Invariant Measures for Uncountable Random Interval Homeomorphisms

A necessary and sufficient condition for the iterated function system {f(·,ω)|ω∈Ω}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{f(\cdot ,\omega )\,|\,\omega \in \Omega \}$$\end{document} with probability P to have exactly one invariant measure μ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _*$$\end{document} with μ∗((0,1))=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _*((0,1))=1$$\end{document} is given. The main novelty lies in the fact that we only require the transformations f(·,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\cdot ,\omega )$$\end{document} to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}. In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven.


Introduction
In this note we study random one-dimensional systems consisting of increasing homeomorphisms defined on the interval [0, 1]. These systems can be looked upon from various points of view. Here we are interested in the existence of a unique invariant measure on (0, 1) (unique ergodicity). Observe that existence of such a measure may not be obtained by the Krylov-Bogolubov theorem since we are looking for a measure on a non-compact interval. On the other hand, every convex combination of the Dirac Alsedá and Misiurewicz studied similar problem for some function systems consisting of piecewise linear homeomorphisms (see [2]). More general iterated function systems were considered by Gharaei and Homburg in [14]. Recently Malicet obtained unique ergodicity as a consequence of the contraction principle for time homogeneous random walks on the topological group of homeomorphisms defined on the circle and interval (see [17]). In turn, Zdunik and the second author [24] defined a class of iterated function systems (the so-called admissible iterated function systems) satisfying unique ergodicity on (0, 1). All the results were formulated for the systems consisting of finitely many transformations.
Here we give a necessary and sufficient condition for the existence and uniqueness of an invariant measure for more general IFS's with probabilities. Our condition is of the same type as that given in [22]. In fact, this condition is expressed in the language of functional equations. Also the proof of uniqueness is based upon some results on existence of solutions of some functional equations.
The second aim of this paper is to study properties of the considered iterated function system with probabilities such as properties of supports of its invariant measures, asymptotic stability, and strong law of large numbers.
The family F = { f ω | ω ∈ } forms an iterated function system (IFS for short) and the pair (F, P), in which we are interested in this paper, forms an iterated function system with probabilities (IFSP for short). Many authors have widely studied such systems (see [8] and the references therein). It is worth mentioning here that we do not make any assumptions on contractivity or at least local contractivity on average of the system as authors usually do (see e.g. [3,19,20]). The contractivity, for instance, does not hold for homeomorphisms. In our setting we do not assume smoothness of homeomorphisms f ω .
Note that f is a Carathéodory function (i.e., continuous with respect to the first variable and A-measurable with respect to the second one), and hence it is measurable with respect to σ -algebra B ⊗ A, where B denotes the σ -algebra of all Borel subsets of [0, 1] (see [1,Lemma 4.51]). Therefore, f is also a random-valued function (rvfunction for short), which iterates were introduced independently in [5] and [10] for different needs, as follows for all x ∈ [0, 1], n ∈ N and ω = (ω 1 , ω 2 , . . .) ∈ ∞ ; here ∞ is defined as the one-sided shift space on , which plays an essential role in topological and symbolic dynamics (for more details see [23]). Observe that for every n ∈ N the iterate f n : [0, 1] × ∞ → [0, 1] is again an rv-function, but now on the product probability space ( ∞ , A ∞ , P ∞ ). More precisely, the iterate f n is measurable with respect to the product σ -algebra B ⊗ A n , where A n denotes the σ -algebra of all sets of the form is Borel and Moreover, if μ is a Borel probability measure, defined on [0, 1], then the function . This justify to the following definition.
A Borel probability measure μ * , defined on [0, 1], is said to be an invariant measure for the IFSP (F, P), if Following the idea from [22] we introduce the following conditions: (a) for any x ∈ (0, 1) there exists a set x − ∈ A with P( x − ) > 0 and such that f ω (x) < x for every ω ∈ x − or for any x ∈ (0, 1) there exists a set x + ∈ A with P( x + ) > 0 and such that x < f ω (x) for every ω ∈ x + ; (b) there exist ρ 1 , ρ 2 ∈ I which are right-continuous, continuous at 1, such that ρ 1 ≤ ρ 2 and for every x ∈ [0, 1].
Note that if the point x ∈ (0, 1) is a common fixed point of all functions of the family F, then the Dirac measure δ x is an invariant measure for (F, P). In such a case, the search for invariant measures for (F, P) reduces to an independent search for invariant measures for (F| [0,x] , P) and (F| [x,1] , P), where families F| [0,x] and F| [x,1] consist of all functions from F restricted to [0, x] and [x, 1], respectively. Condition (A) excludes this situation. In particular, the invariant measure we are looking for cannot be the Dirac measure of any point of the interval (0, 1). Moreover, as we will see later that condition (A) guarantees the atomlessness of the invariant measure we are interested in, as well as its uniqueness. On the other hand, it is well known that to prove the existence of an invariant measure on (0, 1), some conditions on the behaviour in the neighbourhood of 0 and 1 have to be assumed. Condition It is clear that g ω = f −1 ω for every ω ∈ , and hence g satisfies (H 2 ). To see that g satisfies also (H 1 ) it suffices to fix x, a ∈ [0, 1] and note that Therefore, g is a Carathéodory function as well as an rv-function.
In the proofs of our results it will be more convenient to use the function g instead of f .
Proposition 3.1 says that we have a mutual, one-to-one, correspondence between invariant measures for (F, P) and right-continuous and increasing functions satisfying (3.2) taking value 1 at 1. Using for subsets A ⊂ [0, 1] the notation χ A to denote the characteristic function of A defined on [0, 1], we observe that the function χ [0,1] corresponds to the Dirac measure δ 0 and the function χ {1} corresponds to the Dirac measure δ 1 .
The next observation shows that if ϕ * solves (3.2), then there is a fairly large family of pair of functions ρ 1 , ρ 2 satisfying (B).
(⇐) It is enough to apply assertion (ii) of Proposition 3.1.

Remark 3.5 If (F, P) satisfies (A), then (3.3) holds.
In view of Remark 3.5 we see that Corollary 3.4 says that if there is an invariant measure μ for (F, P) satisfying (A) and such that μ((0, 1)) > 0, then there is an extreme invariant measure μ * for (F, P) such that μ * ((0, 1)) = 1; recall that an extreme invariant measure for (F, P) is an invariant measure for (F, P) that cannot be represented as a convex combination of other invariant measures for (F, P).
We now prove that under condition (A) the considered IFSP has at most three extreme invariant measures. Theorem 3.6 Assume that (F, P) satisfies (A). Then there exists at most one invariant measure μ * for (F, P) with μ * ((0, 1)) = 1.
Proof According to Propositions 3.1 and 3.3, and Remark 3.5, it suffices to prove that there exists at most one continuous function ϕ * : Suppose, on the contrary, that there are two different continuous functions Obviously, a = min S ∈ (0, 1) and b = max S ∈ (0, 1). By (A) and (2.2), at least one of the following cases occurs: there exists a set or there exists set a + with P( a + ) > 0 such that g(a, ω) < a for every ω ∈ a + . (3.5) Note that for every x ∈ S we have M = |ϕ(x)| ≤ |ϕ(g(x, ω))|d P(ω) ≤ M, and hence g(x, ω) ∈ S for P-almost all ω ∈ . In consequence, for P-almost all ω ∈ , which contradicts both (3.4) and (3.5).
We end this section with an application of results obtained up to now.

The Case of Three Extreme Measures
We are now in a position to prove the main results of this paper. For this purpose define the operator T : I → I by for every x ∈ [0, 1].
It turns out that condition (B) is not only sufficient, but also necessary for the IFS (F, P) satisfying (A) to have the third extreme measure. More precisely, we have the following consequence of Theorem 5.1. (⇒) Suppose that there exists an invariant measure μ * for (F, P) with μ * ((0, 1)) = 1. Let ϕ * be the function that corresponds to μ * by Proposition 3.1. From Proposition 3.3, which we may apply by Remark 3.5, we see that ϕ * is continuous on (0, 1) and since μ * ({0}) = μ * ({1}) = 0 we conclude that ϕ * is continuous. Hence ϕ * ∈ D. Now, it is enough to apply Remark 3.2.
Returning to the family F from Example 4.3, note that it does not satisfy (B) if (A) holds, which is an immediate consequence of Corollary 5.2 or Theorem 5.1. However, if (A) does not hold, then (B) is satisfied with any right-continuous and continuous at 1 functions ρ 1 , ρ 2 ∈ I. 1 μ((0,1)) μ and let ϕ * be the function that corresponds to μ * by Proposition 3.1. Note that ϕ * is continuous by Proposition 3.3.
The next result gives additional information about supports of invariant measures for (F, P) satisfying (A). If μ is an invariant measure for (F, P),

Stability
Using some martingale techniques developed in [13] (see also [8]), we prove the stability of (F, P). Before formulating our result, let us denote by P the Markov-Feller operator that corresponds to (F, P), i.e.

Pμ(B)
for all μ ∈ M 1 and B ∈ B; here M 1 denotes the family of all probability Borel measures on [0, 1].

Strong Law of Large Numbers
From the uniqueness of an invariant measure and Breiman's law of large numbers we may easily derive the strong law of large numbers. converges weakly to an invariant measure for (F, P). To complete the proof we have to show that this invariant measure is equal to μ * . This, however, easily follows from (7.8) (or its counterpart lim m→∞ P ∞ ({ω ∈ ∞ | f m (x, ω) ∈ (inf supp μ * , 1)} = 1) and the Birkhoff Ergodic Theorem applied to the dynamical system corresponding to the considered Markov process on ( , A, P). In fact, since where β = inf supp μ * and γ = sup supp μ * , with no loss of generality we may (and do) assume that x ∈ (β, γ ); note that, by Theorem 6.2, β = 0 or γ = 1. Choose x 1 , x 2 ∈ (β, γ ) such that x 1 < x < x 2 . Due to the Birkhoff Ergodic Theorem for all i ∈ {1, 2}, c ∈ (β, γ ) and P ∞ -almost all ω ∈ ∞ we have which jointly with (8.2) and (8.3) implies that the weak limit of the sequence (8.1) is equal to μ * .