Ergodicity and central limit theorem for random interval homeomorphisms

i = (i₁,... , i_n,.. .) put


g_n(i) := |f_in (x) − f_in (y)|,
where i_n = (i_n,... , i₁). Then g_n(i) → 0 as n → ∞ for P almost every i ∈ Σ. By the construction of the probability measures P and P_n, n ∈ N, we have
P(B_n × Σ₁ × Σ₁ × ··· ) = P_n(B_n) for B_n ∈ Σ_n.
Since g_n(i) depends only on the first n coordinates, we have

g_n(i)dP(i) =

Σ
^Σn


g_n(i)dP_n(i)

=
⁽ⁱ1 ^,...,in ^)∈Σn
^|f(in ^,...,i1 ^{)(x) − f(i}n ^,...,i1^{)(y)|pi1 ··· pin .}

Therefore, by the Lebesgue theorem we obtain

^n→∞ (i ,...,i )∈Σ
lim
^|f(in ^,...,i1^{)(x) − f(i}n ^,...,i1^{)(y)|pi1 ··· pin = 0}

and inequality (3.9) completes the proof.

n

Remark 1: From uniqueness in Theorem 1 and Theorem 1.3 in [23] it follows that for any f ∈ C([0, 1]) satisfying f (0) = f (1) = 0 and μ_∗,f = 0 we have
U^kf → 0 as n → ∞.

₁

n
k=1
On the other hand, the asymptotic stability in Theorem 2 raises the question whether we have then ∗Uⁿf∗→ 0 as n → ∞. Unfortunately, we have not been able to answer the question, neither affirmatively nor negatively.
Corollary 1: Under the assumptions of Theorem 2 we have:

1. 
   for every f ∈ C([0, 1]) we have the convergence of (Uⁿf (x)) for every
   

x ∈ [0, 1],

2. 
   
   ∗ −
   for every f ∈ L₂(μ_∗) we have
   

Uⁿf
[0,1]
Proof. (i) For x ∈ (0, 1) we have
f dμ_∗∗₂ → 0 as n → ∞.

Uⁿf (x) →


[0,1]
f dμ_∗

by applying Theorem 2 to δ_x. Since Uⁿf (0) = f (0) and Uⁿf (1) = f (1), we have the required convergence (not uniform since the limit may not be continuous).

[0,1]
(ii) Since μ_∗({0, 1}) = 0, we have ∗Uⁿf− f dμ_∗∗₂ → 0 as n →∞ for every
f ∈ C([0, 1]), by (i). Since U is a contraction of L₂(μ_∗), this implies (ii).

4. Auxiliary results

For abbreviation we set, for ε and δ of Lemma 1,

2

2

n

n
ε := (1 − δ) 1 √4 ^n⊗ and γ := (1 − δ) α √4 ^n⊗ for n ∈ N.
Lemma 2: 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. Then for all
positive integers n and k ≥ ≤^√4 n we have

i
P({i ∈ Σ : f^k(x) < ε_n}) ≤ 2Mγ_n for x ∈ [ε_n, 1]

i
and
P({i ∈ Σ : f^k(y) > 1 − ε_n}) ≤ 2Mγ_n for y ∈ [0, 1 − ε_n].

√
Proof. Let α, δ, ε, M be the constants given in Lemma 1. Fix n ∈ N and
x ∈ [ε_n, 1], and let k ≥ ≤ ⁴ n . If x ≥ ε, then we have

i
(4.1) ^{P({i ∈ Σ : fk(x) < εn}) = Pkδx([0, εn)) ≤ Mεαγn < 2Mγn,} where the last inequality follows from the fact that δ_x ∈ P_M⁻_,α and
P (P_M⁻ _,α) ⊆ P_M⁻ _,α,

by Lemma 1.
Assume now that x ∈ [ε_n, ε). Set

1

r

1

m
E := {(i₁,..., i_m) ∈ Σ_∗ : m ≤ ≤^√4 n , ∀_r<mf_{(i ,...,i )}(x) < ε and f<sub>(i ,...,i


)</sub>(x) ≥ ε}

and


F := {(i₁,..., i_k) ∈ Σ_k : ∀_m≤ ^√4 _n⊗f_{(i1,...,im )}(x) < ε}.

Then we have
P^kδ_x =


p_i ··· p<sub>i


P</sub>k−m_δf
(x)⁺


p_i ··· p_i δ_f


(x)^.

1

1

k

m

(i₁ ,...,i_m)

(i₁ ,...,i_k )
By the definition of E we see that δ_{f(i1 ,...,im)(x)} ∈ P_M⁻_,α for (i₁,... , i_m) ∈ E

— _f
and from this also P^{k m}δ
(i₁ ,...,i_s )
_(x) ∈ P_M⁻_,α for (i₁,... , i_m) ∈ E. On the other

n⊗
hand, from Lemma 1 it follows that



p_i1 ··· p_ik ≤ P
(i₁,...,i_k)∈F
Consequently, we have
√₄


j=1
A_x,j(ε)

≤ γ_n.

P({i ∈ Σ : f (x) < ε }) ≤_n
k
i
⁽ⁱ1 ^,...,im^)∈E
α
^pi1
··· p_im
Pk−m_δ

f
(i₁ ,...,i_m )
_(x)([0, ε_n]) + γ_n

≤Mε_nγ_n + γ_n ≤ 2Mγ_n for x ∈ [ε_n, ε).
This and condition (4.1) completes the estimate for x ∈ [ε_n, 1]. The estimate for y ∈ [0, 1 − ε_n] is proved in the same way. The proof is complete.
Observe that if (f₁,..., f_N ; p₁,..., p_N ) is admissible, then there are no non- trivial closed subintervals of (0, 1) invariant by {f₁,... , f_N }. We are now in a position to recall the contracting result obtained by D. Malicet in [26]. This
theorem is crucial in the proof and in our setting may be stated as follows.

Theorem 3 (cf. Corollary 2.13 in [26]): If (f₁,... , f_N ; p₁,... , p_N ) is an admis- sible iterated function system, then there exists q ∈ (0, 1) such that for every x ∈ (0, 1) and P-a.e. i ∈ Σ we have a neighbourhood I of x in (0, 1) satisfying

i
(4.2) |fⁿ(I)|≤ qⁿ for every n ∈ N.

2
Lemma 3: Let (f₁,..., f_N ; p₁,..., p_N ) be an admissible iterated function sys- tem and let a ∈ (0, ¹ ). Then there exists r ∈ N and Ω ⊆ Σ with P(Ω) > 0 such that for J = [a, 1 − a] we have
∞

i

1 − q

n=1
|fⁿ(J)|≤ r + ^q for i ∈ Ω,


where q is the constant given in Theorem 3.



Proof. Fix a ∈ (0, 1/2) and choose x ∈ supp μ_∗ ∩ (0, 1). From Theorem 3 we have q ∈ (0, 1), a neighbourhood I of x and Ω ⊆ Σ with P(Ω) > 0 such that