Ergodicity and central limit theorem for random interval homeomorphisms

i
A_x,k(ε) := {i ∈ Σ : f^k(x) < ε} for k ∈ N,

i
where f^k = f_i|k. Similarly, for ε > 0 and x > 1 − ε we set

i
A^x,k(ε) := {i ∈ Σ : f^k(x) > 1 − ε} for k ∈ N.

3. Invariant measures

Let an admissible iterated function system (f₁,..., f_N ; p₁,..., p_N ) be given and let P be the corresponding Markov operator. The Markov operator P clearly admits two trivial invariant measures: δ₀ and δ₁. In this section we prove the
existence of a unique invariant measure μ ∈ M₁((0, 1)). Let us stress that this
fact was already shown in [15] and our proof of existence of an invariant measure
is just a repetition of the argument. However, the proof of uniqueness is new, more general and elementary. Similarly the proof of stability on the interval (0, 1). We start with the following lemma.

2
Lemma 1: Let (f₁,..., f_N ; p₁,..., p_N ) be an admissible iterated function sys- tem and let P be the corresponding Markov operator. Then there exist con- stants ε ∈ (0, ¹ ), α, δ ∈ (0, 1) and M ≥ 1 such that P (P_M^±_,α) ⊆ P_M^±_,α and

M,α
(3.1) δ_x ∈ P_M⁻_,α
for x ≥ ε and δ_x ∈ P⁺
for x ≤ 1 − ε.

Moreover, for any n ∈ N we have
√₄

A

(ε)

≤ (1 − δ) α 4 ^n⊗ for x ∈ [ε(1 − δ) 1 4 ^n⊗, ε]

(3.2) P
n⊗ _{√ √}

and



√₄

(3.3) P
n⊗

j=1



j=1
_{x,j 2} ²

≤ (1 − δ) α 4 ^n⊗ for x ∈ [1 − ε, 1 − ε(1 − δ) 1 4 ^n⊗].

A^x,j(ε)

2

2
√ √

Proof. By the definition of an admissible iterated function system we can find positive numbers λ₁ < f₁ (0),..., λ_N < f_N (0), λ₁ < f₁ (1),..., λ_N < f_N (1) such that still

(3.4)
N

p_i log λ_i > 0 and
i=1
N

p_i log λ_i > 0.
i=1

2
Using the definition of derivative at points 0 and 1 we can choose ε < ¹ such
that
(3.5) f_i(x) ≥ λ_ix and f_i(1 − x) ≤ λ_i(1 − x)

N
for i = 1,...,N and x ≤ ε. Now we use the definition of derivative for func- tions f₁⁻¹,... ,f⁻¹ at points 0 and 1 and correct the choice of ε to satisfy the

conditions:
(3.6) f⁻¹(x) ≤ ^x and f⁻¹(1 − x) ≥ 1 − ^x
i _λi i _λi
for i = 1,...,N and x ≤ ε.
Writing the Taylor expansion of the function λ^−α at 0 with respect to α we obtain λ^−α = 1 − α log λ + o(α). Therefore, using (3.4) we can fix α ∈ (0, 1) and δ > 0 such that we have

N

N

i

i

i

i
(3.7) p λ ^−α < (1 − δ)^α and p λ ^−α < (1 − δ)^α.

Finally, put M = ε^−α. This immediately implies condition (3.1).



Now we show the invariance of P_M,α. Let μ ∈ P_M⁻_,α and x ∈ (0, 1). We are going to show that Pμ ∈ P_M⁻_,α. If x ≥ ε, then M x^α ≥ Mε^α = 1, hence obviously Pμ([0, x]) ≤ M x^α. If x < ε, then

i
Pμ([0, x]) =


i=1
p_iμ([0,f ⁻¹(x)]) ≤

i

N


i=1
p_iμ 0, _λ


x

≤ i=1
_x α

λ
p_iM
i

= M x^α p_iλ⁻_i ^α < Mx^α(1 − δ)^α ≤ M x^α.
i=1

M,α
Analogous computations for P ⁺
complete this part of the proof.

i

k

k−1

1
We set λ^k := λ_i λ_i ··· λ_i for arbitrary real numbers λ_i, 1 ≤ i ≤ N , and
i = (i₁, i₂,.. .) ∈ Σ. Applying (3.7) we have

i
E((λ √4 ^n⊗)^−α) ≤ (1 − δ)^α √4 ^n⊗ for any n ∈ N.

2
By Chebyshev’s inequality and (3.5) for x ∈ [ε(1 − δ) 1 √4 ^n⊗, ε] we have

n⊗
√₄
P
j=1
A_x,j(ε)


j

i
=P({i ∈ Σ : ∀_j≤ ^√4 _n⊗ f (x) ≤ ε})

_√ ^{j 1 √4} n⊗
≤P({i ∈ Σ : ∀_j≤ 4 _n⊗ f_i (ε(1 − δ) ² ) ≤ ε})
_j 1 ^√4

j≤ ⁴ n⊗

i

2
≤P({i ∈ Σ : ∀ ^√ λ ε(1 − δ) ^n⊗ ≤ ε})

2
≤P({i ∈ Σ : (1 − δ) 1 √4 ^n⊗ < λ⁻ √4 ^n⊗})
i

2
≤(1 − δ)⁻ α √4 ^n⊗E(λ^−α √4 ^n⊗)
i

2

2

≤
(1 − δ)⁻ α √4 ^n⊗(1 − δ)^α √4 ^n⊗ ≤ (1 − δ) α √4 ^n⊗ for n ∈ N.
Estimate (3.3) may be proved in the same way.

Theorem 1: If (f₁,..., f_N ; p₁,..., p_N ) is an admissible iterated function sys- tem, then the corresponding Markov operator P has a unique invariant measure μ_∗ ∈ M₁((0, 1)). Moreover μ_∗ is atomless.

2
Proof. Existence. Let ε, α, δ, M be the positive constants given in Lemma 1. Then δ 1 ∈ P_M,α and, by Lemma 1,

1

2

2
μ_n := _n (δ 1 + Pδ 1 + ··· + P
n−1
δ 1 ) ∈ P_M,α for n ∈ N.

2
Since the family of measures P_M,α is tight, there exists an accumulation point μ ∈ M₁([0, 1]) of the sequence (μ_n) in the weak-∗ topology. It is easy to check that μ ∈ P_M,α, which obviously implies that μ({0, 1}) = 0. Moreover, the operator P is a Feller operator and henceforth the measure μ is invariant for P .
This completes the proof of existence.
Uniqueness. Since for any x ∈ (0, 1) there exist i, j ∈ {1,...,N} such that f_i(x) < x < f_j(x), 0 and 1 belong to the support of every invariant measure μ ∈ M₁((0, 1)).
Assume, contrary to our claim, that there exist at least two different ergodic

i
invariant measures μ₁, μ₂ ∈ M₁((0, 1)). Then we may choose ξ ∈ (0, 1) such that, say, μ₂((0, ξ)) > μ₁((0, ξ)). Since μ₁ is ergodic, μ₁({1}) = 0 and 1 belongs to the support of μ₁, by the Birkhoff Ergodic Theorem we may choose x₁ ∈ (0, 1) such that

lim
n→∞
#{1 ≤ k ≤ n : f^k(x₁) ∈ (0, ξ)} n
= μ₁((0, ξ)) for P-a.e. i ∈ Σ.

On the other hand, since μ₂((x₁, 1)) > 0 (1 belongs to the support of μ₂), we can also choose x₂ ∈ (x₁, 1) such that
#{1 ≤ k ≤ n : f^k(x₂) ∈ (0, ξ)}

n
lim
n→∞
ⁱ = μ₂((0, ξ)) for P-a.e. i ∈ Σ,

again by the Birkhoff Ergodic Theorem. But the functions f₁,... , f_N are in- creasing, x₁ < x₂ and therefore

i

≥
#{1 ≤ k ≤ n : f^k(x₁) ∈ (0, ξ)} n
#{1 ≤ k ≤ n : f^k(x₂) ∈ (0, ξ)} n

ⁱ for all i ∈ Σ.
From this we conclude that μ₁((0, ξ)) ≥ μ₂((0, ξ)), which contradicts our as- sumption.

Finally assume, contrary to our claim, that μ_∗ has an atom. Let a ∈ (0, 1) be such that μ_∗({a}) = sup_x∈(0,1) μ_∗({x}) > 0. Since

⁻μ ({a}) = p μ ({f (a)}),_{∗ i ∗}
N
1
i
i=1

i
we obtain that μ_∗({f⁻¹(a)}) = μ_∗({a}) for all i = 1,...,N and consequently