Ergodicity and central limit theorem for random interval homeomorphisms

i

|fⁿ(I)|≤ qⁿ for i ∈ Ω and n ∈ N.
Set b = inf I. We claim that there exists r ∈ N and a sequence (i₁,..., i_r) ∈ Σ_r such that f_{(i1 ,...,ir )}([a, 1 − a]) ⊆ I. Suppose, contrary to our claim, that for every r ∈ N and (i₁,..., i_r) ∈ Σ_r, if f_{(i1 ,...,ir )}(1 − a) ∈ I, then f_{(i1,...,ir )}(a) ≤ b. This implies that
P^rδ_a([0, b]) ≥ P^rδ_1−a([0, b] ∪ I) = P^rδ_1−a([0, b]) + P^rδ_1−a(I)
for every r ∈ N. Since the iterated function system (f₁,... , f_N ; p₁,... , p_N ) is asymptotically stable and μ_∗ is atomless, we have P^rδ_a([0, b]) → μ_∗([0, b]), P^rδ_1−a([0, b]) → μ_∗([0, b]), P^rδ_1−a(I) → μ_∗(I) as r → ∞, by Alexandrov’s theorem. This is a contradiction with the above inequality, for μ_∗(I) > 0.
Now choose (i₁,..., i_r) ∈ Σ_r such that f_{(i1 ,...,ir )}([a, 1 − a]) ⊆ I and define

Ω = {(i₁,... , i_r)i : i ∈ Ω}∈ Σ.
We see at once that Ω satisfies the desired condition. The proof is complete.

Lemma 4: Let (f₁,..., f_N ; p₁,..., p_N ) be an admissible iterated function sys- tem and let α, δ, ε, M be the positive constants given in Lemma 1. Assume that

J = [a, 1 − a] for some a ∈ (0, ¹ ) such that M = ε^−α < ^a−α . Then there exists
2 6
n₀ ∈ N such that for every integer n ≥ n₀

i

n
P({i ∈ Σ : f √4 ^n⊗([ε , 1 − ε ]) ⊂ J}) ≥ ¹ .
n ₅



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Proof. Let α, δ, ε, M be the positive constants given in Lemma 1. From Lemma 1 it follows that there exists n₀ ∈ N such that for n ≥ n₀ we have
_√ k
P({i ∈ Σ : ∃_k≤ 4 _n⊗f_i (ε_n) ≥ ε}) ≥ ₁₀ .

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From (3.1) and the fact that M = ε^−α < ^a−α we obtain

^√4 n⊗
_√ k α

P({i ∈ Ω : f_i (ε_n) ≥ a}) ≥ P({i ∈ Σ : ∃_k≤ 4 _n⊗f_i (ε_n) ≥ ε}) · (1 − 2Ma )
9 2 3
^≥ 10 ^· 3 ⁼ 5 ^.
In the same way we may prove that for n₀ sufficiently large we have

i

n
P({i ∈ Ω : f √4 ^n⊗(1 − ε
3
) ≤ 1 − a}) ≥ ₅
for n ≥ n₀.

Combining these two estimates we obtain that there exists n₀ ∈ N such that
P({i ∈ Σ : f √4 ^n⊗([ε , 1 − ε ]) ⊂ J}) ≥ 1 − ² + ² = ¹ for n ≥ n .

_i n n

The proof is complete.

5 5 5 ⁰

Definition 3: Let A ⊂ Σ ∪ Σ_∗. We say that a sequence i ∈ Σ_∗ is dominated by A if there exists j ∈A such that i ≺ j. Additionally, we shall assume that the empty sequence is dominated by any A.
Lemma 5: Let A⊂ Σ be such that P(A) ≥ β for some β > 0 and let k, n ∈ N with k < n. Then there exists a set A ⊂ Σ_n such that P_n(Σ_n \ A) ≤ (1 − β)^k and for any i ∈ A there exist i₁, i₂,... , i_k ∈ Σ_∗ such that
i = i₁i₂ ··· i_k
and for j = 1,...,k at least one of the sequences i_j, σi_j ... , σ^k−1i_j is dominated by A.
Proof. Let k, n ∈ N, k < n and let A⊂ Σ with P(A) > 0 be given. Let β > 0 be such that P(A) ≥ β. Define
B₁ := {i ∈ Σ_n : i is dominated by A}

and observe that

P_n(Σ_n \ B₁) ≤ 1 − β.

By C₂ denote the set of all i ∈ Σ_∗ such that i is not dominated by B₁ but i_||i|−1 is dominated. If C₂ is empty, then putting i_m to be the empty sequences for

m = 2,...,k will finish the proof. In the other case define
B₂ := {ij_|n : i ∈ C₂, j is dominated by A},
where ij_|n denotes the natural projection on Σ_n of the concatenation of i and j. Observe that P_n(B₂|Σ_n \ B₁) > β. This gives
P_n(Σ_n \ (B₁ ∪ B₂)) =P_n(Σ_n \ B₁) · P_n(Σ_n \ B₂|Σ_n \ B₁)
=P_n(Σ_n \ B₁)(1 − P_n(B₂|Σ_n \ B₁))

2
≤(1 − β) .
Obviously, if i ∈ B₂, then i = i₁i₂ and for j = 1, 2 either i_j or σi_j is dominated by A.
Further, if B₁,..., B_l−1 are given for some l ≤ k, then
P_n(Σ_n \ (B₁ ∪ ··· ∪ B_l−1)) ≤ (1 − β)^l−1
and if i ∈ B_m for m = 1,... ,l − 1, then i = i₁ ··· i_m and at least one of the sequences i_j, σi_j,..., σ^l−2i_j is dominated by A for j = 1,... , m. Define now the set C_l of all i ∈ Σ_∗ such that for some m = 1,... ,l the se- quences i, i_||i|−1,... , i_||i|−m+1 are not dominated by B_l−1, but i_||i|−m is domi- nated. If C_l is empty, then putting i_m to be the empty sequence for m = l,...,k
will finish the proof. In the other case define
B_l := {ij_|n : i ∈ C_l, j is dominated byA}.
Observe that
P_n(Σ_n\(B₁ ∪ ··· ∪ B_l))
= P_n(Σ_n \ (B₁ ∪ ··· ∪ B_l−1))P_n(Σ_n \ B_l | Σ_n \ (B₁ ∪ ··· ∪ B_l−1))

l
≤ (1 − β) .
Moreover, from the definition of B_l it follows that if i ∈ B_l, then i = i₁ ··· i_l and at least one of the sequences i_m, σi_m,..., σ^l−1i_m for m = 1,... ,l is dominated by A.
Taking l = k and setting A = B₁ ∪ ··· ∪ B_k we finish the proof. In- deed, if i ∈ A there exist i₁, i₂,..., i_k ∈ Σ_∗, some of them possibly empty sequences, such that i = i₁i₂ ··· i_k and for j = 1, 2,...,k at least one of the sequences i_j, σi_j,..., σ^k−1i_j is dominated by A. The proof is complete.

5.