Stability of iterated function systems on the circle

TL;DR

This paper proves that iterated function systems of circle homeomorphisms with at least one dense orbit are asymptotically stable, with the Perron-Frobenius operator satisfying the e-property and establishing a strong law of large numbers.

Contribution

It introduces new stability results for circle homeomorphism systems and demonstrates the e-property and strong law of large numbers for these systems.

Findings

Systems with dense orbit are asymptotically stable

Perron-Frobenius operator satisfies the e-property

Strong law of large numbers holds for trajectories

Abstract

We prove that any Iterated Function System of circle homeomorphisms with at least one of them having dense orbit, is asymptotically stable. The corresponding Perron-Frobenius operator is shown to satisfy the e-property, that is, for any continuous function its iterates are equicontinuous. The Strong Law of Large Numbers for trajectories starting from an arbitrary point for such function systems is also proved.

Equations

$${\int }_{{\mathbf{S}}^1}{P}^{\ast }f(x)\mu (\mathrm{d}x)={\int }_{{\mathbf{S}}^1}f(x)P\mu (\mathrm{d}x)\text{for f∈C(S1), μ∈Mfin.}$$

$$||P^{\ast }(f)||\le ||P^{\ast }(|f|)||\le ||f||,$$

$$\underset{n\to \infty }{\lim }{\int }_{{\mathbf{S}}^1}f(x)P^n\mu (\mathrm{d}x)={\int }_{{\mathbf{S}}^1}f(x){\mu }_{\ast }(\mathrm{d}x)$$

$$\underset{y\to x}{\lim }\underset{n\in \mathbb{N}}{\sup }|P^{\ast n}f(y)-P^{\ast n}f(x)|=0,$$

$$P\mu =\sum _{g\in \Gamma }p(g)\mu \circ g^{-1},$$

$$P^{\ast }f(x)=\sum _{g\in \Gamma }p(g)f(g(x))\text{for f∈C(S1) and x∈S1}$$

$${\xi }_n^f(\omega )={\int }_{{\mathbf{S}}^1}f(g_{i_1,\dots ,i_n}(x))\mu (\mathrm{d}x)\text{for ω=(i1,i2,…)}.$$

$$\underset{n\to \infty }{\lim }{\xi }_n^f(\omega )={\int }_{{\mathbf{S}}^1}f(x)\omega (\mu )(\mathrm{d}x)\text{for every f∈C(S1)}.$$

$$\nu :={\int }_{\Omega }{\delta }_{\upsilon (\omega )}\mathbb{P}(\mathrm{d}\omega ).$$

$${\int }_{{\mathbf{S}}^1}f(x){\mu }_i(\mathrm{d}x)$$

$$\ge \alpha {\int }_{\Omega }f(\upsilon (\omega ))\mathbb{P}(\mathrm{d}\omega )=\alpha {\int }_{{\mathbf{S}}^1}f(x)\nu (\mathrm{d}x)$$

$$h\circ T_r=T_r\circ h.$$

$$(r(z)+\mathrm{d}(z,x))-(r(w)+\mathrm{d}(w,x))=r(z)-r(w)+\mathrm{d}(z,x)-\mathrm{d}(w,x)=r(z)-r(w)+\mathrm{d}(z,w)\ge 0.$$

$$\underset{z\to x^{+}}{\lim }r(z)\ge r(x).$$

$$r(g_1^{-1}(x))\ge r(x),$$

$$P({\mu }_{\ast }\circ R^{-1})(A)=\sum _{i=1}^k{p}_i{\mu }_{\ast }(R^{-1}{g}_i^{-1}(A))$$

$$=\sum _{i=1}^k{p}_i{\mu }_{\ast }(g_{i}R{)}^{-1}(A))=\sum _{i=1}^k{p}_i{\mu }_{\ast }(g_i^{-1}(R^{-1}(A))={\mu }_{\ast }(R^{-1}(A))=({\mu }_{\ast }\circ R^{-1})(A)$$

$$\omega ({\mu }_{\ast })=\frac{1}{M}\sum _{m=0}^{M-1}{\delta }_{R^m(\upsilon (\omega ))},$$

$$U_{m,x}:=\{T_{k\alpha +(m-k)\beta }(x):k=0,\dots m\}=\{T_{k(\alpha -\beta )}(T_{m\beta }(x)):k=0,\dots m\}$$

$$\{T_{k\gamma }(y),k=0,\dots m\}$$

$$P^N{\mu }_i\ge \alpha {\nu }_i\text{for i=1,2}$$

$$\underset{n\to \infty }{\mathrm{limsup}}\left|{\int }_{{\mathbf{S}}^1}f(x)P^n{\nu }_1(\mathrm{d}x)-{\int }_{{\mathbf{S}}^1}f(x)P^n{\nu }_2(\mathrm{d}x)\right|\le \epsilon .$$

$$\left|{\int }_{{\mathbf{S}}^1}f\mathrm{d}{P}^n{\mu }_1-{\int }_{{\mathbf{S}}^1}f\mathrm{d}{P}^n{\mu }_2\right|\le 2\epsilon$$

$$P^{N_1+\cdots +N_k}{\mu }_i$$

$$+\cdots +\alpha (1-\alpha {)}^{k-1}{P}^{N_k}{\nu }_i^k+(1-\alpha {)}^k{\tilde{\mu }}_i^k\text{for i=1,2}$$

$$\underset{n\to \infty }{\mathrm{limsup}}\left|{\int }_{{\mathbf{S}}^1}f(x)P^n{\nu }_1^j(\mathrm{d}x)-{\int }_{{\mathbf{S}}^1}f(x)P^n{\nu }_2^j(\mathrm{d}x)\right|\le \epsilon .$$

$$g_{{\mathbf{i}}_1}(x)<g_{{\mathbf{j}}_1}(y)<g_{{\mathbf{i}}_2}(x)<g_{{\mathbf{j}}_2}(y)<\cdots <g_{{\mathbf{i}}_n}(x)<g_{{\mathbf{j}}_n}(y).$$

$$g_{{\tilde{\mathbf{i}}}_1}(x)<g_{{\tilde{\mathbf{j}}}_1}(y)<g_{{\tilde{\mathbf{i}}}_2}(x)<g_{{\tilde{\mathbf{j}}}_2}(y)<\cdots <g_{{\tilde{\mathbf{j}}}_n}(y)<g_{{\tilde{\mathbf{i}}}_{n+1}}(x).$$

$$g_{{\tilde{\mathbf{i}}}_1}(u)<g_{{\tilde{\mathbf{j}}}_1}(v)<g_{{\tilde{\mathbf{i}}}_2}(u)<g_{{\tilde{\mathbf{j}}}_2}(v)<\cdots <g_{{\tilde{\mathbf{i}}}_n}(u)<g_{{\tilde{\mathbf{j}}}_n}(v).$$

$$x_1<y_1<x_2<y_2<\cdots <x_K<y_K.$$

$$\underset{K\to \infty }{\lim }\underset{n\to \infty }{\mathrm{limsup}}\left|{\int }_{{\mathbf{S}}^1}f(x)P^n{\mu }_1^K(\mathrm{d}x)-{\int }_{{\mathbf{S}}^1}f(x)P^n{\mu }_2^K(\mathrm{d}x)\right|=0.$$

$$\omega ({\mu }_{\ast })(\bigcup _{j=0}^{M-1}{U}_j)=1,$$

Full text

The research partially supported by the Polish NCN grants DEC-2012/07/B/ST1/03320 (T.S.) and 2014/13/B/ST1/04551 (A.Z.). T.S. was also supported by EC Grant RAQUEL

1. Introduction

The action of discrete groups of homeomorphisms (diffeomorphisms) of the circle has been a subject of intensive studies over last decades. See [8] and [16] for a detailed description of recent results.

In this paper, we study the dynamics of discrete, finitely generated semigroups of orientation preserving circle homeomorphisms. After assigning a probability distribution over the set of generators, the system becomes an Iterated Function System. Iterated Function Systems were extensively studied because of their close connections to fractals (see [3]). We study spectral properties of the corresponding Markov operator $P$ and its dual (transfer) operator $P^{*}$ acting on the space $C(\mathbf{S}^{1})$ of continuous functions on the circle. Note that such systems are neither contracting, nor even contracting in average, so the well-known methods elaborated for contracting systems cannot be applied (see [3, 13, 19, 21]).

On the other hand, if, instead of an action of discrete (semi)groups, one considers random maps with some absolutely continuous noise, several strong spectral properties of the corresponding transfer operator, including exponential decay of correlation, can be obtained (see, e.g., [10]). However the case of action of discrete semigroups of homeomorphisms seems to be much more delicate.

We consider arbitrary finitely generated semigroups of orientation preserving homeomorphisms. The only restriction on the system which we assume for all our results is that one of the generators has dense orbits. Such systems will be called admissible.

The paper is organized as follows. In Section 2 we introduce the necessary notation, and we define the objects we deal with. We also introduce the notions of asymptotic stability and the e-property.

Our first, preliminary result is devoted to the uniqueness of an invariant distribution. It is proved in Section 3. We use here some ideas of Furstenberg (see [7]) and Arnold and Crauel (see [2] and the references therein). Recently they have been developed by Kleptsyn et al. in [5] for proving uniqueness of an invariant measure for groups of circle homeomorphisms (see also [4]).

In Section 4 we show some auxiliary properties of semigroups of orientation preserving homeomorphisms which are useful in studying asymptotic stability of our Iterated Function Systems.

Asymptotic stability is proved in Section 5.

Section 6 is devoted to the proof of the e–property. Its usefulness in the study of asymptotic properties of Markov processes may be observed in [12].

Finally, in Section 7 we show that any admissible Iterated Function System on the circle satisfies the Strong Law of Large Numbers (SLLN) for trajectories starting from an arbitrary point.

2. Notation

Let $\mathbf{S}^{1}$ denote the circle with the counterclockwise orientation and let ${\rm d}(x,y)$ denote the normalized distance between $x,y\in\mathbf{S}^{1}$, so that the length of the circle equals $1$. If $I$ is an arc in $\mathbf{S}^{1}$ then $|I|$ denotes the (normalized) length of $I$. Let $x,y\in\mathbf{S}^{1}$ with ${\rm d}(x,y)<1/2$. By $[x,y]$ we denote the oriented (shorter) arc in $\mathbf{S}^{1}$, from $x$ to $y$. We define the following relation on $\mathbf{S}^{1}\times\mathbf{S}^{1}$: we say that $x<y$ if ${\rm d}(x,y)<1/2$ and the orientation of the arc $[x,y]$ coincides with the natural orientation on the circle. Writing $x_{1}<x_{2}<\dots<x_{M}$ we assume that ${\rm d}(x_{1},x_{M})<1/2$ and $x_{i}<x_{i+1},i=1,\dots M-1$.

By $\mathcal{B}(\mathbf{S}^{1})$ we denote the $\sigma$–algebra of Borel sets. Further, $C(\mathbf{S}^{1})$ denotes the space of all continuous functions equipped with the supremum norm $\|\cdot\|$. By $H^{+}$ we shall denote the set of all orientation preserving circle homeomorphisms.

By $\mathcal{M}_{1}$ and $\mathcal{M}_{fin}$ we denote the set of all Borel probability measures and Borel finite measures on $\mathbf{S}^{1}$, respectively. By $\operatorname{supp}\mu$ for $\mu\in\mathcal{M}_{fin}$ we denote the support of $\mu$.

An operator $P:\mathcal{M}_{fin}\to\mathcal{M}_{fin}$ is called a Markov operator if it satisfies the following two conditions:

• •

positive linearity: $P(\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2})=\lambda_{1}P\mu_{1}+\lambda_{2}P\mu_{2}$ for $\lambda_{1},\lambda_{2}\geq 0$; $\mu_{1},\mu_{2}\in\mathcal{M}_{fin}$;

• •

preservation of the norm: $P\mu(\mathbf{S}^{1})=\mu(\mathbf{S}^{1})$ for $\mu\in\mathcal{M}_{fin}$.

A Markov operator $P$ is called a Feller operator if there is a linear operator $P^{*}:C(\mathbf{S}^{1})\to C(\mathbf{S}^{1})$ (dual to $P$) such that

[TABLE]

Note that, if such an operator exists, then $P^{*}(1\!\!1)=1\!\!1$, hence $P^{*}(f)\geq 0$ if $f\geq 0$. As a consequence,

[TABLE]

so $P^{*}$ is a continuous operator. A measure $\mu_{*}$ is called invariant if $P\mu_{*}=\mu_{*}$. Since $\mathbf{S}^{1}$ is compact, every Feller operator on $\mathbf{S}^{1}$ has at least one invariant measure, by the Krylov–Bogolyubov theorem.

A Markov operator $P$ is called asymptotically stable if there exists a unique invariant measure $\mu_{*}\in\mathcal{M}_{1}$ such that

[TABLE]

for $f\in C(\mathbf{S}^{1})$ and $\mu\in\mathcal{M}_{1}$.

Following [12], we say that a Feller operator $P$ satisfies the e–property if for any $x\in\mathbf{S}^{1}$ and a continuous function $f:\mathbf{S}^{1}\to\mathbb{R}$ we have

[TABLE]

i.e. if the family of iterates $\{P^{*n}(f):n\in\mathbb{N}\}$ is equicontinuous. An equivalent notion describing the e-property is the so–called almost periodicity of the dual operator $P^{*}$. Recall that a bounded linear operator $Q:F\to F$ of a Banach space is called almost periodic if for every $b\in F$ the sequence $(Q^{n}(b))_{n\in\mathbb{N}}$ is relatively compact, that is, its closure in $F$ is compact in the norm topology. See, e.g., [14] or [17], for a description of spectral properties of almost periodic operators.

Let $\Gamma=\{g_{1},\dots,g_{k}\}\subset H^{+}$ be a finite collection of homeomorphisms, and let $(p_{1},\dots p_{k})$ be a probability distribution on $\{1,\dots,k\}$. Clearly, it defines a probability distrubution $p$ on $\Gamma$, by putting $p(g_{j})=p_{j}$. We assume that all $p_{i}$’s are strictly positive. Put $\Sigma_{n}=\{1,\dots,k\}^{n}$, and let $\Sigma_{*}=\bigcup_{n=1}^{\infty}\Sigma_{n}$ be the collection of all finite words with entries from $\{1,\dots,k\}$. For a sequence ${\bf i}\in\Sigma_{*}$, ${\bf i}=(i_{1},\dots,i_{n})$, we denote by $|{\bf i}|$ its length (equal to $n$). Finally, denote by $\Omega$ the infinite product $\Omega=\{1,\dots,k\}^{\mathbb{N}}$. Let $\mathbb{P}$ be the product measure distribution on $\Omega$ generated by the initial distribution on $\{1,.\dots,k\}$.

We consider the action of the semigroup generated by $\Gamma$, i.e., the action of all compositions $g_{\bf i}=g_{i_{n},i_{n-1},\ldots,i_{1}}=g_{i_{n}}\circ g_{i_{n-1}}\circ\cdots\circ g_{i_{1}}$, where ${\bf i}=(i_{1},\dots i_{n})\in\Sigma_{*}$. The action of $\Gamma=\{g_{1},g_{2},\ldots,g_{k}\}$ is said to be equicontinuous if the family of homeomorphisms $\{g_{\bf i}\}_{{\bf i}\in\Sigma_{*}}$ is equicontinuous. On the other hand, it is said to be contractive if for each $x\in\mathbf{S}^{1}$, there exists an open interval $I\subset\mathbf{S}^{1}$ containing $x$ and a sequence $({\bf i}_{m})_{m\in\mathbb{N}}$ of elements of $\Sigma_{*}$ such that the length of the intervals $g_{{\bf i}_{m}}(I)$ tends to [math] as $m\to\infty$.

The pair $(\Gamma,p)$ will be called an Iterated Function System. The Markov operator $P:\mathcal{M}_{fin}\to\mathcal{M}_{fin}$ of the form

[TABLE]

where $\mu\circ g^{-1}(A)=\mu(g^{-1}(A))$ for $A\in\mathcal{B}(\mathbf{S}^{1})$, describes the evolution of distribution due to action of randomly chosen homeomorphisms from the collection $\Gamma$. It is a Feller operator, i.e., the operator $P^{*}:C(\mathbf{S}^{1})\to C(\mathbf{S}^{1})$ given by the formula

[TABLE]

is its dual.

We will say that an Iterated Function System $(\Gamma,p)$ is asymptotically stable if the corresponding Markov operator $P$ is asymptotically stable.

3. Uniqueness of an invariant measure

The results in this section are in the spirit of [5] (see also [7]) but we have to point out one substantial difference. Deroin et al. studied a group of circle homeomorphisms so that their family of transformations was richer than in our case. In particular for any homeomorphism the inverse of it, and, of course, the identity, belonged to the class. Since we do not assume this, we have to slightly strengthen our assumption. Indeed, instead of assuming that the action of a semigroup is minimal, we suppose that the system is admissible. The system $\Gamma=\{g_{1},g_{2},\ldots,g_{k}\}\subset H^{+}$ is said to be admissible if one of homeomorphisms, say $g_{1}$, is such that $\{g_{1}^{n}(x):n\geq 1\}$ is dense in $\mathbf{S}^{1}$ for some (and thus all) $x\in\mathbf{S}^{1}$. From now on we shall assume that if $\Gamma$ is admissible, then $g_{1}$ has a dense trajectory. If $\Gamma$ is admissible, then the pair $(\Gamma,p)$ will be called an admissible Iterated Function System.

We are in a position to formulate the main result of this section.

Proposition 1**.**

Let $\Gamma=\{g_{1},g_{2},\ldots,g_{k}\}\subset H^{+}$ be admissible. Then the operator $P$ corresponding to $(\Gamma,p)$ admits a unique invariant measure for any distribution $p$.

Before giving the proof of Proposition 1 we show the following lemma.

Lemma 2**.**

Let $\Gamma=\{g_{1},g_{2},\ldots,g_{k}\}\subset H^{+}$ be admissible. Then the action of $\Gamma$ is either equicontinuous or contractive.

*Proof * We start with a simple observation. Namely, if we assume that $\Gamma$ is admissible, then it is topologically conjugated to some $\tilde{\Gamma}\subset H^{+}$ such that an irrational rotation $\tilde{g}\in\tilde{\Gamma}$ is the conjugate of $g$. Thus, there is no loss of generality in assuming that $\Gamma$ contains an irrational rotation; say, $g_{1}$ is an irrational rotation.

Observe that if ${\rm d}(g_{i}(x),g_{i}(y))={\rm d}(x,y)$ for $x,y\in\mathbf{S}^{1}$ and $i=1,\ldots,k$, then the action of $\Gamma$ is equicontinuous. Otherwise there is an arc $I=[a,b]\subset\mathbf{S}^{1}$ such that $|g(I)|<|I|$. Choose $\alpha\in(0,1)$ such that $|g(I)|<\alpha|I|$. Let $\varepsilon>0$ be such that for $I^{\prime}=[a,b+2\varepsilon]$ we have $|g(I^{\prime})|\leq\alpha|I|$. Fix $x\in\mathbf{S}^{1}$ and let $J$ be an arbitrary arc with $|J|\leq\varepsilon$. Since $g_{1}$ is an irrational rotation, there exist $m$ and $n_{1},\ldots n_{m}\in\mathbb{N}$ such that $g_{1}^{n_{i}}(J)\subset I^{\prime}$ and $g_{1}^{n_{i}}(J)\cap g_{1}^{n_{j}}(J)=\emptyset$ for $i,j\in\{1,\ldots,m\}$ and, moreover, $\sum_{i=1}^{m}|g_{1}^{n_{i}}(J)|=m|J|\geq|I|$. Further $g\circ g_{1}^{n_{1}}(J)\cup\ldots\cup g\circ g_{1}^{n_{m}}(J)\subset g(I^{\prime})$ and consequently $\sum_{j=1}^{m}|g\circ g_{1}^{n_{j}}(J)|\leq\alpha|I|$. Hence $|g\circ g_{1}^{n_{j}}(J)|\leq\alpha|J|$ for some $j\in\{1,\ldots,m\}$. Otherwise, we would have $m\alpha|J|<\sum_{j=1}^{m}|g\circ g_{1}^{n_{j}}(J)|\leq\alpha|I|$, which is impossible. By induction we show that there is a sequence $(i_{1},i_{2},\ldots)\in\Sigma_{\infty}$ such that $|g_{i_{n},i_{n-1},\cdots,i_{1}}(J)|\to 0$ as $n\to\infty$ and we are done.   $\bullet$

*Proof * of Proposition 1. As in the proof of Lemma 2, we can assume that $\Gamma$ contains an irrational rotation.

Let $\Gamma=\{g_{1},\ldots,g_{k}\}$ and let $g_{1}$ be an irrational rotation. If $\Gamma$ is equicontinuous, then the operator $P$ satisfies the e–property. Markov operators with the e–property have been already examined even in the setting of much more general phase spaces, i.e., general Polish spaces (see [11]). In particular, it was proved that such operators may have two different invariant measures $\mu_{1}$ and $\mu_{2}$ only if $\operatorname{supp}\mu_{1}\cap\operatorname{supp}\mu_{2}=\emptyset$ (see Theorem 1 in [11]). This may not be the case if $g_{1}$ is an irrational rotation for the support of any invariant measure is then equal to $\mathbf{S}^{1}$.

Now let $\Gamma$ be contractive. Assume that uniqueness does not hold. Then there exists at least two different ergodic invariant measures. Again, since $\Gamma$ contains an irrational rotation $g_{1}$, every invariant measure is supported on the whole circle $\mathbf{S}^{1}$. Let $\mu_{1}$ and $\mu_{2}$ be two ergodic invariant measures. We shall prove that there exists a positive constant $\alpha$ and a measure $\nu$ such that $\mu_{i}\geq\alpha\nu$ for $i=1,2$. Hence $\mu_{1}=\mu_{2}$ for the fact that two different ergodic invariant measures are mutually singular. Let $J\subset\mathbf{S}^{1}$ be such that $|g_{{\bf i}_{m}}(J)|\to 0$ as $m\to\infty$ for some sequence $({\bf i}_{m})_{m\in\mathbb{N}}$ of elements of $\Sigma_{*}$. Put $\alpha:=\min\{\mu_{1}(J),\mu_{2}(J)\}>0$.

Fix $\mu\in\{\mu_{1},\mu_{2}\}$. Fix $f\in C(\mathbf{S}^{1})$. We define a sequence of random variables $(\xi_{n}^{f})_{n\in\mathbb{N}}$ by the formula

[TABLE]

Since $\mu$ is an invariant measure for $P$, we easily check that $(\xi_{n}^{f})_{n\in\mathbb{N}}$ is a bounded martingale. Note that this martingale depends on the measure $\mu$. From the Martingale Convergence Theorem it follows that $(\xi_{n}^{f})_{n\in\mathbb{N}}$ is convergent $\mathbb{P}$-a.s. and since the space $C(\mathbf{S}^{1})$ is separable, there exists a subset $\Omega_{0}$ of $\Omega$ with $\mathbb{P}(\Omega_{0})=1$ such that $(\xi_{n}^{f}(\omega))_{n\in\mathbb{N}}$ is convergent for any $f\in C(\mathbf{S}^{1})$ and $\omega\in\Omega_{0}$. Therefore for any $\omega\in\Omega_{0}$ there exists a measure $\omega(\mu)\in\mathcal{M}_{1}$ such that

[TABLE]

Now we are ready to show that for any $\varepsilon>0$ there exists $\Omega_{\varepsilon}\subset\Omega$ with $\mathbb{P}(\Omega_{\varepsilon})=1$ satisfying the following property: for every $\omega\in\Omega_{\varepsilon}$ there exists an interval $I$ of length $|I|\leq\varepsilon$ such that $\omega(\mu_{i})(I)\geq\alpha$ for $i=1,2$.

Assume that this fact is proved. Then, using additionally the compactness of $\mathbf{S}^{1}$, we obtain ($\mathbb{P}$ a.s.) that there exists a point $\upsilon(\omega)\in\mathbf{S}^{1}$ such that $\omega(\mu_{i})\geq\alpha\delta_{\upsilon(\omega)}$ for $i=1,2$, where $\delta_{\upsilon(\omega)}$ is the Diract delta measure supported at $\upsilon(\omega)$. It is standard to show that the points $\upsilon(\omega)$, $\omega\in\Omega$, may be chosen in such a way that the function $\Omega\ni\omega\to\upsilon(\omega)\in\mathbf{S}^{1}$ is measurable. This will finish our proof. Indeed, define the measure $\nu\in\mathcal{M}_{1}$ by the formula

[TABLE]

and fix a non–negative function $f\in C(\mathbf{S}^{1})$. We have

[TABLE]

for $i=1,2$. Since $f\in C(\mathbf{S}^{1})$ was an arbitrary non–negative continuous function, we obtain that $\mu_{i}\geq\alpha\nu$ for $i=1,2$.

We now complete our proof by constructing the claimed set $\Omega_{\varepsilon}$. This follows some ideas of Deroin et al. (see [5]). Fix $\varepsilon>0$ and let $l\in\mathbb{N}$ be such that $2/l<\varepsilon$. Since $|g_{{\bf i}_{m}}(J)|\to 0$, and $\Gamma$ contains an irrational rotation, we can require additionally (modifying the sequence ${\bf i}_{m}$ if necessary) that the arcs $J_{m}:=g_{{\bf i}_{m}}(J)$, where $m\leq l$ are mutually disjoint. Put $n^{*}=\max_{m\leq l}|{\bf i}_{m}|$. Now observe that for any sequence ${\bf j}=(j_{1},\ldots,j_{n})\in\Sigma_{*}$ there exists $m\in\{1,\ldots,l\}$ such that $|g_{\bf j}(J_{m})|<1/l<\varepsilon/2$. This shows that for any cylinder in $\Omega$, defined by fixing the first initial $n$ entries $(j_{1},\ldots,j_{n})$, the conditional probability that $(j_{1},\ldots,j_{n},\ldots,j_{n+k})$ are such that $|g_{j_{1},\ldots,{j_{n}},\ldots,j_{n+k}}(J)|\geq\varepsilon$ for all $k=1,\ldots,n^{*}$ is less than $1-q$ for some $q>0$. Hence there exists $\Omega_{\varepsilon}\subset\Omega$ with $\mathbb{P}(\Omega_{\varepsilon})=1$ such that for all $(j_{1},j_{2},\ldots)\in\Omega_{\varepsilon}$ we have $|g_{j_{1},\ldots,j_{n}}(J)|<\varepsilon/2$ for infinitely many $n$’s. Since $\mathbf{S}^{1}$ is compact, we may additionally assume that for infinitely many $n$’s the set $g_{j_{1},\ldots,j_{n}}(J)$ is contained in some set $I$ with $|I|\leq\varepsilon$. This finishes the proof.   $\bullet$

4. Auxiliary results

We start with the following lemma. Recall that we have normalized the arc length so that the length of the circle is equal to $1$.

Lemma 3**.**

Let $h$ be a circle orientation preserving homeomorphism. Assume that there exists $r<1$ such that $h$ maps every arc of length $r$ onto an arc of length at most $r$. If $r$ is irrational, then the map $h$ is a rotation and if $r$ is rational, then $h$ commutes with the rotation by $r$ (denoted by $T_{r}$).

*Proof * First, assume that $r$ is irrational. Denote by $B$ the set of all $\beta\in(0,1)$ such that $h$ maps every arc of length $\beta$ onto an arc of length at most $\beta$. Then $r\in B$. It is easy to see that if $\beta\in B$ then $\{n\beta\}$ (the fractional part) is also in $B$. Since $r$ is irrational, the set $\{n\cdot r\}$ is dense in $[0,1]$.

By continuity of $h$ this implies that $h$ maps every arc $I$ onto an arc of length at most $|I|$. Thus, $h$ must be an isometry – a rotation.

Now, assume that $r$ is rational, say $r=p/q$. Denote by $T_{r}$ the rotation by $r$. Fix some $x_{0}\in S^{1}$ and the arc $I=[x_{0},T_{r}(x_{0})]$.

Observe that $T_{r}^{q}=\operatorname{id}$ and the arcs $\bigcup_{i=1}^{q-1}T_{r}^{i}(I)$ cover the circle $\mathbf{S}^{1}$ exactly $p$ times. The same must be true for $\bigcup_{i=1}^{q-1}h(T^{i}_{r}(I))$. Since the length of each arc $h(T^{i}_{r}(I))$ is not bigger than the length of $T^{i}_{r}(I)$, it implies that $h$ maps every arc of length $r=p/q$ onto an arc of the same length $r=p/q$. Consequently, $h$ and $T_{r}$ commute:

[TABLE]

$\bullet$

Proposition 4**.**

Let $\Gamma$ be contractive. Then there exists a rational rotation $R$ which commutes with the elements of $\Gamma$ and such that for any $x\in\mathbf{S}^{1}$ and any closed arc $I\subset[x,R(x))$ there exists a sequence $({\bf i}_{n})_{n\in\mathbb{N}}$ of elements of $\Sigma_{*}$ such that $|g_{{\bf i}_{n}}(I)|\to 0$ as $n\to\infty$.

*Proof * We will call an arc $I$ contractible if there exists a sequence $({\bf i}_{n})_{n\in\mathbb{N}}$ of elements of $\Sigma_{*}$ such that $|g_{{\bf i}_{n}}(I)|\to 0$ as $n\to\infty$. For $x\in\mathbf{S}^{1}$ we define $R(x)$ in the following way. Consider all positively oriented arcs $[x,y]$ such that the arc $[x,y]$ is also contractible. Clearly, if $[x,y]\subset[x,y^{\prime}]$ and $[x,y^{\prime}]$ is contractible then $[x,y]$ is contractible. Keeping $x$ fixed, take the union of all contractible arcs $[x,y]$. This is an arc of length at least, say, $\varepsilon>0$. One of its endpoints is $x$; the other endpoint is, by definition, $R(x)$. Note that it may happen that $R(x)=x$ (in this case the length of $[x,R(x)]$ is equal to the length of the circle).

For $x\in\mathbf{S}^{1}$ denote by $r(x)$ the length of the (positively oriented) arc $[x,R(x)]$. Now, let $x,z,w\in\mathbf{S}^{1}$; we assume that $x<w<z$. Then $r(z)\geq r(w)-{\rm d}(z,w)$ and $r(w)\geq r(x)-{\rm d}(x,w)$. Consequently:

[TABLE]

Putting $\tilde{r}(z):=r(z)+{\rm d}(x,z)$, we thus have $\tilde{r}(z)\geq\tilde{r}(w)\geq\tilde{r}(x)=r(x)$. Clearly, this implies that the function $r$ is Borel measurable. Moreover, there exists a limit $\lim_{z\to x^{+}}\tilde{r}(z)$, which satisfies $\lim_{z\to x^{+}}\tilde{r}(z)\geq r(x)$, and, consequently, there exists a limit

[TABLE]

Similarly, there exists a limit $\lim_{z\to x^{-}}r(z)\leq r(x)$. Obviously, if an arc $I$ is contractible then for every $g_{i}$, $i\in\{1,\ldots,k\}$, the arc $g_{i}^{-1}(I)$ is also contractible. Thus

[TABLE]

where, recall, $g_{1}$ is an irrational rotation. Since $g_{1}$ is ergodic with respect to the Lebesgue measure, formula (1) implies that the function $r$ is constant (Lebesgue) almost everywhere in $\mathbf{S}^{1}$. Since, for every $x$, we have $\lim_{z\to x^{+}}r(z)\geq r(x)$, and $\lim_{z\to x^{-}}r(z)\leq r(x)$, both limits must be equal to $r(x)$, so the function $r$ is continuous and, since it is $g_{1}$- invariant, it is constant everywhere. So $r(x)\equiv r$ for some $r>0$. Consequently, for every $x\in\mathbf{S}^{1}$ the length of the arc $[x,R(x)]$ is equal to $r$.

Now, choose $g\in\Gamma$ such that $g$ is not a rotation. If an arc $I$ is contractible then $g^{-1}(I)$ is also contractible. Hence, $g^{-1}$ maps any arc of length $r$ onto an arc of length at most $r$. Therefore, $r$ is rational, by Lemma 3. From Lemma 3 we conclude also that either $r=1$ and then $R=\operatorname{id}$ or $r<1$ and all elements of $\Gamma$ commute with $R:=T_{r}$. The proof is complete.   $\bullet$

Lemma 5**.**

Let $\Gamma$ be contractive and let $R$ be a rational rotation that commutes with $\Gamma$. Then the unique invariant measure $\mu_{*}$ is $R$–invariant, i.e. $\mu_{*}\circ R^{-1}=\mu_{*}$.

*Proof * We have

[TABLE]

for any Borel set $A$. Since $P$ possesses a unique invariant measure we conclude that $\mu_{*}\circ R^{-1}=\mu_{*}$.   $\bullet$

The above lemma and the proof of Proposition 1 easily imply the following.

Proposition 6**.**

The measure $\omega(\mu_{*})$ for $\omega=(i_{1},i_{2},\ldots)$ defined $\mathbb{P}$-a.s. is $R$–invariant and, consequently,

[TABLE]

where $\upsilon(\omega)$ are the points defined in the proof of Proposition 1 and $M$ is the order of the rotation $R$, i.e., the smallest integer such that $R^{M}={\rm Id}$.

5. Stability

We start with an easy criterion for stability when $\Gamma=\{g_{1},\ldots,g_{k}\}$ is equicontinuous.

Theorem 7**.**

Let $\Gamma=\{g_{1},\ldots,g_{k}\}$ be equicontinuous and let a probability distribution $p$ be given. Let $P$ be the Markov operator corresponding to $(\Gamma,p)$ and $P^{*}$ its dual. If for any $f\in C({\bf S}^{1})$, $f\geq 0$ and $f\not\equiv 0$, there exists $r\in\mathbb{N}$ such that $P^{*r}f(x)>0$ for $x\in{\bf S}^{1}$, then the Iterated Function System $(\Gamma,p)$ is asymptotically stable.

The proof follows from the results proved for almost periodic primitive operators (for details see Theorem 5.5.3 in [17]. See also Theorem 6 in [20]).

Remark 8**.**

Assume now that $\Gamma$ contains two rotations $g_{1}=T_{\alpha}$ and $g_{2}=T_{\beta}$ such that $(\alpha-\beta)$ is irrational. Then there exists $r\in\mathbb{N}$ such that $P^{*r}f(x)>0$ for $x\in{\bf S}^{1}$ and consequently the corresponding Iterated Function System $(\Gamma,p)$ is asymptotically stable for any probability distribution $p$.

*Proof * Define the set

[TABLE]

for $x\in{\mathbf{S}}^{1}$ and $m\in\mathbb{N}$. Fix $f\in C({\mathbf{S}}^{1})$ and observe that to prove that $P^{*r}f(x)>0$ for $x\in{\mathbf{S}}^{1}$ and some $r\in\mathbb{N}$ it is enough to show that for every $\varepsilon>0$ and $x\in{\mathbf{S}}^{1}$ there is $m\in\mathbb{N}$ such that $U_{m,x}$ forms an $\varepsilon$–net in $\mathbf{S}^{1}$. Indeed, since $f$ is continuous and $f\not\equiv 0$, there exists $\varepsilon>0$ such that for any $\varepsilon$–net there is $y$ from this net such that $f(y)>0$. Further, if $U_{m,x}$ is some $\varepsilon$–net, then $f(T_{i\alpha+(m-i)\beta}(x))>0$ for some $i\in\{0,\ldots,m\}$ and consequently $P^{*m}f(x)\geq p_{1}^{i}p_{2}^{m-i}f(T_{i\alpha+(m-i)\beta}(x))>0$.

Denoting now by $\gamma:=\alpha-\beta$, the problem reduces to the following immediate observation. Assume that $\gamma$ is irrational. Then for every $\varepsilon>0$ there exists $m\in\mathbb{N}$ such that for every $y\in\mathbf{S}^{1}$ the set

[TABLE]

forms an $\varepsilon$–net in $\mathbf{S}^{1}$.   $\bullet$

Now, assume that $\Gamma$ is equicontinuous, and let $g_{1}\in\Gamma$ be a hoemomorphism with dense orbits. After the appropriate change of variables we can assume that $g_{1}$ is an irrational rotation. Using Lemma  2 we see that all elements of $\Gamma$ are rotations. So we can reformulate Remark  8 as follows.

Remark 9**.**

Assume that $\Gamma$ contains an element with dense orbits, and that $\Gamma$ is equicontinuous. If, for some $g_{i},g_{j}\in\Gamma$ the homeomorphism $g_{i}^{-1}\circ g_{j}$ has dense orbits then for any probability distribution $p$ the Iterated Function System $(\Gamma,p)$ is asymptotically stable.

The following lower bound criterion for stability of Markov operators generalizing Doeblin’s theorem (see [6]) will be useful in proving stability when the family $\Gamma$ is contractive.

Theorem 10**.**

Let $P$ be an arbitrary Markov operator. Assume that for any $\varepsilon>0$ and $f\in C(\mathbf{S}^{1})$ there exists $\alpha>0$ such that for every $\mu_{1},\mu_{2}\in\mathcal{M}_{1}(\mathbf{S}^{1})$ there are $\nu_{1},\nu_{2}\in\mathcal{M}_{1}(\mathbf{S}^{1})$ and $N\in\mathbb{N}$ satisfying

[TABLE]

and

[TABLE]

Then the operator $P$ is asymptotically stable.

*Proof * Fix $\varepsilon>0$ and $f\in C(\mathbf{S}^{1})$. Fix $\mu_{1},\mu_{2}\in\mathcal{M}_{1}(\mathbf{S}^{1})$. We shall show that

[TABLE]

for all $n$ sufficiently large. To do this choose $k\in\mathbb{N}$ such that $2(1-\alpha)^{k}\|f\|<\varepsilon$, where $\alpha>0$ is such that condition (2) holds for given $\varepsilon$ and $f$. Using (2) we have $P^{N_{1}}\mu_{i}=\alpha\nu_{i}+(1-\alpha)\tilde{\mu}_{i}^{1}$, where $\tilde{\mu}_{i}^{1}$ are some probability measures. Proceeding inductively, and using (2) at every step we may find sequences of probability measures $\nu_{i}^{1},\ldots,\nu_{i}^{k}$ and $\tilde{\mu}_{i}^{k}$ for $i=1,2$ and a sequence of positive integers $N_{1},\ldots,N_{k}$ such that

[TABLE]

and

[TABLE]

for $j=1,\ldots,k$ (for details see Theorem 5.3 in [18]). Hence, by the definition of $k$, condition (3) follows. Since $P$ admits an invariant measure, the proof is complete.   $\bullet$

Lemma 11**.**

Let $\Gamma=\{g_{1},\ldots,g_{k}\}$ be admissible and contractive. For any $n\in\mathbb{N}$ and $x,y\in\mathbf{S}^{1}$ there exists $N\in\mathbb{N}$ and two collections of elements of $\Sigma_{N}$: ${\bf i}_{1},\dots,{\bf i}_{n}$, ${\bf j}_{1},\dots,{\bf j}_{n}$ such that

[TABLE]

Moreover, one can require that the length of the arcs $(g_{{\bf i}_{m}}(x),g_{{\bf j}_{m}}(y))$ and $(g_{{\bf j}_{m}}(y),g_{{\bf i}_{m+1}}(x))$ for $m=1,\ldots,n$ is bigger than some constant, say $\tau$, depending on $n$ but independent of $x$ and $y$.

*Proof * After a necessary change of variables one can assume that $g_{1}\in\Gamma$ is an irrational rotation. Fix $x,y\in\mathbf{S}^{1}$ and let $I$ be such an arc that $x\in I$ and $|g_{{\bf l}_{m}}(I)|\to 0$ as $m\to\infty$ for some sequence $({\bf l}_{m})_{m\in\mathbb{N}}$ of elements of $\Sigma_{*}$. Since $g_{1}$ is an irrational rotation, we can additionally assume that there is an open arc $I_{0}$, with $I\setminus{\rm cl}I_{0}\neq\emptyset$ such that $g_{{\bf l}_{m}}(I)\subset I_{0}$ for all $m\in\mathbb{N}$. We shall proceed by induction on $n$. The case $n=1$ is obvious but we show that we may additionally require that $g_{{\bf i}_{1}}(x),g_{{\bf j}_{1}}(y)\in I$. Indeed, set $h_{1}:=g_{{\bf l}_{m}}$, $h_{2}=g_{1}^{|{\bf l}_{m}|}$ and observe that $h_{2}$ is an irrational rotation again. Hence there exists $l\in\mathbb{N}$ such that $h_{2}^{l}(y)\in I\setminus\operatorname{cl}I_{0}$ and $h_{2}^{l}(y)>z$ for every $z\in I_{0}$. Since $h_{1}^{l}(x)\in I_{0}$, we are done.

Now let the statement of our lemma hold for some $n$. Denote by $\tilde{I}=(g_{{\bf i}_{1}}(x),g_{{\bf j}_{n}}(y))\subset I$. We have $h_{1}(\tilde{I})\subset I_{0}$. Analogously as in the previous step we find $l$ such that $h_{2}^{l}(x)\in I\setminus\operatorname{cl}I_{0}$ and $h_{2}^{l}(x)>z$ for every $z\in I_{0}$. Hence there exist $N$ and $\tilde{\bf i}_{m}\in\Sigma_{N}$, $m=1,\ldots,n+1$, $\tilde{\bf j}_{m}\in\Sigma_{N}$, $m=1,\ldots,n$ such that

[TABLE]

Replacing $\tilde{I}$ with $\hat{I}=(g_{{\bf i}_{1}}(x),g_{{\bf i}_{n+1}}(x))$ and $x$ with $y$, and repeating the above procedure we show our lemma for $n+1$.

Now we observe that the length of the arcs $(g_{{\bf i}_{m}}(x),g_{{\bf j}_{m}}(y))$ and $(g_{{\bf j}_{m}}(y),g_{{\bf i}_{m+1}}(x))$ for $m=1,\ldots,n$ may be bigger than some constant $\tau$ (depending on $n$) but independent of $x,y\in\mathbf{S}^{1}$. Fix $z\in\mathbf{S}^{1}$. From what we have proved above it follows that for any $u,v\in\mathbf{S}^{1}$ there exist two sequences $({\bf p}_{n})_{n\in\mathbb{N}}$ and $({\bf q}_{n})_{n\in\mathbb{N}}$ of elements of $\Sigma_{*}$ such that $|{\bf p}_{n}|=|{\bf q}_{n}|$ for $n\in\mathbb{N}$ and ${\rm d}(g_{{\bf p}_{n}}(u),g_{{\bf q}_{n}}(v))\to 0$ as $n\to\infty$. Since $g_{1}$ is an irrational rotation we may assume additionally that $g_{{\bf p}_{n}}(u)\to z$ and $g_{{\bf q}_{n}}(v)\to z$ as $n\to\infty$. Applying now first part of our consideration we obtain that the constant $\tau$, chosen for $x=y=z$, will be also a lower bound for the length of the arcs $(g_{{\bf\tilde{i}}_{m}}(u),g_{{\bf\tilde{j}}_{m}}(v))$ and $(g_{{\bf\tilde{j}}_{m}}(v),g_{{\bf\tilde{i}}_{m+1}}(u))$ for $m=1,\ldots,n$ and some ${\bf\tilde{i}}_{1},\dots,{\bf\tilde{i}}_{n}$, ${\bf\tilde{j}}_{1},\dots,{\bf\tilde{j}}_{n}$ (with the same length) such that

[TABLE]

This completes the proof.   $\bullet$

Lemma 12**.**

Let $\Gamma$ be admissible and contractive and let $(\mu^{K}_{1})_{K\in\mathbb{N}}$ and $(\mu^{K}_{2})_{K\in\mathbb{N}}$ be two sequences of probability distributions such that for any $K\in\mathbb{N}$ the measures $\mu^{K}_{1}$ and $\mu^{K}_{2}$ are uniformly distributed on $\{x_{1},x_{2},\ldots,x_{K}\}\subset\mathbf{S}^{1}$ and $\{y_{1},y_{2}\ldots,y_{K}\}\subset\mathbf{S}^{1}$, respectively and

[TABLE]

Let a probability distribution $p$ be given and let $P$ be the Markov operator corresponding to the Iterated Function System $(\Gamma,p)$. Then for an arbitrary $f\in C(\mathbf{S}^{1})$ we have

[TABLE]

*Proof * Fix $f\in C(\mathbf{S}^{1})$ and let $\omega=(i_{1},i_{2},\ldots)$ be such that $\omega(\mu_{*})$ is defined. Let $z\in{\mathbf{S}^{1}}$ be such $\omega(\mu_{*})=(\delta_{z}+\delta_{R(z)}+\ldots+\delta_{R^{M-1}(z)})/M$, by Proposition 6. Fix $\varepsilon>0$ and let $U_{1},\ldots U_{M-1}$ be mutually disjoint open arcs such that $R^{i}(z)\in U_{i}$ and $|f(u)-f(v)|\leq\varepsilon/2$ for $u,v\in U_{i}$, $i=0,\ldots,M-1$. Fix $K\in\mathbb{N}$. All the arcs $(x_{i},y_{i})$ and $(y_{i},x_{i+1})$ for $i=1,\ldots,K$ have positive $\mu_{*}$ measure, by the fact that the support of $\mu_{*}$ is equal to $\mathbf{S}^{1}$. Since $\mu_{*}\circ(g_{i_{1}}\cdots g_{i_{n}})^{-1}$ converges weakly to $\omega(\mu_{*})$ and

[TABLE]

for any arc $(u,v)$ and $n$ arbitrary large (depending on $\mu_{*}((u,v))$) there exists $w\in(u,v)$ such that $g_{i_{1}}\cdots g_{i_{n}}(w)\in\bigcup_{j=0}^{M-1}U_{j}$. If this is not the case, we obtain that $\omega(\mu_{*})(\bigcup_{j=0}^{M-1}U_{j})\leq 1-\mu_{*}((u,v))$, contrary to condition (5).

Due to the above observation we find points $z^{n}_{i},w^{n}_{i}$ such that

[TABLE]

and

[TABLE]

for $i=1,\ldots,K$ and all $n\geq n_{0}$, where $n_{0}$ depends on the measure $\mu_{*}$ (or: on length) of the arcs $(x_{1},y_{1}),(y_{1},x_{2}),\ldots,(y_{K-1},x_{K}),(x_{K},y_{K})$. In other words, it is independent of locations of the points if the distance between them is bigger than some $\tau>0$. From this it follows that there are at most $2M$ pairs of $(x_{i},y_{i})$ such that $g_{i_{1}}\cdots g_{i_{n}}(x_{i}),g_{i_{1}}\cdots g_{i_{n}}(y_{i})$ are not in the same $U_{j}$. Hence

[TABLE]

for all $n\geq n_{0}$. We may find $n_{0}$ such that the above condition holds for all $\omega$ from some set $\tilde{\Omega}$ with $\mathbb{P}(\tilde{\Omega})\geq 1-\varepsilon/(2M)$. Then we obtain

[TABLE]

for all $n\geq n_{0}$. Taking limit as $K\to\infty$ completes the proof.   $\bullet$

Remark 13**.**

The above proof also shows the following: Fix some $f\in C(\mathbf{S}^{1})$. For every $\varepsilon>0$ and $\tau>0$ there exists $n_{0}=n_{0}(\tau,\varepsilon,f)$ such that

[TABLE]

for $n\geq n_{0}$ and any probability measures $\mu_{1}^{K},\mu_{2}^{K}$ distributed uniformly on $\{x_{1},x_{2},\ldots,x_{K}\}$ and $\{y_{1},y_{2}\ldots,y_{K}\}$, respectively, such that

[TABLE]

and the length (measure $\mu_{*}$) of the arcs $(x_{1},y_{1}),(y_{1},x_{2}),\ldots,(y_{K-1},x_{K}),(x_{K},y_{K})$ is bounded from below by $\tau>0$ (Here and below $M$ denotes the constant defined in Proposition 6).

Theorem 14**.**

Let $\Gamma=\{g_{1},\ldots,g_{k}\}$ be admissible and contractive and let $p$ be a probability distribution. Then the Iterated Function System $(\Gamma,p)$ is asymptotically stable.

*Proof * Recall that we may assume that $g_{1}$ is an irrational rotation. We are going to show that the assumptions of Theorem 10 are satisfied. Fix $\varepsilon>0$ and let $K\in\mathbb{N}$ be so large that

[TABLE]

Let $(x,y)\in{\mathbf{S}^{1}}\times{\mathbf{S}^{1}}$. It follows from Lemma 11 that there exists $N=N_{x,y}$ and ${\bf i}_{l},{\bf j}_{l}\in\Sigma_{N}$ for $l=1,\ldots,K$ and such that

[TABLE]

Further, we may find open neighbourhood $U_{x}$, $U_{y}$ of $x,y$, respectively, such that

[TABLE]

Now, choose a collection of pairs $(x_{1},y_{1}),\ldots,(x_{r},y_{r})\in{\mathbf{S}^{1}}\times{\mathbf{S}^{1}}$ such that ${\mathbf{S}^{1}}\times{\mathbf{S}^{1}}=\bigcup_{i=1}^{r}U_{x_{i}}\times U_{y_{i}}$. Fix $\mu_{1},\mu_{2}\in\mathcal{M}_{1}(\mathbf{S}^{1})$. Set $p=:\min_{i}p_{i}$. Take the product measure $\mu_{1}\times\mu_{2}$ and observe that there exists $j\in\{1,\dots,r\}$ such that $(\mu_{1}\times\mu_{2})(U_{x_{j}}\times U_{y_{j}})\geq 1/r$. It is now easy to check that

[TABLE]

and

[TABLE]

where $m_{1}(x)$ and $m_{2}(y)$ are probability measures as in the hypothesis of Lemma 12, i.e., the measures uniformly distributed over the points

[TABLE]

and

[TABLE]

respectively. Set

[TABLE]

and observe that estimates (6) and (7) can be rewritten as:

[TABLE]

with $\alpha:=p^{N}Kr^{-1}$, where $N=\max_{1\leq l\leq r}N_{x_{l},y_{l}}$. Since, by Lemma 12 and Remark 13

[TABLE]

for $(x,y)\in U_{x_{j}}\times U_{y_{j}}$ and $n$ sufficiently large, we obtain that

[TABLE]

This completes the proof.   $\bullet$

6. E–property

The e–property plays an important role in proving asymptotic properties of Markov processes. Usually it is a necessary step when we want to justify that the studied process has a unique invariant measure and is asymptotically stable. Here we inverted the order of our considerations. First we showed the uniqueness of an invariant measure and its stability and now we shall prove independently that our operator also satisfies the e–property. This fact seems to be surprising because the transformations under considerations are neither contractions, nor average contractions. Here the e–property is forced by the geometry.

Proposition 15**.**

Let $\Gamma=\{g_{1},\ldots,g_{k}\}$ be admissible and let $p$ be a probability distribution. Then the operator $P$ corresponding to the Iterated Function System $(\Gamma,p)$ satisfies the e–property.

*Proof * If $\Gamma=\{g_{1},\ldots,g_{k}\}$ is equicontinuous, the e–property follows immediately. So we may assume that $\Gamma$ is contractive. Fix a function $f\in C(\mathbf{S}^{1})$ and $\varepsilon>0$. Let $K\in\mathbb{N}$ be such that $2\|f\|M/K\leq\varepsilon/2$. (Here again $M$ is the constant coming from Proposition  6). Take an arbitrary point $z\in\mathbf{S}^{1}$. From the proof of Theorem 14, applied for both $x:=z$ and $y:=z$ (see also Lemma 11), it follows that we may find $\alpha>0$ and $N_{1},\ldots,N_{m}$ such that $P^{N_{1}+\cdots+N_{m}}\delta_{z}$ admits two representations below:

[TABLE]

where, for every $j=1,\dots m$ the pair of probability measures $(\nu_{1}^{j},\nu_{2}^{j})$ is a convex combination of pairs of measures such that each pair is uniformly distributed over some collections of points $\{x_{1},\ldots,x_{K}\}$ and $\{y_{1},\ldots,y_{K}\}$, respectively, and

[TABLE]

where the length of the arcs $(x_{1},y_{1}),(y_{1},x_{2}),\ldots,(y_{K-1},x_{K}),(x_{K},y_{K})$ is bounded from below by some $\tau>0$ depending only on $K$. Furthermore, $\mu_{1}$, $\mu_{2}$ are some probability measures.

Now, let $m\in\mathbb{N}$ be so large that $(1-\alpha)^{m}\leq\varepsilon(4||f||)^{-1}$. Since every element from $\{x_{1},\ldots,x_{K}\}$ and $\{y_{1},\ldots,y_{K}\}$ (on which the measures defining $\nu_{1}^{1},\nu_{2}^{1},\ldots,\nu_{1}^{m},\nu_{2}^{m}$ are supported) is of the form $g_{\bf j}(z)$ for some ${\bf j}\in\Sigma_{*}$, we may find $\eta>0$ such that for any $w\in\mathbf{S}^{1}$ with ${\rm d}(z,w)<\eta$ we have

[TABLE]

and each pair of probability measures $(\nu_{1}^{j},\tilde{\nu}_{2}^{j})$ is a convex combination of pairs of measures that are uniformly distributed over some collections of points $\{x_{1},\ldots,x_{K}\}$ and $\{\tilde{y}_{1},\ldots,\tilde{y}_{K}\}$, respectively, such that

[TABLE]

and the length of the arcs $(x_{1},\tilde{y}_{1}),(\tilde{y}_{1},x_{2}),\ldots,(\tilde{y}_{K-1},x_{K}),(x_{K},\tilde{y}_{K})$ is bounded from below by $\tau/2$. From Remark 13 it follows now that there exists $n_{0}\in\mathbb{N}$ such that for any $n\geq n_{0}$ and every $j\in\{1,\dots,m\}$ we have

[TABLE]

Hence, we obtain that for any $w\in\mathbf{S}^{1}$ such that ${\rm d}(w,z)<\eta$ we have $|P^{*n}f(z)-P^{*n}f(w)|\leq\varepsilon$ for $n\geq N_{1}+\ldots+N_{m}+n_{0}$ and we are done.   $\bullet$

7. Strong Law of Large Numbers

Let $\Gamma=\{g_{1},\dots g_{k}\}\subset H^{+}$ and let $\tilde{\Gamma}=\{g_{1}^{-1},\dots g_{k}^{-1}\}$. We define the probability distribution on $\tilde{\Gamma}$ by putting $p(g_{i}^{-1})=p_{i}$.

Proposition 16**.**

Let $\Gamma=\{g_{1},\dots g_{k}\}\subset H^{+}$ be admissible and let $p$ be a probability distribution. Then the Strong Law of Large Numbers for trajectories starting from an arbitrary point holds. More precisely: let $\phi\in C(\mathbf{S}^{1})$. For every $x\in\mathbf{S}^{1}$ there exists a subset $\Omega^{\prime}\subset\Omega$ with $\mathbb{P}(\Omega^{\prime})=1$ such that for every $\omega=(i_{1},i_{2},\dots)\in\Omega^{\prime}$

[TABLE]

*Proof * From Lemma 2 it follows that $\Gamma$ is either equicontinuous or contractive. If $\Gamma$ is equicontinuous then the theorem holds simply by the Birkhoff ergodic theorem. Indeed, since the invariant measure $\mu_{*}$ is unique it is also ergodic. From Birkhoff’s theorem it follows then that formula (9) holds for $\mathbb{P}\times\mu_{*}$-almost every pair $(\omega,z)$. Since the support of $\mu_{*}$ is equal to $\mathbf{S}^{1}$, for any $x\in\mathbf{S}^{1}$ and any $k\in\mathbb{N}$ we may find a point $z_{k}\in\mathbf{S}^{1}$ and a set $\Omega_{k}\subset\Omega$ with $\mathbb{P}(\Omega_{k})=1$ such that condition (9) holds with $x$ replaced with $z_{k}$ and $|\phi(g_{i_{n},i_{n-1},\cdots,i_{1}}(x))-\phi(g_{i_{n},i_{n-1},\cdots,i_{1}}(z_{k}))|<1/k$ for $n\in\mathbb{N}$ and $(i_{1},i_{2},\ldots)\in\Omega_{k}$, by the fact that $\Gamma$ is equicontinuous and $\varphi$ is uniformly continuous. Let $\Omega^{\prime}=\bigcap_{k=1}^{\infty}\Omega_{k}$. Since $\mathbb{P}(\Omega^{\prime})=1$ and $x$ satisfies condition (9) for $(i_{1},i_{2},\ldots)\in\Omega^{\prime}$, we are done.

Now assume that $\Gamma$ is contractive and that the theorem does not hold. Then there exists $x\in\mathbf{S}^{1}$ and a set $\Omega_{x}\subset\Omega$ such that $\mathbb{P}(\Omega_{x})>0$ and for every $\omega=(i_{1},i_{2},\dots)\in\Omega_{x}$ formula (9) does not hold. Taking a subset of $\Omega_{x}$, if necessary, we may assume that

[TABLE]

for all $\omega=(i_{1},i_{2},\dots)\in\Omega_{x}$ and some $\varepsilon>0$.

Now consider the system generated by $\tilde{\Gamma}$ and the probability distribution $p$. Let $\tilde{\mu}_{*}$ be its unique invariant measure. Since $\tilde{\mu}_{*}(\{x\})=0$ we may find $\delta>0$ and a subset $\tilde{\Omega}_{x}\subset\Omega_{x}$ with $\mathbb{P}(\tilde{\Omega}_{x})>0$ such that $\operatorname{supp}(\omega(\tilde{\mu}_{*}))\cap(x-\delta,x+\delta)=\emptyset$ for $\omega\in\tilde{\Omega}_{x}$.

Let $\theta>0$ be such that $|\phi(u)-\phi(v)|\leq\varepsilon/2$ for $|u-v|<\theta$. Set $\gamma:=\inf_{x\in\mathbf{S}^{1}}\tilde{\mu}_{*}((x-\theta/2,x+\theta/2))$. Obviously $\gamma>0$. Since $g_{i_{1}}^{-1}\circ g_{i_{2}}^{-1}\circ\cdots\circ g_{i_{n}}^{-1}\circ\tilde{\mu}_{*}$ converges weakly to $\omega(\tilde{\mu}_{*}))$ for $\mathbb{P}$-a.s. $\omega=(i_{1},i_{2},\dots)\in\tilde{\Omega}_{x}$, we have $\tilde{\mu}_{*}((g_{i_{1}}^{-1}\circ g_{i_{2}}^{-1}\circ\cdots\circ g_{i_{n}}^{-1})^{-1}((x-\delta,x+\delta))<\gamma$ for all $n$ sufficiently large. From this and from the definition of $\gamma$ it follows that $|(g_{i_{1}}^{-1}\circ g_{i_{2}}^{-1}\circ\cdots\circ g_{i_{n}}^{-1})^{-1}((x-\delta,x+\delta))|<\theta$ for $n$ sufficiently large and all $\omega=(i_{1},i_{2},\dots)\in\tilde{\Omega}_{x}$. This gives $|g_{i_{n},i_{n-1},\cdots,i_{1}}(u)-g_{i_{n},i_{n-1},\cdots,i_{1}}(v)|\leq\theta$ and consequently

[TABLE]

for all $n$ sufficiently large and $u,v\in(x-\delta,x+\delta)$. Together with condition (10), this gives that

[TABLE]

for $\omega=(i_{1},i_{2},\ldots)\in\tilde{\Omega}_{x}$ and all $u\in(x-\delta,x+\delta)$, contrary to the Birkhoff theorem. This contradiction completes the proof.   $\bullet$

Remark 17**.**

In [1] the authors introduce the notion of an essentially contracting system, prove this property for some special system of piecewise linear maps of an interval, and obtain several interesting consequences. The proof of Proposition 16 shows, in particular, that our system is essentially contracting. Contrary to [1], we do not need special estimates; we simply use the properties of the ”conjugate” system generated by $\tilde{\Gamma}$.

Remark 18**.**

Observe that the known criteria for the Central Limit Theorem and Law of the Iterated Logarithm require more than it was proved in Theorem 14. To apply the results by M. Maxwell and M. Woodroofe (see [15]) we need to know the rate of convergence to the invariant measure. On the other hand, finding new sufficient conditions for the Central Limit Theorem in the setting of considered IFS’s would be an interesting question worthy of further study.