A GENERALIZATION OF STRAUBE’S THEOREM: EXISTENCE OF ABSOLUTELY CONTINUOUS INVARIANT MEASURES FOR RANDOM MAPS

A random map is a discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. In this paper, we study random maps. The main result provides a necessary and sufficient condition for the existence of absolutely continuous invariant measure for a random map with constant probabilities and position-dependent probabilities.


Introduction
Random dynamical systems provide a useful framework for modeling and analyzing various physical, social, and economic phenomena.A random dynamical system of special interest is a random map where the process switches from one map to another according to fixed probabilities [5] or, more generally, position-dependent probabilities [1,3,4].The existence and properties of invariant measures for random maps reflect their longterm behavior and play an important role in understanding their chaotic nature.
It is well known that if a map τ : I → I, I = [0,1], is piecewise expanding, then it possesses an absolutely continuous invariant measure (ACIM) [2].This result can be generalized to random maps where the condition of piecewise expanding is replaced by an average expanding condition where the weighting coefficients are the probabilities of switching [3,4,5].Such results have been generalized in [1].There are a number of interesting examples which do not fall into the average expanding condition for which the conditions of this paper may present a possible approach.
Consider the following simple random maps on I: an ACIM, but the "average expanding" sufficiency condition for existence of an ACIM for the random map based on τ 1 and τ 2 fails since τ 2 has regions of arbitrarily small slope.Hence, in general, we cannot conclude that even such a simple random map admits an ACIM.
The foregoing suggests the need for results that can establish existence of an ACIM directly for random maps.To this end we generalize a theorem of Straube [7], which provides a necessary and sufficient condition for existence of an ACIM of a nonsingular map, to random maps.We consider both random maps with constant probabilities and random maps with position-dependent probabilities.
In Section 2, we present the notation and summarize the results that we will need in the sequel.In Section 3, we prove the main result.

Preliminaries
Let (X,Ꮾ,λ) be a measure space, where λ is an underlying measure and let τ k : X → X, k = 1,2,...,K be nonsingular transformations.A random map T with constant probabilities is defined as where {p 1 , p 2 ,..., p K } is a set of constant probabilities.For any x ∈ X, T(x) = τ k (x) with probability p k and for any nonnegative integer N, T N (x A T-invariant measure satisfies the following condition [6]: for any E ∈ Ꮾ.
A position-dependent random map T is defined as where {p 1 (x), p 2 (x),..., p K (x)} is a set of position-dependent probabilities, that is, with probability p k (x) and for any nonnegative integer N, T N (2.4) In [2], it was proved that a T-invariant measure µ is given by for any measurable set E ∈ Ꮾ.
We now recall some definitions and results from [6,7] which will be used to prove our main results in Section 3.
Theorem 2.5 [7].Let φ be a finitely additive positive measure on a σ-algebra Ꮾ and let ν be a countably additive positive measure on Ꮾ.Then, there exists a decreasing sequence Theorem 2.6 [6].Let (X,B,λ) be a measure space with normalized measure λ, and let f : X → X be a nonsingular transformation.Then, the following conditions are equivalent: (i) there exists an f -invariant normalized measure µ which is absolutely continuous with respect to λ; (ii) there exists δ > 0, and α,0 < α < 1 such that (2.6)

Existence of absolutely continuous invariant measures
In this section, we prove necessary and sufficient conditions for existence of an absolutely continuous invariant measure for random maps.For notational convenience, we consider K = 2, that is, we consider only two transformations τ 1 ,τ 2 .The proofs for larger number of maps are analogous.We first consider random maps with constant probabilities, then random maps with position-dependent probabilities.
Theorem 3.1.Let (X,Ꮾ,λ) be a measure space with normalized measure λ and let τ i : X → X, i = 1,2 be nonsingular transformations.Consider the random map T = {τ 1 ,τ 2 ; p 1 , p 2 } with constant probabilities p 1 , p 2 .Then, there exists a normalized absolutely continuous (w.r.t.λ) T-invariant measure µ if and only if there exists δ > 0 and 0 < α < 1 such that for any measurable set E and any positive integer k, λ(E) < δ implies To prove this theorem, we first prove the following two lemmas.
Lemma 3.2.Let (X,Ꮾ,λ) be a probability measure space and let µ be absolutely continuous with respect to λ, µ = f • λ, for f an L 1 (X,Ꮾ,λ) function.Then, there exists a constant M ≥ 0 and a measurable set A 0 such that µ(A 0 ) ≤ 1/10 and f ≤ M on X \ A 0 .
Proof.Consider the following sets: Clearly, {B n } are disjoint measurable sets and For any measure φ, any integer k, and any measurable set E, define Proof.Let M and A 0 be as in the previous lemma.We have We want to prove that there exist δ > 0, 0 < α < 1 such that for any E ∈ Ꮾ and for any positive integer k, Md. Shafiqul Islam et al. 137 Suppose not.Then, for any α, 0 < α < 1, there exists E ∈ Ꮾ and there exists a positive integer n 0 such that where E ∈ Ꮾ.
Choose δ > 0 such that Mδ + 1/10 < 1/4, where M is the constant of Lemma 3.2.Let n 0 be the index corresponding to δ in formula (3.6).Then by Lemma 3.2, we have, for From our choice of δ, we get

.11)
Thus, .12) a contradiction.Conversely, suppose that there exists δ > 0 and 0 < α < 1 such that for any measurable set E and any positive integer k, λ(E) < δ implies We want to show that there exists a measure µ such that µ Consider the measures λ n defined by It can be shown that for all n, λ n are normalized measures.Moreover, if λ(E) = 0, then by nonsingularity of τ 1 and τ 2 .Hence λ n λ.We imbed λ n in the dual space L ∞ (λ) * of L ∞ (λ) in the following way: For every n, Hence, for each n, g n ≤ 1.Thus, the λ n can be thought of as elements of the unit ball of L ∞ (λ) * .This unit ball is weak * -compact by Alaoglu's theorem [7].Let ν be a cluster point in the weak * -topology of L ∞ (λ) * of the sequence (λ n ) n≥1 .Define a set function µ on Ꮾ by We claim that µ is finitely additive, bounded and it vanishes on sets of λ-measure zero: µ(∅) = ν(χ ∅ ) = ν(0) = 0, since ν is a linear functional.For any E ∈ Ꮾ, since Λ λ k is a measure.Thus, where µ c is countably additive and µ c ≥ 0 and µ p is purely additive and µ p ≥ 0. We claim that µ c = 0. Suppose µ c = 0. Then by Theorem 2.5 there exists a decreasing sequence {E n } n≥1 of elements of Ꮾ such that lim n→∞ λ(E n ) = 0 and µ(E n ) = µ(X) = 1.Thus, there exists an integer n 0 such that for all n ≥ n 0 , λ(E n ) < δ and, as a consequence of our hypothesis, we have, for all k, is a countably additive measure, and m ≤ µ.Thus, by Lemma 2.4, we have m ≤ µ c and hence is a positive measure.But this measure has total mass zero.Hence, it is a zero measure.Thus µ c is T-invariant.Because µ vanishes on sets of λ-measure zero and 0 ≤ µ c ≤ µ, we have µ c λ. Finally, γ(E) = µ c (E)/µ c (X) is normalized, T-invariant, and absolutely continuous with respect to λ.
We now state the analogous result for position-dependent random maps.Theorem 3.4.Let (X,B,λ) be a measure space with normalized measure λ and let τ i : X → X, i = 1,2 be nonsingular transformations.Consider the random map T = {τ 1 ,τ 2 ; p 1 , p 2 } with position-dependent probabilities p 1 , p 2 .Then there exists a normalized absolutely continuous (w.r.t.λ) T-invariant measure µ if and only if there exists δ > 0 and 0 < α < 1 such . . . (3.31) Proof.The proof is analogous to the proof of Theorem 3.1.

Md.
Shafiqul Islam et al. 141 that for any measurable set E and any positive integer k, λ(E) < δ implies x)p 2 τ 2 (x) dλ < α; Let M be the constant from the previous lemma and let δ be such that Mδ + 1/10 < 1/4.Then, for any n ≥ 1, and any measurable set A, Λ λ n