\cdot \|_{d^{\ast }}$ in part (4).

Remark 6.4. The first part of the above proposition implies that the map $(\sigma , \omega ) \mapsto \mu _{\sigma , \omega }$ is Lipschitz continuous with respect to the metric

$$ \begin{align*} d((\sigma, \omega), (\tilde{\sigma},\tilde{\omega})) := \min\{ s^{k\wedge l}: {_k[\sigma]}={_k[\tilde{\sigma}]}, [\omega]_l=[\tilde{\omega}]_l\}. \end{align*} $$

In particular, the image of each compact subset of $\Sigma ^-\times \Sigma $ is a compact subset of the space of probability measures.

Moreover, by fixing an order on ${\mathcal W}$ , the associated adic flow $h_t$ on $\Sigma ^-\times \Sigma $ is uniquely ergodic (see [Reference Fisher, Bandt, Mosco and Zähle15]) and, in particular, for any Hölder continuous $f:X \to \mathbb R$ , the continuity of $(\sigma , \omega ) \to \int f \,d\mu _{\sigma , \omega }$ implies that

$$ \begin{align*} \frac1T \int_0^T\int f(x) \,d\mu_{h_t(\sigma, \omega)}(x) \,dt \xrightarrow{T \to \infty} \iint f(x) \,d\nu_{\sigma, \omega}(x) \,dm(\sigma, \omega) \end{align*} $$

uniformly, where m refers to the Parry measure (or measure of maximal entropy). The analogue of this statement holds for $\omega \to \int f \,d\mu _{\omega , \omega }$ and Birkhoff sums with respect to the odometer on $ \Sigma $ , or with respect to uniformly ergodic adic flows or adic transformations acting on compact subsets of $\Sigma ^-\times \Sigma $ or $\Sigma $ , respectively.

The result provides the following link to invariant measures and equilibrium states. A finite word w generates a periodic infinite word $\overline {w}:= (ww\ldots ) \in \Sigma $ and a periodic left-infinite word $\underline w:=(\ldots ww)\in \Sigma ^-$ . Then, by Proposition 6.3, the measure $\mu _{\underline {w},\overline {w}}$ is $T_w$ -invariant, $d \mu _{\underline {w},\overline {w}} = h_{\underline {w},\overline {w}}\,d\mu _{\overline {w}}$ and

$$ \begin{align*} L_{w}(h_{\underline{w},\overline{w}}) = \unicode{x3bb}_{w,\overline{w}} h_{\underline{w},\overline{w}}. \end{align*} $$

Here, $\unicode{x3bb} _{w, \overline w}$ is given as in Proposition 6.1.

The following result identifies $\mu _{\underline {w},\overline {w}}$ as the unique equilibrium state of $T_w$ with respect to the Hölder potential $\varphi _w$ . Note that the statement avoids the notion of pressure as X might be non-compact. However, if X is compact, then $\log \unicode{x3bb} _{w,\overline {w}}$ is equal to the pressure [Reference Ruelle28] and one obtains the usual notion of equilibrium state. In the proposition, $H_{\mu }(T_w)$ refers to Kolmogorov’s entropy.

Proposition 6.5.

$$ \begin{align*} \log \unicode{x3bb}_{w,\overline{w}\,} & {=} H_{\mu_{\underline{w},\overline{w}}}(T_w) + \int \varphi_w \,d\mu_{\underline{w}, \overline{w}} \\ \nonumber & = \sup \bigg\{ H_{\nu}(T_w) + \int \varphi_w \,d\nu: \nu \in \mathcal{M}_1(X), \nu = \nu \circ T_w^{-1} \bigg\}. \end{align*} $$

Furthermore, $\mu _{\underline {w},\overline {w}}$ is the unique measure which realizes the supremum.

Proof. As $T_w$ is Ruelle expanding, the restriction $T_w|_U$ to a ball U of radius a is bimeasurable. Hence, $A \mapsto \mu _{\underline {w},\overline {w}} \circ T_w(A)$ defines a measure on U which is, as a consequence of Propositions 6.1 and 6.3, absolutely continuous with respect to $\mu _{\underline {w},\overline {w}}|_U$ . Hence, $J_{\mu _{\underline {w},\overline {w}}} := {d \mu _{\underline {w},\overline {w}}\circ T_w}/{d \mu _{\underline {w},\overline {w}}}$ is a well-defined function on X, sometimes referred to as the Jacobian of $T_w$ with respect to $\mu _{\underline {w},\overline {w}}$ . In fact, it follows from the construction of $\mu _{\underline {w},\overline {w}}$ that $J_{\mu _{\underline {w},\overline {w}}} = \exp (-\tilde {\varphi }_w)$ , where

$$ \begin{align*}\tilde{\varphi}_w := \varphi_w + \log h_{\underline{w},\overline{w}} - \log h_{\underline{w},\overline{w}}\circ T_w - \log \unicode{x3bb}_{w,\overline{w}}. \end{align*} $$

By construction, $J_{\mu _{\underline {w},\overline {w}}} = \exp (-\tilde {\varphi }_w)$ and, as $T_w$ is Ruelle expanding, Rokhlin’s formula for entropy (see, e.g. Theorem 9.7.3 in [Reference Viana and Oliveira35]) implies that

$$ \begin{align*} H_{\mu_{\underline{w},\overline{w}}}(T_w) & = \int \log J_{\mu_{\underline{w},\overline{w}}} \,d\mu_{\underline{w},\overline{w}} \\ & = \log \unicode{x3bb}_{w,\overline{w}} - \int (\varphi_w + \log h_{\underline{w}, \overline{w}} - \log h_{\underline{w}, \overline{w}} \circ T_w) \,d\mu_{\underline{w}, \overline{w}} \\ & = \log \unicode{x3bb}_{w,\overline{w}} - \int \varphi_w \,d\mu_{\underline{w}, \overline{w}}. \end{align*} $$

This proves the first identity. Now suppose that $\nu $ is an invariant probability measure with $ H_{\nu }(T_w) + \int \varphi _w \,d\nu \geq \log \unicode{x3bb} _{w,\overline {w}}$ . Then, by Rokhlin’s formula, the invariance of $\nu $ and the definition of the transfer operator of $T_w$ with respect to $\nu $ , denoting by $J_{\nu }=d\nu \circ T_w/d\nu $ ,

$$ \begin{align*} 0 & \leq H_{\nu}(T_w) + \int \varphi_w \,d\nu - \log \unicode{x3bb}_{w,\overline{w}} \\ & = \int (\log J_{\nu} + \varphi_w + \log h_{\underline{w},\overline{w}} - \log h_{\underline{w},\overline{w}}\circ T_w - \log \unicode{x3bb}_{w,\overline{w}}) \,d\nu \\ & = \int \log \frac{J_{\nu}}{J_{\mu_{\underline{w},\overline{w}}} } \,d\nu = \int \sum_{T_w(y)=x} \frac1{J_{\nu}(y)} \log \frac{J_{\nu}(y)}{J_{\mu_{\underline{w},\overline{w}}} (y) } \,d\nu(x). \end{align*} $$

As $\nu $ is invariant, it follows that $\sum _{T_w(y)=x} 1/{J_{\nu }(y)} =1$ for all $x \in X$ . Hence, by Jensens’s inequality,

$$ \begin{align*} 0 & \leq H_{\nu}(T_w) + \int \varphi_w \,d\nu - \log \unicode{x3bb}_{w,\overline{w}} \stackrel{\ast}{\leq} \int \log \sum_{T_w(y)=x} \frac1{J_{\nu}(y)} \frac{J_{\nu}(y)}{J_{\mu_{\underline{w},\overline{w}}} (y) }\,d\nu(x) = 0. \end{align*} $$

Moreover, equality holds in $(\ast )$ if and only if ${J_{\nu }(y)}/{J_{\mu _{\underline {w},\overline {w}}}(y) }=1$ almost surely.

Remark 6.6. By usual normalization procedure, replacing the potential $\varphi _w$ with $\tilde {\varphi }_w $ , one then obtains a new operator $\tilde {L}_w$ with $\tilde {L}_w(\mathbf {1}) = \mathbf {1}$ , that is, $\tilde {L}_w$ is normalized and $\tilde {L}_w^{\ast }(\mu _{\underline w, \overline w})=\mu _{\underline w, \overline w}$ . In particular, part (2) of Proposition 6.1 applied to the semigroup generated by $T_w$ implies that $\tilde {L}_w$ has a spectral gap. However, the construction depends on the specific periodic word $\overline {w}$ and is in general not functorial, that is, $\tilde {L}_{vw} \neq \tilde {L}_{w}\circ \tilde {L}_{v}$ .

7 Annealed exponential decay

So far, we have considered only quenched operators, which are determined by iterations in $\mathcal S$ tracked by certain finite words and their limiting behaviour. As stated in the introduction, another objective is to study annealed operators, which are averages of all the quenched operators tracked by finite words of given lengths. To be more precise, suppose that the one-sided full shift of finite alphabet $(\Sigma , \theta )$ is endowed with a non-singular probability measure $\rho $ . For every $k\in \mathbb N$ , define the averaged transfer operator

$$ \begin{align*} {\mathcal A}_k(f)(x) := \int_{\Sigma} L_{[\omega]_k}(f)(x) \,d\rho(\omega) \end{align*} $$

for $f\in {\mathcal H}_{\alpha }$ . One can do so for more general shifts, but we keep $\Sigma $ to be a topological mixing subshift of finite type for simplicity. Naturally, one would need some properties of the shift space $(\Sigma , \theta , \rho )$ to study the operator $\mathcal A_k$ . We summarize them below.

Since $\rho $ is non-singular, for a finite word $ u$ , let $p_u:\Sigma \to \mathbb R_+$ be defined by

$$ \begin{align*} p_u(\omega) := \frac{d\rho}{d\rho\circ\theta^{|u|}}(u\omega), \quad \omega\in\Sigma. \end{align*} $$

With the usual distance given on the shift, denote by ${{\mathcal H}}(\Sigma )$ the space of Hölder continuous functions on $\Sigma $ and by ${\mathcal C}(\Sigma )$ the space of continuous functions on $\Sigma $ . Recall that $\unicode{x3bb} _{u,\omega }=\int L_u(\mathbf 1)\,d\mu _{\omega }$ , as in Proposition 6.1. Note that $\log \unicode{x3bb} _{i,\cdot }\in {\mathcal H}(\Sigma )$ by Proposition 6.1. Suppose that $\log p_i\in {\mathcal H}(\Sigma )$ as well. Define a linear operator $\iota $ acting on ${\mathcal C}(\Sigma )$ by

$$ \begin{align*} \iota(g)(\omega) := \sum_{i\in {\mathcal W}} \unicode{x3bb}_{i,\omega} p_i(\omega) g(i\omega),\quad g\in {\mathcal C}(\Sigma). \end{align*} $$

As $u \mapsto p_u$ and $u \mapsto \unicode{x3bb} _{u,\omega }$ are multiplicative cocycles with respect to $\theta $ , it can be shown that for every $k\in \mathbb N$ ,

$$ \begin{align*} \iota^k(g)(\omega) =\sum_{u\in{\mathcal W}^k}\unicode{x3bb}_{u,\omega}p_u(\omega) g(u\omega).\end{align*} $$

In view of the duality with $\theta $ , we have that for any $g_1, g_2\in {\mathcal C}(\Sigma )$ ,

(7.1) $$ \begin{align} \int \iota^k(g_1)\cdot g_2 \,d\rho= \int \unicode{x3bb}_{[\omega]_k, \theta^k\omega}\cdot g_1\cdot g_2\circ \theta^k \,d\rho. \end{align} $$

Since $\log \unicode{x3bb} _{i,\omega }$ and $\log p_i$ are both Hölder continuous, Ruelle’s Perron–Frobenius theorem implies that there are $\beta>0, m\in \mathcal M_1(\Sigma )$ and $g_o\in {\mathcal C}(\Sigma ), g_o>0$ such that

(7.2) $$ \begin{align} \iota^*m=\beta m,\quad \iota(g_o)=\beta g_o,\quad m(g_o)=1. \end{align} $$

Furthermore, there exists $t \in (0,1)$ such that for any $g\in {\mathcal H}(\Sigma )$ and $k\in \mathbb N$ ,

(7.3) $$ \begin{align} \bigg\|\, \beta^{-k}\iota^k(g) - g_o\int g \,dm \bigg\|_{\scriptscriptstyle \Sigma} \ll t^k \|g\|_{\scriptscriptstyle \Sigma}, \end{align} $$

where $\|\cdot \|_{\scriptscriptstyle \Sigma }=D_{\scriptscriptstyle \Sigma }(\cdot )+\|\cdot \|_{\infty }$ , the sum of the Hölder norm and the supremum norm over the shift. Note that $g_o$ is uniformly bounded from above and away from $0$ as $\Sigma $ is compact.

Remark 7.2. The associated skew product

$$ \begin{align*}F: X \times \Sigma \to X \times \Sigma, \quad (x, i_1 i_2 \ldots) \mapsto (T_{i_1}(x), i_2i_3 \ldots)\end{align*} $$

reflects the time evolution along a given path in $\Sigma $ with a distribution on the space of possible paths, that is, the probability of the event of applying $T \in \mathcal {S}$ in time n is given by $\rho (\{ \omega \in \Sigma : F^n(\cdot \,,\omega ) =(T(\,\cdot \,), \theta ^n(\omega )) \})$ .

We proceed to prove that the family $\{\mathcal A_n\}$ has exponential decay of correlations. Fix $k_0\in \mathbb N$ and $s\in (0,1)$ , as given in Theorem 5.1. With m defined as in equation (7.2), let $\pi \in \mathcal M_1(X)$ be given by

$$ \begin{align*} d\pi : = d\mu_{\omega} dm(\omega). \end{align*} $$

For $f\in \mathcal H_{\alpha }$ , let

$$ \begin{align*} \|f\|_m:=\|\mu_{\cdot}(|f|)\|_{\infty} \end{align*} $$

be the supremum norm with respect to m of the map $\omega \mapsto \mu _{\omega }(|f|)$ over the shift.

Theorem 7.3. Suppose the Ruelle expanding semigroup $\mathcal {S}$ is jointly topologically mixing and finitely aperiodic, and that every potential $\varphi _i$ is $\alpha $ -Hölder and summable. Suppose that every $\log p_i$ , $i\in {\mathcal W}$ is Hölder continuous on $\Sigma $ . Then there exists $r\in (0,1)$ such that for all $f \in \mathcal {H}_{\alpha }$ and $n \geq 2k_0$ ,

$$ \begin{align*} \bigg| \frac{\mathcal{A}_n(f)(x)}{\mathcal{A}_n(\mathbf{1})(x)} - \int f \,d\pi \bigg| \ll r^n (\overline D(f) +\|f\|_m).\end{align*} $$

Moreover, there exists a positive function $h\in {\mathcal H}_{\alpha }$ such that for all $f\in {\mathcal H}_{\alpha }$ and $n\ge 2k_0$ ,

$$ \begin{align*}\bigg|\frac{\mathcal A_n(f)(x)}{\beta^{n} h(x)}-\int f \,d\pi\bigg|\ll r^n(\overline D(f)+\|f\|_m),\end{align*} $$

with $\beta>0$ given by equation (7.2).

Proof. In the first step of the proof, we derive the first decay. Proposition 6.1 implies that for any $n\geq 2k_0, \omega \in \Sigma $ and $x\in X, f \in \mathcal {H}_{\alpha }$ ,

$$ \begin{align*} | L_{[\omega]_n}(f)(x) - \mu_{\omega}(f) L_{[\omega]_n}(\mathbf{1})(x) | \ll s^n \overline{D}(f) L_{[\omega]_n}(\mathbf{1})(x). \end{align*} $$

After integration, it yields that

(7.4) $$ \begin{align} \bigg| \mathcal{A}_n(f)(x) - \int \mu_{\omega}(f) L_{[\omega]_n}(\mathbf{1})(x) \,d\rho(\omega) \bigg| \ll s^n \overline{D}(f) \mathcal{A}_n(\mathbf{1})(x). \end{align} $$

It remains to analyse $\int \mu _{\omega }(f) L_{[\omega ]_n}(\mathbf {1}) \,d\rho (\omega )$ as $n \to \infty $ . To do so, write $n=k+l$ with $l=[n/2]+1$ . Observe that by equation (6.1),

(7.5) $$ \begin{align} |L_{[\omega]_n}(\mathbf{1}) - \unicode{x3bb}_{[\omega]_k, \theta^k\omega} L_{[\theta^k\omega]_l}(\mathbf{1}) | \ll s^l \unicode{x3bb}_{[\omega]_k, \theta^k\omega} L_{[\theta^k\omega]_l}(\mathbf{1}). \end{align} $$

Note that it follows from Proposition 6.1 that $\omega \mapsto \mu _{\omega }(f)$ is Hölder continuous on $\Sigma $ and its Hölder coefficient is bounded by a constant times $\overline {D}(f)$ . Hence,

$$ \begin{align*} & \bigg| \int \mu_{\omega}(f) L_{[\omega]_n}(\mathbf{1}) \,d\rho(\omega) - \int \mu_{\omega}(f) \unicode{x3bb}_{[\omega]_k,\theta^k \omega} L_{[\theta^k\omega]_l}(\mathbf{1}) \,d\rho(\omega)\bigg| \\ &\qquad\kern8pt\ll\ s^l \int \mu_{\omega}(|f|) \unicode{x3bb}_{[\omega]_k, \theta^k \omega} L_{[\theta^k \omega]_l}(\mathbf{1}) \,d\rho(\omega)\\ &\quad\overset{\mathrm{equation}~(7.1)}{=}s^l\int \iota^k (\mu_{\omega}(|f|))\cdot L_{[\omega]_l}(\mathbf 1) \,d\rho(\omega)\\ &\qquad\kern8pt=\ s^l\int (\beta^{-k}g_o^{-1}\iota^k (\mu_{\omega}(|f|))-\pi(|f|)+\pi(|f|))\cdot \iota^k(g_o) L_{[\omega]_l}(\mathbf 1) \,d\rho(\omega)\\ &\quad\overset{\mathrm{equation}~(7.3)}{\ll} s^l(t^k(\overline D(f) + \|f\|_m)+\pi(|f|))\int \iota^k(g_o) L_{[\omega]_l}(\mathbf{1}) \,d\rho(\omega)\\ &\quad\overset{\mathrm{equation}~(7.1)}=s^l(t^k(\overline D(f) + \|f\|_m)+\pi(|f|)) \int g_o \cdot \unicode{x3bb}_{[\omega]_k, \theta^k\omega} L_{[\theta^k\omega]_l}(\mathbf{1}) \,d\rho(\omega) \\ &\quad\overset{\mathrm{equation}~(7.5)} \ll s^l (t^k(\overline D(f) + \|f\|_m)+\pi(|f|))\int L_{[\omega]_n}(\mathbf{1}) \cdot g_o \,d\rho(\omega) \\ &\qquad\kern7pt\ll\ s^l (t^k\overline D(f)+\|f\|_m) \mathcal{A}_{n}(\mathbf{1}). \end{align*} $$

Observe that in the previous estimate, we have also shown that

(7.6) $$ \begin{align} \int \iota^k(g_o)L_{[\omega]_l}(\mathbf 1)\,d\rho(\omega)\ll \mathcal A_n(\mathbf 1). \end{align} $$

Then one can extract $\pi (f)$ by

$$ \begin{align*} & \bigg| \int \mu_{\omega}(f) \unicode{x3bb}_{[\omega]_k, \theta^k\omega} L_{[\theta^k\omega]_l}(\mathbf{1}) \,d\rho(\omega)- \pi(f) \int \unicode{x3bb}_{[\omega]_k, \theta^k\omega} L_{[\theta^k\omega]_l}(\mathbf{1}) \,d\rho(\omega) \bigg| \\ &\quad\overset{\mathrm{equation}~(7.1)}= \bigg| \int \iota^k(\mu_{\omega} (f)) L_{[\omega]_l}(\mathbf{1}) \,d\rho(\omega) - \pi(f) \int \iota^k(1) L_{[\omega]_l}(\mathbf{1}) \,d\rho(\omega)\bigg|\\ &\qquad\kern9.5pt = \bigg|\int ((\beta^{-k}g_o^{-1}\iota^k(\mu_{\omega}(f))-\pi(f))-(\beta^{-k}g_o^{-1}\iota^k(1)-1)\pi(f))\iota^k(g_o) L_{[\omega]_l}(\mathbf 1)\,d\rho(\omega)\bigg|\\ &\quad\overset{\mathrm{equation}~(7.3)}\ll t^k(\overline D(f) + \|f\|_m) \int \iota^k(g_o) L_{[\omega]_l}(\mathbf{1}) \,d\rho(\omega) \ll t^k(\overline D(f) + \|f\|_m) \mathcal{A}_{n}(\mathbf{1}). \end{align*} $$

Finally, equation (7.5) induces that

$$ \begin{align*} \bigg| \pi(f)\int \unicode{x3bb}_{[\omega]_k, \theta^k\omega} L_{[\theta^k\omega]_l}(\mathbf{1}) \,d\rho(\omega) - \pi(f) \mathcal{A}_{n}(\mathbf{1}) \bigg| \ll s^l |\pi(f)| \mathcal{A}_{n}(\mathbf{1}). \end{align*} $$

Combining the above estimates, one obtains that

$$ \begin{align*}\bigg|\int \mu_{\omega}(f)L_{[\omega]_n}(\mathbf 1)\,d\rho(\omega)-\pi(f)\mathcal A_n(\mathbf 1)\bigg|\ll (t^k\overline D(f)+t^k\|f\|_m+s^l\|f\|_m)\mathcal A_n(\mathbf 1). \end{align*} $$

The first statement now follows from equation (7.4) with $r=\max \{\sqrt s, \sqrt [3]t\}$ .

We now proceed with proving the existence of h. To do so, let

$$ \begin{align*}\tilde {\mathcal A}_n(x):=\int L_{[\omega]_n}(\mathbf 1)(x)\cdot g_o({\omega}) \,d\rho(\omega).\end{align*} $$

We first show that $\tilde I_n(x):=\beta ^{-n}\tilde {\mathcal A}_n(x)$ converges uniformly and exponentially fast to a positive function $h(x)\in {\mathcal H}_{\alpha }$ .

It follows from equation (7.5) that for any $n=k+l$ with $l\ge k_0$ ,

$$ \begin{align*}L_{[\omega]_n}(\mathbf 1)\asymp \unicode{x3bb}_{[\omega]_k, \theta^k\omega} L_{[\theta^k\omega]_{l}}(\mathbf{1}),\end{align*} $$

so that

$$ \begin{align*} \tilde {\mathcal A}_n\asymp \int \unicode{x3bb}_{[\omega]_k, \theta^k\omega} L_{[\theta^k\omega]_{l}}(\mathbf{1})\cdot g_o \,d\rho \overset{\mathrm{equation}~(7.1)}=\int \iota^k(g_o) L_{[\omega]_{l}}(\mathbf 1) \,d\rho=\beta^k \tilde {\mathcal A}_l,\end{align*} $$

and hence, $\tilde I_n\asymp \tilde I_l$ , especially $\tilde I_n\asymp \tilde I_{k_0}$ for all $n\ge k_0$ . Since equation (7.5) also implies that

$$ \begin{align*} |\tilde {\mathcal A}_n- \beta^k \tilde {\mathcal A}_l|\ll s^{l} \beta^k\tilde {\mathcal A}_l, \end{align*} $$

one has

$$ \begin{align*} |\tilde I_n-\tilde I_l|\ll s^l \tilde I_l. \end{align*} $$

Hence, $\{\tilde I_n(\cdot )\}$ is a Cauchy sequence. Denote the limit of $\tilde I_n(x)$ by $h(x)$ . Then $\tilde I_n(x)$ converges uniformly to $h(x)$ since for $n\ge l\ge k_0$ ,

$$ \begin{align*} |\tilde I_n-\tilde I_l|\ll s^l\tilde I_{k_0}\ll s^l. \end{align*} $$

Then because $\tilde I_n$ are all Hölder, h is Hölder as well. That h is positive and $\|h\|_{\infty }$ is finite can be seen from $h\asymp \tilde I_{k_0}$ . To see that the rate of convergence is exponential, for $n\geq k_0$ , choose $j\in \mathbb N$ such that $|\tilde I_{jn}- h|\le s^n,$ then

$$ \begin{align*} |\tilde I_n -h|\le |\tilde I_n-\tilde I_{2n}|+\cdots +|\tilde I_{(j-1)n}- \tilde I_{jn}|+|\tilde I_{jn}-h|\ll s^n. \end{align*} $$

Moreover, Lemma 4.2 infers that $\inf _{x\in X} \tilde I_{k_0}(x)>0$ , and so are $\tilde I_n$ for $n\ge k_0$ and so is h. It follows that $\tilde I_n/h$ converges to $1$ uniformly and exponentially fast.

Next we show that $I_n(x):=\beta ^{-n}{\mathcal A}_n(\mathbf 1)(x)$ also tends to $h(x)$ . For $n=k+l$ with $l\ge k_0$ , because

$$ \begin{align*} \bigg|{\mathcal A}_n(\mathbf 1)-\int \iota^k(1)L_{[\omega]_l}(\mathbf 1) \,d\rho\bigg|\ll s^l\int \iota^k(1)L_{[\omega]_l}(\mathbf 1) \,d\rho \end{align*} $$

obtained from integrating equation (7.5) and because

$$ \begin{align*} \bigg|\int (\iota^k(1)-\iota^k(g_o))L_{[\omega]_l}(\mathbf 1)\,d\rho\bigg|&=\bigg|\int (\beta^{-k}g_o^{-1}\iota^k(1)-1)\iota^k(g_o)L_{[\omega]_l} \,d\rho\bigg|\\ &\hspace{-19pt}\overset{\mathrm{equation}~(7.3)}\le t^k\int \iota^k(g_o)L_{[\omega]_l} \,d\rho = t^k\beta^k \tilde A_l, \end{align*} $$

one can deduce that

$$ \begin{align*} |{\mathcal A}_n(\mathbf 1)-\beta^k\tilde {\mathcal A}_l|\ll (s^l+t^k)\beta^k\tilde A_l, \end{align*} $$

and hence

$$ \begin{align*} |{I_n}- \tilde I_l|\ll (s^l+t^k)\tilde I_l, \end{align*} $$

so that

$$ \begin{align*} |I_n-h|\ll (s^l+t^k) h. \end{align*} $$

Lastly, applying Theorem 7.3, one has that for all $f\in {\mathcal H}_{\alpha }$ and $n\ge 2k_0$ ,

$$ \begin{align*} |\beta^{-n}{\mathcal A}_n(f)-\pi(f)h|&\le \beta^{-n}|{\mathcal A}_n(f)-\pi(f){\mathcal A}_n(\mathbf 1)|+ \pi(f)|\beta^{-n}{\mathcal A}_n(\mathbf 1)- h|\\ &\ll r^n(\overline D(f) +\|f\|_m) I_n+ \pi(f)|I_n-h|\\ &\ll r^n(\overline D(f)+ \|f\|_m) h. \end{align*} $$

The second assertion on the decay follows from this.

The next result reveals an annealed version of the decay of correlations.

Theorem 7.4. Now suppose that the assumptions of the above theorem hold and that, in addition, $\rho $ is $\theta $ -invariant. Then there exist a probability measure $\tilde {\pi }$ on $\Sigma \times X$ , $r \in (0,1)$ and $k_1 \in \mathbb {N}$ such that

$$ \begin{align*} &\bigg| \int \sum_{v \in {\mathcal W}^n} \mathbf{1}_{[v]}(\omega) f (T_v(x)) g(x) \,d\mu_{\omega}(x) \,d\rho(\omega) - \int f \,d\tilde{\pi} \int g \,d\mu_{\omega} \,d\rho \bigg| \\ &\quad\leq r^n \int |f| \,d\mu_{\omega} \,d\rho \bigg( \overline D(g) + \int |g| \,d\mu_{\omega} \,d\rho \bigg) \end{align*} $$

for all $n\ge k_1, g \in {\mathcal H}_{\alpha }$ and $f: X \to \mathbb {R}$ integrable with respect to $d\mu _{\omega }(x) \,d\rho (\omega )$ .

Proof. For $\omega = (\omega _1 \omega _2 \ldots ) \in \Sigma $ , set $\unicode{x3bb} _{n,\omega } := \unicode{x3bb} _{\omega _1 \ldots \omega _n, \theta ^n\omega }$ and $h_{n,\omega } := h_{\omega _1 \ldots \omega _n, \theta ^n\omega }$ , where $\unicode{x3bb} _{\cdot }$ and $h_{\cdot }$ are given by Proposition 6.1. Moreover, Proposition 6.1 and Lemma 4.2 imply for n sufficiently large that

(7.7) $$ \begin{align} & \int \sum_{v \in {\mathcal W}^n} \mathbf{1}_{[v]} f \circ T_v g \,d\mu_{\omega} \,d\rho = \int \sum_{v \in {\mathcal W}^n} \mathbf{1}_{[v]} f \frac{L_v(g)}{\unicode{x3bb}_{n,\omega}} \,d\mu_{\theta^n\omega} \,d\rho \nonumber\\ &\quad= \int \sum_{v} \mathbf{1}_{[v]} f \mu_{\omega}(g) \frac{L_v(\mathbf{1})}{\unicode{x3bb}_{n,\omega}} \,d\mu_{\theta^n\omega} \,d\rho \pm 2 s^n \overline D(g) \int \sum_{v} \mathbf{1}_{[v]} |f| \frac{L_v(\mathbf{1})}{\unicode{x3bb}_{n,\omega}} \,d\mu_{\theta^n\omega} \,d\rho\nonumber\\ &\quad= \int \sum_{v \in {\mathcal W}^n} \mathbf{1}_{[v]} f \mu_{\omega}(g) h_{n,\omega} \,d\mu_{\theta^n\omega} \,d\rho \pm C s^n \overline D(g) \int \sum_{v \in {\mathcal W}^n} \mathbf{1}_{[v]} |f| \,d\mu_{\theta^n\omega} \,d\rho \nonumber\\ &\quad= \int f \mu_{\omega}(g) h_{n,\omega} \,d\mu_{\theta^n\omega} \,d\rho \pm C s^n \overline D(g) \int |f| \,d\mu_{\omega} \,d\rho, \end{align} $$

where $C/2$ is given by Lemma 4.2, and the last equality follows from $\theta $ -invariance of $\rho $ . Now assume that n is even and $n = 2m$ . Then, by item (4) of Proposition 6.3, there exists C such that

$$ \begin{align*} & \int f \mu_{\omega}(g) h_{n,\omega} \,d\mu_{\theta^n\omega} \,d\rho \\ &\quad = \int f \mu_{\omega}(g) h_{m,\theta^m\omega} \,d\mu_{\theta^n\omega} \,d\rho \pm C s^m \int \mu_{\theta^n\omega}{(|f|)} |\mu_{\omega}(g)| \,d\rho. \end{align*} $$

However, as $\omega \to \mu _{\omega }(g)$ is Lipschitz continuous by Proposition 6.1, the exponential decay of correlations, say with rate $t \in (0,1)$ and the same constant $C>0$ , applied to the error term implies that

(7.8) $$ \begin{align} \nonumber & \int f \mu_{\omega}(g) h_{m,\theta^m\omega} \,d\mu_{\theta^n\omega} \,d\rho \pm C s^m \int \mu_{\theta^n\omega}{(|f|)} |\mu_{\omega}(g)| \,d\rho \\ &\quad= \int f \mu_{\omega}(g) h_{m,\theta^m\omega} \,d\mu_{\theta^n\omega} \,d\rho \pm C^2 s^m \int \mu_{\omega}(|f|) \,d\rho \int \mu_{\omega}(|g|) \,d\rho. \end{align} $$

A further application of invariance and the exponential decay of correlations of $\theta $ to the main term and Lemma 4.2 gives that

(7.9) $$ \begin{align} \nonumber & \int f \mu_{\omega}(g) h_{m,\omega} \,d\mu_{\theta^n\omega} \,d\rho = \int \mu_{\omega}(g) \mu_{\theta^{2m}\omega} ( f \,h_{m,\theta^m\omega} ) \,d\rho \\ &\quad= \int \mu_{\omega}(g)\,d\rho \int f h_{m,\omega} \,d\mu_{\theta^{m}\omega} \,d\rho \pm C^2 t^m \int \mu_{\omega}(|f|) \,d\rho \overline D(g) \end{align} $$

Hence, it remains to analyse $\int \! f h_{m,\omega } \,d\mu _{\theta ^{m}\omega }$ . To do so, let $(\hat \Sigma ,\hat \theta ,\hat \rho )$ refer to natural extension of $\theta $ . Then, again by item (4) of Proposition 6.3, it follows that

(7.10) $$ \begin{align} \nonumber & \int f h_{m,\omega} \,d\mu_{\theta^{m}\omega} \,d\rho(\omega) = \int f h_{m,\omega} \,d\mu_{{\theta}^{m}\omega} \,d\hat\rho (\tilde{\omega},\omega)\\ \nonumber &\quad= \int f h_{\tilde{\omega}_{-m}\cdots \tilde{\omega}_{-1},\omega} \,d\mu_{\omega} d\hat\rho (\tilde{\omega},\omega) = \int f h_{\tilde{\omega},\omega} \,d\mu_{\omega} \,d\hat\rho (\tilde{\omega},\omega) \pm Cs^m \int \mu_{\omega}(|f|) \,d\rho \\ &\quad= \int f \,d\mu_{\tilde{\omega},\omega} \,d\hat\rho (\tilde{\omega},\omega) \pm Cs^m \int \mu_{\omega}(|f|) \,d\rho. \end{align} $$

Let $d\tilde {\pi }(x) := d\mu _{\tilde {\omega },\omega }(x) d\hat \rho (\tilde {\omega },\omega )$ . The theorem now follows by combining equations (7.7), (7.8), (7.9) and (7.10).

8 An almost sure invariance principle

Exponential decay of correlations has many implications on the statistical behaviour of the dynamical system. A large deviation principle, a relativized central limit theorem and laws of iterated logarithm for random dynamical systems generated by expanding dynamics follow from the works by Kifer [Reference Kifer21, Reference Kifer22]. For sequential dynamical systems of expanding maps of the interval, first versions of central limit theorems were obtained by Heinrich [Reference Heinrich19] and Conze and Raugi [Reference Conze and Raugi9]. We now show an almost sure invariance principle in the setting of Ruelle expanding maps. It is worth mentioning that almost sure invariance principles have been obtained in the context of quenched random dynamical systems (see e.g. [Reference Dragičević, Froyland, González-Tokman and Vaienti13] and references therein). Let $\mathcal B$ be the Borel $\sigma $ -algebra on X. With respect to the measure $\mu _{uv\omega }$ , where $u,v $ are finite words and $\omega $ is an infinite word, $\mathbb P_u^v$ can be seen as a conditional expectation in the following way.

Lemma 8.1. For any $f\in {\mathcal H}_{\alpha }$ ,

$$ \begin{align*}\mathbb E_{\mu_{uv\omega}}(f\circ T_u|T^{-1}_{uv}\mathcal B)=\mathbb P_u^v(f)\circ T_{uv}.\end{align*} $$

Proof. For any $A\in \mathcal B$ , using item (3) of Proposition 6.1,

$$ \begin{align*} \int_{T^{-1}_{uv}A} f\circ T_u \,d\mu_{uv\omega}&=\int \mathbf 1_A\circ T_v\cdot f \,d\mu_{uv\omega}\circ T^{-1}_u=\int \mathbf 1_A\circ T_v\cdot f \,d\mu_{u, v\omega}\\ &=\int \mathbf 1_A\circ T_v\cdot f \,d{\mathbb P_u^v}^{\ast}(\mu_{uv,\omega})=\int \mathbb P_u^v(\mathbf 1_A\circ T_v\cdot f) \,d\mu_{uv,\omega}\\[12pt] &=\int \mathbf 1_A\cdot \mathbb P_u^v(f)\,d\mu_{uv, \omega}=\int_A \mathbb P_u^v(f) \,d\mu_{uv\omega}\circ T^{-1}_{uv}\\ &=\int_{T_{uv}^{-1}A}\mathbb P_u^v(f)\circ T_{uv}\,d\mu_{uv\omega}.\\[-3.7pc] \end{align*} $$

The almost sure invariance principle we are going to show is similar to the one in [Reference Stadlbauer and Zhang32] for non-stationary shift. Both are based on the almost sure invariance principle for reverse martingale differences by Cuny and Merlevède.

Theorem 8.2. [Reference Cuny and Merlevède10, Theorem 2.3]

Let $(U_n)_{n\in \mathbb N}$ be a sequence of square integrable reverse martingale differences with respect to a non-increasing filtration $(\mathcal G_n)_{n\in \mathbb N}$ . Assume that $\sigma _n^2:=\sum _{k=1}^n\mathbb E(U_k^2)\to \infty $ and that $\sup _n\mathbb E(U_n^2)<\infty $ . Assume that

$$ \begin{align*} &\sum_{k=1}^n(\mathbb E(U_k^2|\mathcal G_{k+1})-\mathbb E(U^2_k))=o(\sigma_n^2) \quad \mbox{almost surely},\\ &\quad\ \sum_{n\geq 1}\sigma_n^{-2t}\mathbb E(|U_n|^{2t})<\infty \quad \text{for some } 1\leq t\leq 2. \end{align*} $$

Then, enlarging our probability space if necessary, it is possible to find a sequence $(Z_k)_{k\geq 1}$ of independent centred Gaussian variables with $\mathbb E(Z_k^2)=\mathbb E(U_k^2)$ such that

$$ \begin{align*}\sup_{1\leq k\leq n}\bigg|\sum_{i=1}^k U_i-\sum_{i=1}^k Z_i\bigg|=o(\sqrt{\sigma_n^2\log\log \sigma_n^2})\quad \mbox{almost surely}.\end{align*} $$

We need to make another assumption.

Definition 8.1. An $(a,\unicode{x3bb} )$ -Ruelle expanding map T is finitely expanding if

$$ \begin{align*}\sup_{\substack{x,y\in X \\ 0<d(x,y)<a}}\frac{d(T(x), T(y))}{d(x,y)}<\infty.\end{align*} $$

We refer to $\mathcal S$ as finitely Ruelle expanding if every $T_i, i\in {\mathcal W}$ satisfies this property.

Theorem 8.3. Suppose the finitely Ruelle expanding semigroup $\mathcal {S}$ is jointly topologically mixing and finitely aperiodic, and that every potential $\varphi _i$ is $\alpha $ -Hölder and summable. Suppose $\omega \in \Sigma $ , $f\in {\mathcal H}_{\alpha }$ . Let $f_n=f -\int f\circ T_{[\omega ]_n} \,d\mu _{\omega }$ for every $n\in \mathbb N_0$ and let $s_n^2 = \mathbb E_{\mu _{\omega }}(\sum _{k=0}^{n-1} f_k\circ T_{[\omega ]_k})^2$ for $n\ge 1$ . Assume that

(8.1) $$ \begin{align} \quad \sum_n s_n^{-4}<\infty. \end{align} $$

Then, enlarging our probability space if necessary, there exists a sequence $(Z_n)$ of independent centred Gaussian random variables such that

$$ \begin{align*} &\qquad\qquad \ \sup_n\bigg|\sqrt{\textstyle \sum_{k=0}^{n-1} \mathbb E_{\mu_{\omega}} Z_k^2}-s_n\bigg|<\infty,\\ \sup_{0\leq k \leq n-1} &\bigg| \sum_{i=0}^k f_i\circ T_{[\omega]_i} - \sum_{i=0}^k Z_i \bigg| = o\bigg(\sqrt{s^2_n \log \log s^2_n}\bigg)\quad \mu_{\omega}\mbox{-almost surely}. \end{align*} $$

Proof. Denote $\mathcal B_n=T_{[\omega ]_n}^{-1}\mathcal B$ for $n\in \mathbb N$ and let $\mathcal B_0=\mathcal B$ , then $\mathcal B_n$ is a non-increasing filtration. Let $h_0=0$ and define $h_n\in \mathcal {H}_{\alpha }$ recursively by $h_{n+1}=\mathbb P_{[\omega ]_n}^{[\theta ^n\omega ]_1}(f_n+h_n)$ . Then equation (4.4) implies that $h_n=\sum _{k=0}^{n-1}\mathbb P_{[\omega ]_k}^{[\theta ^k\omega ]_{n-k}}f_k\in {\mathcal H}_{\alpha }$ . It follows from Proposition 6.1 that $\mu _{\omega }\circ T^{-1}_{[\omega ]_k}=\mu _{[\omega ]_k, \theta ^k\omega }$ , then

$$ \begin{align*}\mathbb P_{[\omega]_k}^{[\theta^k\omega]_{n-k}}f_k=\mathbb P_{[\omega]_k}^{[\theta^k\omega]_{n-k}}f-\int f\circ T_{[\omega]_k}\,d\mu_{\omega}=\mathbb P_{[\omega]_k}^{[\theta^k\omega]_{n-k}}f-\int f \,d\mu_{[\omega]_k,\theta^k\omega}\end{align*} $$

and that, with $k_0\in \mathbb N$ and $s\in (0,1)$ given by Theorem 5.1,

$$ \begin{align*} \|h_n\|&\le\sum_{k=0}^{n-k_0}2s^{n-k}\overline D(f)+\sum_{k=n-k_0+1}^{n-1}\|\mathbb P_{[\omega]_k}^{[\theta^k\omega]_{n-k}}f_k\|\\ &\le \sum_{k=0}^{n-k_0}2s^{n-k}\overline D(f)+\sum_{k=n-k_0+1}^{n-1}C\|f\|\ll \|f\|, \end{align*} $$

where C is a uniform bound for all $\|\mathbb P_u^v\|$ (Lemma 4.1).

Let

$$ \begin{align*} U_n:=f_n\circ T_{[\omega]_n}+h_n\circ T_{[\omega]_n}-h_{n+1}\circ T_{[\omega]_{n+1}}. \end{align*} $$

Here, $U_n$ is $\mathcal B_n$ -measurable and square integrable. Moreover, apply Lemma 8.1 to get that

$$ \begin{align*} \mathbb E_{\mu_{\omega}}(U_n|\mathcal B_{n+1})=\mathbb P_{[\omega]_n}^{[\theta^n\omega]_1} f_n\circ T_{[\omega]_{n+1}}+\mathbb P_{[\omega]_n}^{[\theta^n\omega]_1} h_n\circ T_{[\omega]_{n+1}}-h_{n+1}\circ T_{[\omega]_{n+1}}=0. \end{align*} $$

So $(U_n)_{n\in \mathbb N_0}$ is a sequence of square integrable reverse martingale differences. Let

$$ \begin{align*} \sigma_n^2:=\sum_{k=0}^{n-1}\mathbb E_{\mu_{\omega}}U_k^2=\mathbb E_{\mu_{\omega}}\bigg(\sum_{k=0}^{n-1}U_k\bigg)^2. \end{align*} $$

We check the conditions of Theorem 8.2. Note that $\mathbb E$ in the rest of the proof stands for $\mathbb E_{\mu _{\omega }}$ .

First we show $\sigma _n^2\to \infty $ and $\sup _n\mathbb EU_n^2<\infty $ . It follows from

$$ \begin{align*} |\sigma_n-s_n|&=\bigg|\mathbb E^{1/2}\bigg(\sum_{k=0}^{n-1}U_k\bigg)^2-\mathbb E^{1/2}\bigg(\sum_{k=0}^{n-1}f_k\circ T_{[\omega]_k}\bigg)^2\bigg|\\ &\leq \mathbb E^{1/2}\bigg(\sum_{k=0}^{n-1}U_k-\sum_{k=0}^{n-1}f_k\circ T_0^k\bigg)^2=\mathbb E^{1/2}(h_n\circ T_{[\omega]_n})^2\\ &\ll \|f\| \end{align*} $$

that $|\sigma _n-s_n|$ is uniformly bounded. So $s_n^2\to \infty $ implies that $\sigma _n^2\to \infty $ . Since $\|U_n\|_{\infty }$ is uniformly bounded, $\sup _n\mathbb E U_n^2<\infty $ .

Next we show that

$$ \begin{align*}\sum_{k=0}^{n-1}(\mathbb E(U_k^2|\mathcal B_{k+1})-\mathbb E(U^2_k))=o(\sigma^2_n) \quad \mathbb \mu_{\omega}\text{-almost surely}. \end{align*} $$

Let $u_n=f_n+h_n-h_{n+1}\circ T_{[\theta ^n \omega ]_1}$ and let $\tilde u_n=u_n^2-\mathbb E U_n^2$ . Then $\|\tilde u_n\|_{\infty }\ll \|f\|^2.$ Moreover, the Hölder coefficient of $\tilde u_n$ is also uniformly bounded because, denoting $[\theta ^{n-1}\omega ]_1=i\in {\mathcal W}$ ,

$$ \begin{align*} D_{\alpha}(h_{n}\circ T_i)&=\sup_{x\neq y\in X}\frac{|h_{n}\circ T_i (x)-h_{n}\circ T_i(y)|}{d(x, y)^{\alpha}}\\ &\le D_{\alpha}(h_n)\cdot \sup_{0<d(x,y)<a}\bigg(\frac{d(T_i(x), T_i(y))}{d(x,y)}\bigg)^{\alpha}+2a^{-\alpha}\|h_n\|_{\infty}, \end{align*} $$

which is uniformly bounded by assumption. Let

$$ \begin{align*} F_n =\sigma_n^{-2}\sum_{k=0}^{n-1}\mathbb E(U_k^2|\mathcal B_{k+1}), \end{align*} $$

then

$$ \begin{align*}\sum_{k=0}^{n-1}(\mathbb E(U_k^2|\mathcal B_{k+1})-\mathbb E(U_k^2))=\sum_{k=0}^{n-1}\mathbb P_{[\omega]_k}^{[\theta^k\omega]_1} \tilde u_{k}\circ T_{[\omega]_{k+1}}=\sigma_n^2(F_n-1). \end{align*} $$

Applying Proposition 6.1, we have

$$ \begin{align*} & \mathbb E\bigg(\sum_{k=0}^{n-1} \mathbb P_{[\omega]_k}^{[\theta^k\omega]_1}\tilde u_k\circ T_{[\omega]_{k+1}}\bigg)^2 \\ &\quad\ll \sum_{0\leq k\leq l\leq n-1}\mathbb E (\mathbb P_{[\omega]_k}^{[\theta^k\omega]_1}\tilde u_k\circ T_{[\omega]_{k+1}}\cdot \mathbb P_{[\omega]_l}^{[\theta^l\omega]_1}\tilde u_l\circ T_{[\omega]_{l+1}})\\ &\quad=\sum_{0\leq k\leq l\leq n-1}\int \mathbb P_{[\omega]_k}^{[\theta^k\omega]_{l-k+1}}\tilde u_k\cdot \mathbb P_{[\omega]_l}^{[\theta^l\omega]_1}\tilde u_l ~ \,d\mu_{[\omega]_{l+1},\theta^{l+1}\omega}\\ &\quad\ll \sum_{l-k+1\ge k_0}s^{l-k+1} \overline D\tilde u_k\cdot \mathbb EU_l^2 +\sum_{l-k+1<k_0}\| \tilde u_k\|_{\infty}\cdot \mathbb EU_l^2\\ &\quad\ll k_0\cdot \sum_{l=0}^{k_0-2} \mathbb EU_l^2+(s^{k_0}+k_0)\cdot \sum_{l=k_0-1}^{n-1}\mathbb EU_l^2, \end{align*} $$

where in the last inequality, we have used that $\|\tilde u_k\|$ is uniformly bounded. Therefore,

$$ \begin{align*}\mathbb E(F_n-1)^2=\sigma_n^{-4}\mathbb E\bigg(\sum_{k=0}^{n-1} \mathbb P_{[\omega]_k}^{[\theta^k\omega]_1} \tilde u_{k}\circ T_{[\omega]_{k+1}}\bigg)^2\ll \sigma_n^{-4}\sum_{l=0}^{n-1}\mathbb E U_l^2= \sigma_n^{-2}.\end{align*} $$

As $\sigma _n\to \infty $ , $\mathbb E(F_n-1)^2\to 0$ . We need to show that it is almost sure convergence. Let $ C = \sup _n \mathbb E U^2_n$ and let $k_n=\inf \{k: \sigma _k^2\geq n^2 C\}.$ Then $k_n<\infty , k_n\to \infty $ and

$$ \begin{align*} n^2C\leq\sigma_{k_n}^2\leq(n^2+1)C. \end{align*} $$

Since

$$ \begin{align*} \sum_n \mathbb E(F_{k_n}-1)^2\ll\sum_n\sigma_{k_n}^{-2}<\infty, \end{align*} $$

$F_{k_n}\to 1$ almost surely by the Borel–Cantelli lemma. Let $m=m(n)\to \infty $ be such that $k_{m}\leq n\leq k_{m+1}$ , then

$$ \begin{align*} F_{k_m}\frac{m^2}{(m+1)^2+1}\leq F_{k_m}\frac{\sigma^2_{k_m}}{\sigma^2_{k_{m+1}}}\leq F_n\leq F_{k_{m+1}}\frac{\sigma^2_{k_{m+1}}}{\sigma_{k_{m}}^2}\leq F_{k_{m+1}}\frac{(m+1)^2+1}{m^2}. \end{align*} $$

Hence, $F_n\to 1$ almost surely. Lastly, $\sum _{n}\sigma _n^{-2}\mathbb E U_n^{2}<\infty $ because $\|U_n\|_{\infty }$ is uniformly bounded, $|\sigma _n-s_n|\ll \|f\|$ and $\sum _n s_n^{-4}<\infty $ by assumption.

Now we can use Theorem 8.2 to find a sequence of independent centred Gaussian variables $\{Z_k\}$ with $\mathbb EZ_k^2=\mathbb EU_k^2$ such that

$$ \begin{align*} \sup_{0\leq k\leq n-1}\bigg|\sum_{i=0}^k U_i-\sum_{i=0}^k Z_i\bigg|=o\bigg(\sqrt{\sigma^2_n \log\log \sigma^2_n}\bigg)\quad \text{almost surely}. \end{align*} $$

Since $|\sum _{i=0}^{k} f_i\circ T_{[\omega ]_i}-\sum _{i=0}^{k} U_i|$ and $|\sigma _n-s_n|$ are both uniformly bounded, the statement of the theorem follows.

Remark 8.4. One can verify condition (8.1) on total variance $s_n$ by verifying the inequality

$$ \begin{align*} \liminf_{n\to\infty} \frac1n\sum_{k=0}^{n-1}\mathbb E_{\mu_{\omega}}(f^2_k\circ T_{[\omega]_k})>2\sup_{k,m\in\mathbb N_0}\bigg|\sum_{l=k+1}^{k+m} \mathbb E_{\mu_{\omega}}(f_k\circ T_{[\omega]_k}\cdot f_l\circ T_{[\omega]_l})\bigg|. \end{align*} $$

Assuming that the Ruelle expanding semigroup $\mathcal S$ and the potentials $\varphi _i$ satisfy the conditions of Theorem 5.1, a priori the left-hand side of this inequality is positive and the right-hand side is finite for every $f\in \mathcal H_{\alpha }$ . A more explicit sufficient condition for f under which this inequality (and equation (8.1)) holds is yet unknown to us.

In that regard, it is also worth noting that the applications of Theorem 2.3 in [Reference Cuny and Merlevède10] (cf. Theorem 8.2) by Cuny and Merlevède to the iteration of a single, weakly expanding map give rise to explicit function spaces and stronger rates of approximation. However, their results rely on a moderate deviation result for stationary Markov chains by Wu and Zhao in [Reference Wu and Zhao36], which seems not to be available for inhomogeneous Markov chains. Moreover, Dragičević and Hafouta [Reference Dragičević and Hafouta14] and Hafouta [Reference Hafouta16] obtained a vector valued almost sure invariance principle for the sequential iteration of non-uniformly expanding maps. There, the authors obtain a better rate of approximation by assuming an abstract condition on the characteristic functions of the associated process. Finally, we also would like to mention the almost sure invariance principle in [Reference Stadlbauer and Zhang32]. There, it was possible to determine an explicit class of functions and sometimes their asymptotic variance such that the almost sure invariance principle holds with respect to sequential systems associated with the continued fraction expansion.

9 Applications

In this section, we illustrate some possible applications of our main results, both for conformal iterated function systems and the thermodynamic formalism of free semigroup actions by expanding maps.

9.1 Non-autonomous conformal iterated function systems

The class of non-autonomous conformal iterated function system was introduced and studied in [Reference Rempe-Gillen and Urbański27], and is defined as follows.

Definition 9.1. We refer to $\{X,(\Phi _i:1\leq i \leq k)\}$ as a non-autonomous conformal iterated function system if X is a convex, compact subset of $\mathbb {R}^d$ for some $d \in \mathbb {N}$ with $\overline {\mbox {int}(X)} =X$ , and $(\Phi _i)$ is a collection $\{ \varphi _{i,1},\ldots ,\varphi _{i,k(i)}\}$ of maps from X to X such that:

1. (1) the following conformality condition holds—there exists an open connected set $V \supset X$ such that each $\varphi _{i,j}$ extends to a continuously differentiable conformal diffeomorphism from V into V;

2. (2) the open set condition holds— $\varphi _{i,j}(\mbox {int}(X)) \cap \varphi _{i,\tilde {j}}(\mbox {int}(X)) = \emptyset $ , for all $1 \leq j < \tilde {j} \leq k(i)$ and $i= 1,\ldots k$ ;

3. (3) the following conditions on bounded distortion and uniform contraction hold—there exist constants $K \geq 1$ and $\eta \in (0,1)$ such that for any $n \in \mathbb {N}$ and any choice $(i_1,j_1),\ldots , (i_n,j_n)$ , with $i_l \in \{1,\ldots , k\}$ and $1\leq j_l \leq k(l)$ and all $x,y \in X$ , for $\varphi := \varphi _{i_n,j_n} \circ \cdots \circ \varphi _{i_1,j_1}$ , we have that

   $$ \begin{align*} \|D \varphi(x)\| \leq K \|D \varphi(y)\|, \quad \|D \varphi(x)\| \leq K \eta^n. \end{align*} $$

As X is assumed to be compact and $k(i) < \infty $ for all $i= 1,\ldots k$ , it follows for any compact set $A \subset K$ that $\Phi _i(A):= \bigcup _{j=1}^{k(i)} \varphi _{i,j}(A)$ is compact. Hence, for a given $\omega \in \Sigma $ , where $\Sigma = \{(\omega _1\omega _2\ldots ): 1 \leq \omega _i \leq k\}$ , $(\Phi _{\omega _1} \circ \cdots \circ \Phi _{\omega _n} (X))_n$ is a decreasing sequence of compact sets which then implies that the limit set $J_{\omega }$ , defined by

$$ \begin{align*}J_{\omega} := \lim_{n \to \infty} \Phi_{\omega_1} \circ \Phi_{\omega_2} \circ \cdots \circ \Phi_{\omega_n} (X), \end{align*} $$

is non-empty and compact.

We now derive an averaged version of Bowen’s formula to have access to the Hausdorff dimension of these limit sets. To do so, we have to adapt the semigroup setting to the intuitionistic fuzzy set (IFS). First observe that equation (1) in Definition 9.1 implies that $\varphi := \varphi _{i_n,j_n} \circ \cdots \circ \varphi _{i_1,j_1}$ is a well-defined conformal diffeomorphism for any $n \in \mathbb {N}$ and $(i_1,j_1),\ldots , (i_n,j_n)$ , with $i_l \in \{1,\ldots , k\}$ and $1\leq j_l \leq k(l)$ . Furthermore, by equation (3), $\varphi $ is a contraction with rate $K\eta ^n$ and, by a standard argument, $x \mapsto \log \|D \varphi (x)\|$ is Lipschitz continuous with respect to a uniform constant.

For $\delta \geq 0$ , we now consider the operators, for $w=(\omega _1 \ldots \omega _n)$ ,

$$ \begin{align*} L^{\delta}_{\omega_i}(f) &:= \sum_{j=1}^{k(\omega_i)} \|D\varphi_{\omega_i,j} (\,\cdot\,)\|^{\delta} f\circ\varphi_{\omega_i,j}, \\ L^{\delta}_{w}(f) &:= \sum_{j_1, \ldots, j_n} \|D(\varphi_{\omega_1,j_1} \cdots \varphi_{\omega_n,j_n}) (\,\cdot\,)\|^{\delta} f \circ \varphi_{\omega_1,j_1} \cdots \varphi_{\omega_n,j_n} \\ &\phantom{:}= L^{\delta}_{\omega_1} \circ L^{\delta}_{\omega_2} \circ \cdots \circ L^{\delta}_{\omega_n} (f) \end{align*} $$

for f in a suitable function space (the last equality follows from conformality). Now assume that $\rho $ is a probability measure on $\Sigma $ which satisfies the conditions of Theorem 7.3, that is, $\log d\rho /d\rho \circ \sigma $ is Hölder continuous and the support of $\rho $ is a topological mixing SFT, and, for $n \in \mathbb {N}$ ,

$$ \begin{align*} \mathcal{A}^{\delta}_n := \sum_{w \in \{1,\ldots k\}^n} \rho([w]) L^{\delta}_w. \end{align*} $$

Here $[w]$ represents the cylinder set $\{\omega \in \Sigma : [\omega ]_n=w\}$ . Observe that the arguments in the proofs of Theorems A and C apply straightforwardly in this context through an interpretation of $\varphi _{\omega _1,j_1} \cdots \varphi _{\omega _n,j_n}$ as an inverse branch of an expanding map. Hence, we obtain uniform and exponential convergence of $L^{\delta }_w$ as $|w| \to \infty $ and of $\mathcal {A}^{\delta }_n$ as $n \to \infty $ . In particular, for each $\delta \geq 0$ , there exists $\unicode{x3bb} _{\delta }$ such that $\mathcal {A}^{\delta }_n(\mathbf {1}) \asymp \unicode{x3bb} _{\delta }^n$ . Thus, the annealed pressure function $P:[0,\infty ) \to \mathbb {R}$ given by

$$ \begin{align*} P(\delta) := \lim_{n \to \infty} \frac{1}{n} \log \mathcal{A}^{\delta}_n(\mathbf{1}) = \log \unicode{x3bb}_{\delta} \end{align*} $$

is well defined.

Lemma 9.1. The function P is continuous and strictly decreasing. Furthermore, $\lim _{\delta \to +\infty } P(\delta ) = -\infty $ and $P_0 = \log \unicode{x3bb} _0 \geq \log (\min _i k(i))$ , where $\unicode{x3bb} _0$ is the spectral radius of the operator defined by

$$ \begin{align*} \iota(f) = \sum_{i=1}^{k} k(i) \frac{d\rho}{d\rho\circ\sigma}(i\,\cdot \,) f(i\,\cdot \,). \end{align*} $$

Proof. It follows from the definition and the finiteness of the generating IFS that there exist $\eta _+,\eta _- \in (0,1)$ such that $\eta _-^n \ll \|D(\varphi _{\omega _1,j_1} \cdots \varphi _{\omega _n,j_n})\| \ll \eta _+^n$ . Hence, for $\epsilon> 0$ , we have that

$$ \begin{align*} \eta_-^{n\epsilon}\mathcal{A}^{\delta}_n(\mathbf{1}) \ll \mathcal{A}^{\delta+\epsilon}_n(\mathbf{1}) \ll \eta_+^{n\epsilon} \mathcal{A}^{\delta}_n(\mathbf{1}), \end{align*} $$

which implies that $ \epsilon \log \eta _- \leq P(\delta +\epsilon ) -P(\delta ) \leq \epsilon \log \eta _+ $ . Hence, P is continuous and strictly decreasing. To determine $\lim _{\delta \to +\infty } P(\delta ) = -\infty $ , observe that

$$ \begin{align*} \mathcal{A}^{\delta}_{m+n}(\mathbf{1})(x) &\leq \sum_{|v|=m} \sum_{|w|=n} \rho([vw]) L^{\delta}_v \circ L^{\delta}_w (\mathbf{1}) (x) \\ & \leq \sum_{|v|=m} \rho([v]) L^{\delta}_v \bigg( \sum_{|w|=n} \frac{\rho([vw])}{\rho([v])\rho([w])} \rho([w]) L^{\delta}_w (\mathbf{1}) \bigg) (x) \\ & \leq C \mathcal{A}^{\delta}_{m} \circ \mathcal{A}^{\delta}_{n}(\mathbf{1})(x), \quad \text{for all } m,n \ge 1 \end{align*} $$

as there is a uniform bound C for ${\rho ([v])\rho ([w])}/{\rho ([vw])}$ by bounded distortion of $\rho $ . Hence, for every fixed $n\ge 1$ ,

$$ \begin{align*}\unicode{x3bb}_{\delta} = \lim_{l} \sqrt[ln]{\mathcal{A}^{\delta}_{ln}(\mathbf{1})} \leq \sqrt[n]{ C\|\mathcal{A}^{\delta}_{n}(\mathbf{1})\|_{\infty}} \xrightarrow{\delta \to +\infty} 0.\end{align*} $$

To determine $P(0)$ , we employ Theorem 7.3 as follows. For $\delta =0$ , $L_i(\mathbf {1}) = k(i)\mathbf {1}$ . Hence, by the proof of Theorem 7.3, $\unicode{x3bb} _0$ is the spectral radius of $\iota $ which is bigger than or equal to $\log (\min _i k(i))$ .

As an immediate corollary, it follows that there exists a unique $\delta _0> 0$ such that ${P(\delta _0)=0}$ , provided that $P(0)> 0$ , e.g. if $\min _i k(i)> 1$ .

Theorem 9.2. Assume that $P(0)>0$ . Then, for $\rho $ -almost every $\omega $ , the Hausdorff dimension $\dim _H(J_{\omega })$ of $J_{\omega }$ is equal to the unique root $\delta _0$ of P.

Proof. Fix $x\in X$ . In analogy to the above pressure function, for $\omega = (\omega _i)$ , set

$$ \begin{align*} P_{\omega}(\delta) := \limsup_{n \to \infty} \frac{1}{n} \log L^{\delta}_{\omega_1\ldots \omega_n} (\mathbf{1}) (x). \end{align*} $$

To prove almost sure convergence, we employ Kingman’s subadditive ergodic theorem. To do so, observe that the shift is $\rho $ -ergodic, and that there exists an equivalent invariant probability measure. Set

$$ \begin{align*} g_n(\omega) := \sup \{ \log L^{\delta}_{\omega_1\ldots \omega_n} (\mathbf{1}) (x): {x \in X}\}.\end{align*} $$

By construction, $g_{m+n}(\omega ) \leq g_m(\omega ) + g_n(\sigma ^n(\omega ))$ . As $g_n(\omega ) \asymp \log L^{\delta }_{\omega _1\ldots \omega _n} (\mathbf {1}) (x)$ , it now follows from Kingman’s subadditive ergodic theorem that $P_{\omega }(\delta )$ exists almost everywhere and in $L^1(\rho )$ , that $P_{\omega }(\delta )$ is almost surely constant and that the $\limsup $ in the definition in fact is a limit. It follows from these observations that $P_{\omega }(\delta ) = P(\delta )$ almost surely, but for $\delta $ fixed. However, by the same argument for Lipschitz continuity of P in the proof above, one obtains that the maps $P_{\omega }$ are equi-Lipschitz continuous. Hence, by choosing a countable and dense set $\{\delta _i\}$ , one obtains a set of full measure $\Omega $ such that $P_{\omega }(\delta ) = P(\delta )$ for all $\omega \in \Omega $ and $\delta \geq 0$ .

We now show that $\dim _H(J_{\omega }) = \delta _0$ for each $\omega = (\omega _i) \in \Omega $ . To do so, we first recall some consequences of conformality. As $\varphi := \varphi _{\omega _1,j_1} \cdots \varphi _{\omega _n,j_n}$ is conformal, it follows that the diameter $\mbox {diam}(\varphi (X))$ satisfies $\mbox {diam}(\varphi (X)) \asymp \|D\varphi \| \cdot \mbox {diam}(X)$ . Furthermore, covers by sets of type $\varphi (X)$ are optimal in the following sense. By Lemma 2.7 in [Reference Mauldin and Urbański24], or from the proof of Theorem 3.2 in [Reference Rempe-Gillen and Urbański27], there exists $M\in \mathbb {N}$ such that for each ball B of radius $r>0$ , there exist a subset $W(B)$ of $\{((\omega _1,j_1), \cdots (\omega _n,j_n)) : n \in \mathbb {N}, 1 \leq j_i \leq k(i)\}$ of at most M elements such that:

1. (1) the elements of $\{\varphi _{\omega _1,j_1} \cdots \varphi _{\omega _n,j_n} (\mbox {int}(X)) :((\omega _1,j_1), \ldots (\omega _n,j_n)) \in W(B) \}$ are pairwise disjoint;

2. (2) $\mbox {diam}(\varphi _{\omega _1,j_1} \cdots \varphi _{\omega _n,j_n} (X)) \asymp \mbox {diam}(B)$ for $((\omega _1,j_1), \ldots (\omega _n,j_n)) \in W(B)$ ;

3. (3) $B \cap J_{\omega } \subset \bigcup _{((\omega _1,j_1), \ldots (\omega _n,j_n)) \in W(B)} \varphi _{\omega _1,j_1} \cdots \varphi _{\omega _n,j_n} (X) $ .

The result now provides access to the $\delta $ -Hausdorff measure of $J_{\omega }$ as follows. Assume that $\mathcal {U}$ is a finite cover of $J_{\omega }$ by closed balls. By replacing each $B \in \mathcal {U}$ by $\{\varphi _{\omega _1,j_1} \cdots \varphi _{\omega _n,j_n} (X) :((\omega _1,j_1), \ldots (\omega _n,j_n)) \in W(B) \}$ , we obtain a further cover $\mathcal {V}$ which satisfies

$$ \begin{align*} \sum_{B \in \mathcal{U}} \mbox{diam}(B)^{\delta} \asymp \sum_{A \in \mathcal{V}} \mbox{diam}(A)^{\delta}. \end{align*} $$

Hence, to estimate the right-hand side, we may assume without loss of generality that for each $B \in \mathcal {U}$ , there exist $(\omega _i,j_i)$ such that $B = \varphi _{\omega _1,j_1} \kern-1.5pt\cdots \varphi _{\omega _n,j_n} (X)$ . However, Proposition 6.1 implies that for an arbitrary $x \in \mbox {int}(X)$ ,

$$ \begin{align*} \mu_{\omega}(B) & = \lim_{l \to \infty} \frac{L^{\delta}_{\omega_{n+1} \ldots \omega_{n+l}}\circ L^{\delta}_{\omega_1 \ldots \omega_{n}}(\mathbf{1}_B)(x)}{L^{\delta}_{\omega_1 \ldots \omega_{n+l}}(\mathbf{1})(x)} \\ & \asymp \|D\varphi_{\omega_1,j_1} \cdots \varphi_{\omega_n,j_n}\|^{\delta} \lim_{l \to \infty} \frac{L^{\delta}_{\omega_{n+1} \ldots \omega_{n+l}}(\mathbf{1})(x)}{L^{\delta}_{\omega_1 \ldots \omega_{n+l}}(\mathbf{1})(x)} \asymp \mbox{diam}(B)^{\delta} \unicode{x3bb}_{\omega_1 \ldots \omega_{n},\sigma^n\omega}^{-1}. \end{align*} $$

Setting $|B|=n$ , this implies that

$$ \begin{align*} \sum_{B \in \mathcal{U}} \mbox{diam}(B)^{\delta} \asymp \sum_{B \in \mathcal{U}} \unicode{x3bb}_{\omega_1 \ldots \omega_{|B|},\sigma^{|B|}\omega} \mu_{\omega}(B). \end{align*} $$

Now assume that the interiors of the elements of $\mathcal {U}$ are disjoint. Then $\sum \mu _{\omega }(B) =1$ and the asymptotics of $\sum \mbox {diam}(B)^{\delta } $ as $\max \mbox {diam}(B) \to 0$ are determined by the asymptotics of $\unicode{x3bb} _{\omega _1 \ldots \omega _{n},\sigma ^{n}\omega }$ as $n \to \infty $ . Hence, if $\delta> \delta _0$ , then the $\delta $ -Hausdorff measure of $J_{\omega }$ is $0$ and if $\delta < \delta _0$ , then the $\delta $ -Hausdorff measure of $J_{\omega }$ is $\infty $ . This implies that $\dim _H(J_{\omega }) = \delta _0$ .

9.2 Thermodynamic formalism of semigroup actions

In this subsection, we will provide some applications of our results to the setting of finitely generated free semigroup actions.

Let X be a compact metric space, $\varphi : X \to \mathbb R$ be a continuous potential and let $G_1=\{g_1, g_2, \ldots , g_k\}$ be a finite set of continuous self maps on X, for some $k\geq 2$ . The semigroup $\mathcal {S}$ generated by $G_1$ induces a continuous semigroup action given by

$$ \begin{align*} \mathbb{S} :\, & \mathcal{S} \times X \to X \\ & (g,x) \,\mapsto g(x), \end{align*} $$

meaning that for any $\underline {g},\,\underline {h} \in \mathcal {S}$ and every $x \in X$ , we have $\mathbb {S}(\underline {g}\,\underline {h},x)=\mathbb {S}(\underline {g}, \mathbb {S}(\underline {h},x)).$ The thermodynamic formalism of semigroup actions faces several difficulties. On one hand, while probability measures which are invariant by all generators may fail to exist, in opposition to the case of group actions, there are evidences that the stationary measures seem not sufficient to describe the dynamics. On the other hand, the existence of some distinct concepts of topological pressure for group and semigroup actions makes it necessary to test their effectiveness to describe the dynamics. In the case of free semigroup actions, the coding of the dynamics by the full shift suggests to consider the skew-product

(9.1) $$ \begin{align} \begin{array}{lll} F :\!\!\!\! & \{1,2,\ldots, k\}^{\mathbb N} \times X\!\!\!\! &\to \{1,2,\ldots, k\}^{\mathbb N} \times X \\ & (\omega,x)\!\!\!\! &\mapsto (\sigma(\omega), g_{\omega_1}(x)). \end{array} \end{align} $$

Moreover, a random walk on the semigroup $\mathcal S$ can be modelled by a Bernoulli probability measure $\mathbb P$ on $\{1,2,\ldots , k\}^{\mathbb N}$ . The pressure $P_{\text {top}}(\mathbb S, \phi , \mathbb P)$ of the semigroup action determined by that random walk coincides with the annealed topological pressure $P^{(a)}_{\text {top}}(F, \tilde \phi , \mathbb P)$ of the random dynamical system determined by F, associated to the potential $\tilde \phi : \{1,2, \ldots , k\}^{\mathbb N} \times X \to \mathbb R$ given by $\tilde \phi ({\omega },x)=\phi (x)$ (cf. Proposition 4.1 in [Reference Carvalho, Rodrigues and Varandas7]). In particular, $P_{\text {top}}(\mathbb S, \phi , \mathbb P)$ coincides with the logarithm of the spectral radius of the averaged transfer operator

$$ \begin{align*} {\mathcal A}_1(f) = \int L_{g_{{\omega}}}(f)\,d\mathbb P(\omega). \end{align*} $$

Furthermore, if $P_{\text {top}}(\mathbb S,0,\mathbb P)<\infty $ , then entropy and invariant measures can be defined through a functional analytic approach, which culminates in the variational principle

(9.2) $$ \begin{align} P_{\text{top}}(\mathbb{S},\phi,\mathbb P) = \sup_{\{\nu \, \in \, \mathcal{M}(X)\,\colon\, \Pi(\nu,\sigma) \neq \emptyset\}} \bigg\{h_{\nu}(\mathbb{S},\mathbb P) + \int \phi \,d\nu\bigg\} \end{align} $$

(we refer the reader to [Reference Carvalho, Rodrigues and Varandas7] for the definitions and more details). If all generators are Ruelle expanding maps and $\phi $ is Hölder continuous, then there exists a unique equilibrium state for the semigroup action $\mathbb S$ with respect to $\phi $ and this can be characterized either as a marginal of the unique equilibrium state for the annealed random dynamics or as the unique probability on X obtained as the limit of the equidistribution along pre-orbits associated to the semigroup dynamics by

$$ \begin{align*} e^{-n P_{\text{top}}(\mathbb S, \phi, \mathbb P)} {\mathcal A}_1^{*n} \delta_x = e^{-n P_{\text{top}}(\mathbb S, \phi, \mathbb P)} \int_{{\mathcal W}_n} \; \bigg[ \sum_{g_{\omega}(y)=x} \delta_y \, \bigg] \,d\mathbb P(\omega) \end{align*} $$

(we refer the reader to [Reference Carvalho, Rodrigues and Varandas6, §9] and [Reference Carvalho, Rodrigues and Varandas7, Theorem B] for more details). A more general formulation, considering more general probabilities on semigroup actions rather than random walks, was not available up to now as the thermodynamic formalism of the associated annealed dynamics needed to be described through a sequence of transfer operators instead of a single averaged operator.

Our results allow not only to consider the thermodynamic formalism of semigroup actions with respect to more general probabilities in the base, but also to provide important asymptotic information on the convergence to equilibrium states. Indeed, in general, if one endows the semigroup $\mathcal S$ with a probability generated by a Markov measure $\mathbb P$ on $ \{1,2, \ldots , k\}^{\mathbb N}$ , then it is natural to define the topological pressure of the semigroup action $\mathbb S$ by

(9.3) $$ \begin{align} P_{\text{top}}(\mathbb S, \phi, \mathbb P) =\limsup_{n\to\infty} \frac1n \log \|{\mathcal A}_n(1)\|_{\infty} \end{align} $$

where, as before, ${\mathcal A}_n(f) = \int _{{\omega } \in {\mathcal W}_n} L_{g_{{\omega }_1 {\omega }_2 \ldots {\omega }_n}}(f)\,d\mathbb P({\omega })$ (compare to the definition of topological pressure of a semigroup action in [Reference Carvalho, Rodrigues and Varandas7, §2.6]). Our main results have the following immediate consequences.

Corollary 9.3. Given $x\in X$ , the sequence of probability measures on X defined as

$$ \begin{align*} \nu^x_n:=\frac{\mathcal{A}^*_n(\delta_x)}{\mathcal{A}_n(\mathbf{1})(x)}, \quad n\ge 1 \end{align*} $$

is weak $^*$ convergent to some probability $\nu =h d\pi $ on X (independently of x). Moreover, the convergence is exponentially fast with respect to the Wasserstein distance.

9.3 A boundary of equilibria

As in the section before, we now assume that X is compact and that there is only one potential $\varphi : X \to \mathbb {R}$ . However, in contrast to the approach via the free semigroup, we are now interested in identifying elements in the semigroup $\mathcal {S}$ which are dynamically close and use this information to define a compactification of the discrete set $\mathcal {S}$ . However, as the topology will rely on the associated equilibrium states, we have to extend the semigroup by considering also the potential function. That is, for $\mathbb {G}_1 := \{ (g_1,\varphi ),(g_2,\varphi ),\ldots (g_k,\varphi )\}$ , we consider

$$ \begin{align*} \mathbb{G} := \{\kern-0.5pt (g,\psi) : \text{there exists } n \kern1.5pt{\in}\kern1.5pt \mathbb{N}, j_1, \ldots, j_n \mbox{ such that } (g,\psi) \kern1.5pt{=}\kern1.5pt (g_{i_1},\varphi) \ast \cdots \ast (g_{i_n}, \varphi) \}, \end{align*} $$

where

$$ \begin{align*}(g_1,\psi_1) \ast (g_2,\psi_2) := (g_1 \circ g_2,\psi_2 +\psi_1\circ g_2)\end{align*} $$

is also the product on $\mathbb G$ .

As a first step, we begin with the definition of a metric on the countable set ${\mathcal W}^{\ast } := \{w : |w|< \infty \}$ of finite words. For finite words $v=(v_1 \ldots v_m)$ and $w=(w_1 \ldots w_n)$ in ${\mathcal W}^{\ast }$ , set $d_{{\mathcal W}^{\ast }}(v,w) =0$ for $v=w$ and

$$ \begin{align*} d_{{\mathcal W}^{\ast}}(v,w) :=&\, 2^{-\min\{ k: v_k \neq w_k \mbox{ or } k> \min\{m,n\} \} }\\ & + 2^{-\min\{ k: v_{m+1-k} \neq w_{n+1-k} \mbox{ or } k > \min\{m,n\} \} }, \end{align*} $$

for $v \neq w$ . Observe that $d_{{\mathcal W}^{\ast }}$ is a metric, that ${\mathcal W}^{\ast }$ is discrete with respect to this metric and that two words are close if they have the same beginning and ending. In particular, Cauchy sequences either have to be eventually constant or have to grow from the interior of a word. The reason for this construction is based on the following observation. Let $\underline {w}$ and $\overline {w}$ refer to the periodic extensions of w to the left and the right, respectively, as defined in Remark 6.4. Then, by Proposition 6.3, the map $w \to \mu _{\underline {w},\overline {w}}$ is Hölder continuous with respect to $d_{{\mathcal W}^{\ast }}$ . In particular, $d_{{\mathcal W}^{\ast }}$ can be seen as a metric on the free semigroup which is compatible with the Wasserstein distance of the associated equilibrium states.

Second, we define a metric on $\mathbb {G}$ which does not depend on the choice of $w \in {{\mathcal W}^{\ast }}$ for the representation of $(g,\psi )= (T_w, \varphi _w)$ . To do so, define for $g \in \mathcal {S}$ ,

$$ \begin{align*} \kappa(g) : = \lim_{\epsilon \to 0} \inf\bigg\{ \frac{d(g(x),g(y))}{d(x,y)} :0< d(x,y) < \epsilon \bigg\},\end{align*} $$

and note that as the semigroup is Ruelle expanding with parameter $\unicode{x3bb} \in (0,1)$ , we have that $\kappa (T_w) \geq \unicode{x3bb} ^{-|w|}$ . Furthermore, for $(g,\psi ) \in \mathbb {G}$ , let $\mu _{g,\psi }$ be the unique equilibrium state for the potential $\psi $ and the map g, that is, if $(g, \psi )= (T_w, \varphi _w)$ , then $\mu _{g,\psi } = \mu _{\underline {w},\overline {w}}$ . Now set

$$ \begin{align*} d_{\mathbb{G}}((g,\psi_1),(h,\psi_2)) := \begin{cases} \overline{W}(\mu_{g,\psi_1},\mu_{h,\psi_2}) + \dfrac{1}{\kappa(g)} + \dfrac{1}{\kappa(h)}\!\!\!\! &, (g,\psi_1)\neq (h,\psi_2),\\ 0\!\!\!\! &, (g,\psi_1)=(h,\psi_2). \end{cases} \end{align*} $$

The following proposition summarizes the basic topological facts. The proof is omitted as the assertions almost immediately follow from the definitions and Proposition 6.3.

Proposition 9.4. Assume that $g_1, \ldots , g_k$ are Ruelle expanding and jointly topological mixing, and that $\varphi $ is Hölder continuous. Then, for the objects defined above, the following hold.

1. (1) $({\mathcal W}^{\ast }, d_{{\mathcal W}^{\ast }})$ and $({\mathbb {G}}, d_{\mathbb {G}})$ are discrete, metric spaces.

2. (2) The map $w \mapsto (T_w,\varphi _w)$ is Hölder continuous.

3. (3) A sequence $((g_n,\psi _n))_n$ in $\mathbb {G}$ is a Cauchy sequence if and only if $\kappa (g_n) \to \infty $ and $(\mu _{g_n,\psi _n})$ converges in the weak $^{\ast }$ -topology. Moreover, two Cauchy sequences have the same limit if and only if their sequences of equilibrium states have the same limit.

4. (4) For the boundary $\partial \mathbb {G}$ of the completion with respect to $d_{\mathbb {G}}$ , identified with limits of Cauchy sequences $((g_n,\psi _n))_n$ in $\mathbb {G}$ , we have that the map

   $$ \begin{align*} \partial \mathbb{G} \to \{ \mu_{\sigma,\omega} : \sigma \in \Sigma^-, \omega \in \Sigma\}, \; (({g_n,\psi_n}))_n \mapsto \lim_{n \to \infty} \mu_{g_n,\psi_n} \end{align*} $$
   is Lipschitz continuous and onto.

Observe that the result provides a description of $\partial \mathbb {G}$ as a set of equivalence classes of Cauchy sequences, that is, two sequences are considered to be equivalent if they have the same limit. However, it seems to be impossible to obtain an explicit description of $\partial \mathbb {G}$ in general. We close with two examples where this is possible. In the first example, $\partial \mathbb {G}$ is trivial whereas in the second example, $\partial \mathbb {G}$ is equal to $\Sigma ^{-}$ .

Proposition 9.5. If $\mathbb {G}$ is Abelian, then $\partial \mathbb {G}$ is a point.

Proof. Assume that $(g_1,\psi _1), (g_2,\psi _2) \in \mathbb {G}$ , and denote by $L_i$ the corresponding Ruelle operators. As $\mathbb {G}$ is Abelian, it immediately follows that $L_1L_2 =L_2L_1$ . Now assume that the $h_i$ are the unique positive Hölder functions (up to colinearity) and $\unicode{x3bb} _i>0$ such that $L_i(h_i) = \unicode{x3bb} _i h_i$ , given by Ruelle’s theorem. Hence, $L_2(L_1 (h_2)) = L_1(L_2 (h_2)) = \unicode{x3bb} _2 L_1(h_2)$ . As $L_1(h_2)$ is positive, it follows that $L_1(h_2)$ and $h_1$ are colinear, that is, $L_1(h_2)$ is a multiple of $h_1$ and $\unicode{x3bb} _1 = \unicode{x3bb} _2$ . The same argument then shows that the $L_i^{\ast }$ -eigenmeasures coincide. Hence, after normalizing, we obtain that $\mu _{g_1,\psi _1} = \mu _{g_2,\psi _2}$ . In particular, $\{ \mu _{\sigma ,\omega } : \sigma \in \Sigma ^-, \omega \in \Sigma \}$ is a singleton.

Example 9.6. Let $T:[0,1] \to [0,1]$ , $x \mapsto 4x (\mbox {mod} 1)$ and $S = U^{-1} T U$ , where

$$ \begin{align*} U:[0,1] \to [0,1], \quad x \mapsto \begin{cases} 3x/2, & 0 \leq x < 1/8, \\ x + 1/16, & 1/8 \leq x < 3/8, \\ x/2 + 1/4, & 3/8 \leq x < 1/2, \\ x, & 1/2 < x \leq 1. \end{cases} \end{align*} $$

Proposition 9.7. The semigroup $\mathcal {S}$ generated by $\{S,T\}$ is a free semigroup, that is, two elements in $\mathcal {S}$ coincide if and only if they have the same representation as a product of the generators. Moreover, $\partial \mathbb {G} \cong \Sigma ^{-}$ , where $\mathbb {G}$ is the semigroup generated by $(T,0)$ and $(S,0)$ .

Proof. The proof relies on the construction of a family of renormalization operators acting on the set of orientation-preserving homeomorphisms f in such a way that

$$ \begin{align*} T^n \circ \Xi_n(f) = f \circ T^n, \end{align*} $$

as this allows to associate to each element $g=S^{m_k}T^{n_k} \cdots S^{m_1}T^{n_1}$ in $\mathcal {S}$ a uniquely determined normal form $T^{m_1 + n_1 + \cdots m_k+ n_k} \circ f_g$ , where $f_g$ is an orientation-preserving homeomorphism. The uniqueness of the normal form is a consequence of the choice of U as the compositions with U and $U^{-1}$ act as markers in the following way. For an orientation-preserving homeomorphism f, it is shown below that $\|\Xi ^n(f) - \mathrm {id}\|_{\infty } = 4^{-n} \| f- \mathrm {id}\|_{\infty }$ , and that the composition $\Xi _n(f)\circ U^{\pm 1}$ leaves invariant the right half of $\Xi _n(f)$ , whereas the left half is marked by a positive or negative bump of size bigger than $\|\Xi ^n(f) - \mathrm {id}\|_{\infty }$ .

Construction and properties of $\Xi _n$ . Let $f: [0,1] \to [0,1]$ be a homeomorphism which fixes $0$ and $1$ and define for $x \in [k/4^n, (k+1)/4^n]$ ,

$$ \begin{align*} \Xi_n(f)(x) := ( T^n|_{[k/4^n, (k+1)/4^n]} )^{-1} \circ f \circ T^n(x) = 4^{-n}(f(4^n x - k) + k). \end{align*} $$

Then, as it can be easily seen, $ T^n \circ \Xi _n(f) = f \circ T^n $ and $\Xi _n(f)(k/4^n) = k/4^n$ for all $k= 0, \ldots , 4^n$ . In particular, as $\Xi _n(f)|_{[k/4^n, (k+1)/4^n]}$ is a homeomorphism, $\Xi _n(f)$ is a homeomorphism. Moreover, for $x \in [k/4^n, (k+1)/4^n]$ , we have

$$ \begin{align*}\Xi_n(f)(x) - x & = 4^{-n}(f(4^n x - k) + k) - x \\ & = 4^{-n}(f(4^n x - k) - (4^{n}x-k)) = 4^{-n}(f\circ T^n(x) - T^n(x) ). \end{align*} $$

That is, $\Xi _n$ contracts the distance to the identity by the factor $4^{-n}$ . We now proceed with an analysis of the concatenations $\Xi _n(f)\circ U$ and $\Xi _n(f)\circ U^{-1}$ , where f is a homeomorphism with $\|f - \textrm {id} \|_{\infty } \leq 1/12$ . First note that

$$ \begin{align*} U(x) - x =\begin{cases} x/2, & x \in [0,\frac{1}{8}), \\ 1/16, & x \in [\frac{1}{8},\frac{3}{8}), \\ -x/2 + 1/4, & x \in [\frac{3}{8},\frac{1}{2}), \\ 0, & x \in [\frac{1}{2},1], \end{cases} \quad U^{-1}(x) - x =\begin{cases} - x/3, & x \in [0,\frac{3}{16}), \\ - 1/16, & x \in [\frac{3}{16},\frac{7}{16}), \\ x - 1/2, & x \in [\frac{7}{16},\frac{1}{2}),\\ 0, & x \in [\frac{1}{2},1], \end{cases} \end{align*} $$

and observe that, by construction, $\Xi _n(f) - \textrm {id}$ is periodic with period $4^{-n}$ . However, as $[\tfrac 18,\tfrac 38)$ , $ [{3}/{16},{7}/{16})$ and $[\tfrac 12,1]$ are all of length bigger than or equal to $1/4$ , we obtain that

$$ \begin{align*} \max_{x \in [0,1]} (\Xi_n(f)(U(x)) - x ) & = \max_{x \in [{1}/{8},{3}/{8})} (\Xi_n(f)(U(x)) - U(x) + U(x)-x ) \\ & = 4^{-n} \max_{x \in [0,1]} (f(x) - x ) + \frac{1}{16} = \frac{1}{4^n \cdot 12} + \frac{1}{16} \leq \frac{1}{12}, \end{align*} $$

and, repeating the argument, $\|\Xi _n(f)\circ U^{j} - \textrm {id}\|_{\infty } \leq 1/12$ , for $j = \pm 1$ .

In other words, the space $\mathfrak {H}$ of orientation-preserving homeomorphisms with $\|f - \textrm {id}\|_{\infty } \leq 1/12$ is invariant under the operation $f \mapsto \Xi _n(f)\circ U^{j}$ . Moreover, we have that

(9.4) $$ \begin{align} \|\Xi_n(f)\circ U^j - U^j\|_{\infty} = 4^{-n} \| \Xi_n(f) - \textrm{id} \|_{\infty} = 4^{-n} \| f - \textrm{id} \|_{\infty} \leq \tfrac{1}{48}. \end{align} $$

Coding of $\mathbb {G}$ . Assume that $g = S^{m_k}T^{n_k} \cdots S^{m_1}T^{n_1}$ for some $k \in \mathbb {N}$ and $m_i,n_i \in \mathbb {N} \cup \{0\}$ . As $U, U^{-1} \in \mathfrak {H}$ , it follows from an iterated application of $ \Xi _n(\cdot )\circ U^{j}$ that there exists a homeomorphism $f_g \in \mathfrak {H}$ such that $g = T^n \circ f_g$ , where $n = \sum _{i=1}^k m_i + n_i$ . Moreover, as $T^n$ is a local homeomorphism, $f = f_g$ is uniquely determined.

Now assume that $g = S^{m_k}T^{n_k} \cdots S^{m_1}T^{n_1} \in \mathcal {S}$ where, without loss of generality, $m_1,\ldots ,m_{k-1} \neq 0$ and $n_2,\ldots ,n_{k} \neq 0$ . We now show how to determine $m_1$ and $n_1$ from f in a unique way.

Case 1. If $m_1 = 0$ , then $k=1$ , $g=T^{n_1}$ and $f = \textrm {id}$ .

Case 2. If $m_1 \neq 0$ and $n_1 \neq 0$ , then $k> 1$ and for $\bar {f}:= f_{S^{m_k}T^{n_k} \cdots S^{m_1}}$ , we have that $f = \Xi _{n_1}(\bar {f})$ . It now follows from equation (9.4) that $\bar {f} - \textrm {id}$ is strictly positive on $[ {1}/{8},{3}/{8}]$ and has zeros in $[1/2,1]$ . Therefore, $n_1$ is determined by the periodicity of $f - \textrm {id}$ , and $\bar {f}(x) = f(2^{n_1})(x)$ . The value of $m_1$ is then determined by applying Case 3 to $S^{m_k}T^{n_k} \cdots S^{m_1}$ and $\bar {f}$ .

Case 3. If $m_1 \neq 0$ and $n_1 =0$ , then $k \geq 1$ and for $\bar {f}:= f_{S^{m_k}T^{n_k} \cdots T^{m_2}}$ , we have that $f = \Xi _{m_1}(\bar {f} \circ U^{-1})\circ U$ or, equivalently, $f\circ U^{-1} = \Xi _{m_1}(\bar {f})$ . Hence, to repeat the above argument based on periodicity, we have to show that the left half of $\bar {f} - \textrm {id}$ is somehow marked. If $k=1$ , then $\bar {f} = U^{-1}$ and, in particular, $\bar {f}$ is strictly negative on $[{3}/{16}, {7}/{16}]$ and has zeros in $[1/2,1]$ . Hence, $m_1$ can be determined through the period of $f\circ U^{-1}$ . However, if $k>1$ , then $n_2> 0$ and the same argument is applicable as equation (9.4) implies that $\bar {f}$ is strictly negative on $[{3}/{16}, {7}/{16}]$ and has zeros in $[1/2,1]$ .

By iterating this procedure, one then recovers $m_2, \ldots , m_k$ and $n_2, \ldots , n_k$ from f. Furthermore, as the $m_i$ and $n_i$ only depend on the period, it follows that the relation between f and these values is one-to-one. This then implies that the map

$$ \begin{align*} \mathcal S \to \{f_g : g \in \mathcal{S}\}, \quad (w_1 \ldots w_n) \mapsto f_{w_n \circ \cdots \circ w_1} \end{align*} $$

is a bijection, and, as an immediate corollary, $\mathcal {S}$ is a free semigroup.

The associated measures of maximal entropy. Now fix a Hölder function h, an element $g \in \mathcal {S}$ and let $n\in \mathbb {N}$ be given by $g = T^n\circ f_g$ . Then the Ruelle operators $L_g$ and $L_T$ associated to g and T, respectively, satisfy

$$ \begin{align*} L_g(h)(x) & = \sum_{g(y)=x} h(y) = \sum_{T^n z=x} h(f_g^{-1}(z)) = L_T^n(h\circ f_g^{-1})(x),\\ \frac{L_g(h L_g(\mathbf{1}))}{L_{g^2}(\mathbf{1})} & = \frac{L_g(4^n h )}{4^{2n}} = \frac{1}{4^n} L_T^n(h\circ f^{-1}). \end{align*} $$

By Proposition 6.3, the measures of maximal entropy $\mu _g$ and $\mu _T$ of g and T, respectively, satisfy $\overline {W}(\mu _g,\mu _T\circ f_g) \ll s^n$ . Hence, $\mu _g = \lim _{l \to \infty } \mu _T\circ f_{g^l}$ . However, this result also implies that for an infinite word $ (v_i) \in \{S,T\}^{\mathbb N}$ , the sequence $ \mu _{g_{v_l \cdots v_1}}$ is a Cauchy sequence and therefore convergent. It remains to show that the mapping from $ (v_i) $ to this limit is injective. To do so, let $ (v_i) \neq (w_i) $ be different elements in $\{S,T\}^{\mathbb N}$ . Then, by applying the construction of the $n_i$ and $m_i$ above to infinite words, it follows that $\mu _{g_{v_l \cdots v_1}} \neq \mu _{g_{w_l \cdots w_1}}$ for all l sufficiently large. Furthermore, it can be deduced from the recursive construction of $f_g$ that there exists an open set A and $\epsilon> 0$ such that $f_{v_l \cdots v_1}(x) - f_{w_l \cdots w_1}(x)> \epsilon $ for all $x \in A$ and all l sufficiently large. Hence, $\lim _l \mu _{g_{v_l \cdots v_1}} \neq \lim _l \mu _{g_{w_l \cdots w_1}}.$ $\Box $