Ergodicity and central limit theorem for random interval homeomorphisms

Notation

Let (S, d) be a metric space. By M(S) we denote the set of all finite measures on the σ-algebra B(S) of all Borel subsets of S and by M₁(S) ⊆ M(S) we denote the subset of all probability measures on S. By C(S) we denote the family of

all bounded continuous functions equipped with the supremum norm ∗· ∗. We shall write μ, f for _S f dμ.

An operator P : M(S) → M(S) is called a Markov operator if it satisfies the following two conditions:

(1) P (λ₁μ₁ + λ₂μ₂) = λ₁Pμ₁ + λ₂Pμ₂ for λ₁, λ₂ ≥ 0, μ₁, μ₂ ∈ M(S),
(2) Pμ(S) = μ(S) for μ ∈ M(S).
A Markov operator P is called a Feller operator if there is a linear operator
U : C(S) → C(S) such that

i.e.,
U^∗ = P,


μ, Uf = Pμ, f for f ∈ C(S), μ ∈ M(S).

A measure μ_∗ is called invariant for a Markov operator P if
Pμ_∗ = μ_∗.
If S is a compact metric space, then every Feller operator P has an invariant probability measure. For example, let μ ∈ M₁(S) and define ν ∈ C(S)^∗ by

k

ν(f ) = LIM

P μ, f

,
_{1 n}

n


k=1

where LIM denotes a Banach limit. By the Riesz Representation Theorem

ν(f ) = ν, f ,

where ν ∈ M₁(S) is invariant.

An operator P is called asymptotically stable if it has a unique invariant
measure μ_∗ ∈ M₁(S) such that the sequence (Pⁿμ) converges in the weak-∗ topology to μ_∗ for any μ ∈ M₁(S), i.e.,

n→∞
lim Pⁿμ, f = μ_∗,f for any f ∈ C(S).
In this paper we shall consider a special type of Feller operators. Assume
that f_i : [0, 1] → [0, 1] for i = 1,...,N are continuous transformations and let (p₁,... , p_N ) be a probability vector, i.e., p_i ≥ 0 for all i = 1,...,N and

N
p_i = 1.
i=1
The family
(f₁,... , f_N ; p₁,... , p_N )

N

generates a Markov operator P : M([0, 1]) → M([0, 1]) of the form
(2.1) Pμ(A) = p_iμ(f⁻¹(A)) for A ∈ B([0, 1]).

i
i=1
This Markov operator is a Feller operator and its predual operator
U : C([0, 1]) → C([0, 1])
is given by the formula

N
U ϕ(x) = p_iϕ(f_i(x)) for ϕ ∈ C([0, 1]) and x ∈ [0, 1].
i=1
By induction we check that
N N

1
(2.2) Uⁿϕ(x) = p_i
··· p_in
ϕ(f_i1
◦ ··· ◦ f_in
(x))

ⁱ1 ^{=1 i}n ⁼¹
for n ∈ N, ϕ ∈ C([0, 1]) and x ∈ [0, 1].
Markov operators corresponding to random transformations have been inten-
sively studied for many years. In particular, W. Doeblin and R. Fortet in [12] considered the case when the maps f_i were strict contractions but the prob- abilities p_i were dependent on position, but Lipschitz functions. S. R. Foguel and B. Weiss in [13] considered convex combinations of commuting contrac- tions in Banach spaces. R. Sine in [29] studied random rotations of the unit circle with position-dependent probabilities p_i. In turn, the connections of ran- dom transformations to fractals have been discovered by J. Hutchinson in [18].
M. Barnsley and S. Demko coined the term iterated function systems for systems with contractions (see [5]). In [6] the authors considered function sys- tems contractive on the average in the case where the state space S is locally compact (see also [25]). Their result on asymptotic stability was extended to Polish spaces in [30]. Random transformations, more general than iterated func- tion systems, have been also studied, but for more details we refer the reader to Kifer’s book (see [19]).
We start with the following definitions.
Definition 1: Let H⁺ be the space of homeomorphisms f : [0, 1] → [0, 1] satis- fying the following properties:

1. 
   f is increasing,
   
2. 
   f is differentiable at 0 and 1.
   

Definition 2: Let {f₁,... , f_N }⊆ H⁺ be a finite collection of homeomorphisms and let (p₁,..., p_N ) be a probability vector such that p_i > 0 for all i = 1,...,N .
The family (f₁,... , f_N ; p₁,... , p_N ) is called an admissible iterated function system if

1. 
   for any x∈ (0, 1) there exist i, j∈{1,...,N} such that f_i(x) _j (x),
   
2. 
   f_i (0) > 0 and f_i (1) > 0 and the Lyapunov exponents at both points
   

0, 1 are positive, i.e.,

N
p_i log f_i (0) > 0 and
i=1
N

p_i log f_i (1) > 0.
i=1

We set Σ = {1,...,N}^N and Σ_n = {1,...,N}ⁿ for n ∈ N. Put

∞
Σ_∗ = Σ_n.
n=1
Clearly, a probability vector (p₁,... , p_N ) on {1,...,N} defines the product measures P, P_n on Σ and Σ_n for n ∈ N, respectively. The expected value with respect to P is denoted by E. For any n ∈ N and i = (i₁, i₂,.. .) ∈ Σ we set i_|n = (i₁, i₂,..., i_n). In the same way we define i_|n for i = (i₁,..., i_k) ∈ Σ_k with k ≥ n. Additionally, we assume that i_|0 is the empty sequence for any i ∈ Σ ∪ Σ_∗. For a sequence i ∈ Σ_∗, i = (i₁,..., i_n), we denote by |i| its length (equal to n). We shall write
f_i = f_in ◦ f_in−1 ◦ ··· ◦ f_i1
for any sequence i = (i₁,... , i_n) ∈ Σ_n, n ∈ N.
Let σ : Σ → Σ denote the shift transformation, i.e.,
σ((i₁, i₂,... .)) = (i₂, i₃ .. .)
for (i₁, i₂,.. .) ∈ Σ. If i = (i₁,... , i_n) ∈ Σ_∗ and j = (j₁,... , j_k) ∈ Σ_∗, then by ij we denote the concatenation of i and j, i.e., the sequence (i₁,... , i_n, j₁,..., j_k) ∈ Σ_∗. If i ∈ Σ_∗ and j ∈ Σ, then we can define concatenation ij of sequences i and j in the same way obtaining the sequence from the space Σ. We write i ≺ j for i ∈ Σ_∗, j ∈ Σ ∪ Σ_∗ if there exists k ∈ Σ ∪ Σ_∗ such that ik = j.
Let an admissible iterated function system (f₁,..., f_N ; p₁,..., p_N ) be given
and let P be the corresponding Markov operator defined by formula (2.1). For every measure ν ∈ M₁ the law of the Markov chain (X_n) with transition prob-
ability π(x, A) = Pδ_x(A) for x ∈ [0, 1], A ∈ B([0, 1]) and initial distribution ν,

is the probability measure P_ν on ([0, 1]^N, B([0, 1])^⊗N) such that:

P_ν[X_n+1 ∈ A|X_n = x] = π(x, A) and P_ν[X₀ ∈ A] = ν(A),

·

· ·
where x ∈ [0, 1], A ∈ B([0, 1]). The existence of P_ν follows from the Kolmogorov Extension Theorem. The expectation with respect to P_ν is denoted by E_ν . For ν = δ_x, the Dirac measure at x ∈ [0, 1], we write just P_x and E_x. Obviously,

P_ν( ) =
[0,1]
P_x( )ν(dx) and E_ν ( ) =
[0,1]
E_x(·)ν(dx).

Observe that for n ∈ N and A₀,..., A_n ∈ B([0, 1]) we have

P_x((X₀,... , X_n) ∈ A₀ × ··· × A_n))

=

⁽ⁱ1^,...,in ^)∈Σn
(1_{A1×···×An} (f_i1 (x),... , f_{(in,...,i1 )}(x))p_i1 ··· p_in )

Σ
= 1_{A1×···×An} (f_i1 (x),..., f_{(in ,...,i1)}(x))P_n(di)

n

Σ
= 1_{A1×···×An} (f_i1 (x),... , f_{(in,...,i1 )}(x))P(di).

Consequently,


(2.3) E_x(H(X₀,..., X_n)) =
Σ


H(f_i1 (x),... , f_{(in,...,i1 )}(x))P(di)

and
(2.4) E_ν (H(X₀,..., X_n)) =
[0,1]


H(f_i1 (x),..., f_{(in ,...,i1)}(x))P(di)ν(dx)
Σ

for an arbitrary bounded Borel-measurable function H : [0, 1]ⁿ → C.

, P
For α ∈ (0, 1) and M ≥ 1 we define the sets P_M⁻_,α
+
M,α
, P_M,α as follows:

P_M⁻_,α := {μ ∈ M₁([0, 1]) : μ([0, x]) ≤ M x^α for all x ∈ [0, 1]},
+ α
P_M,α := {μ ∈ M₁([0, 1]) : μ([1 − x, 1]) ≤ Mx for all x ∈ [0, 1]},

+
P_M,α := P_M⁻ _,α ∩ P_M,α.
For ε > 0 and x < ε we set