The central limit theorem for function systems on

Proposition 6. Let Γ = {g₁,...,g_k} be a familiy of homeomorphisms on S¹ such that there is no measure invariant by Γ. Let p = (p₁,...p_k) be a probability vector. If Γ acts minimally, then there exists q ∈ (0,1) such that:

• 
  for every x ∈ S¹ there exists an open neighbourhood I of x and Σ˜ ⊂ Σ with P(Σ)˜ > 0 such that for i = (i₁,i₂,...) ∈ Σ˜ we have
  

|gin,...,i1(I)| ≤ qn;

• 
  (asymptotic stability) for any x ∈ S¹ the sequence (P ⁿδ_x)_n≥1, where P is the Markov operator corresponding to (Γ,p), converges in the ∗–weak topology to the unique invariant measure µ_∗.
  

First, let us note that Proposition 6 implies the e-property:
Proposition 7. Under the hypothesis of Proposition 6 the operator P corresponding to (Γ,p) satisfies the e–property. Moreover, for any open interval I ⊂ S¹ there exists m ∈ N such that
.
Proof From Proposition 6 it follows that P is asymptotically stable. Since suppµ_∗ = S¹, in particular Int_S1 suppµ_∗ =6 ∅, from Theorem 4.8 in [4] we obtain that P satisfies the e–property.
Now fix an open interval I ⊂ S¹. Since the operator P corresponding to (Γ,p) is asymptotically stable and it satisfies the e–property, for any x ∈ S¹ there exists N_x ∈ N such that
(3) P ⁿδ_x(I) > µ_∗(I)/2 > 0
for all n ≥ N_x. On the other hand, from the e–property it follows that for every x ∈ S¹ we may choose some neighbourhood U_x of x such that the above property will be satisfied if we replace x with an arbitrary y from U_x. By compactness of S¹ we may find x₁,...,x_k ∈ S¹ such that S . Then for any m ≥ max{N_x1,...,N_xk} we have

and the proof is completed. •
We are going to introduce the following notation: for ϕ : S¹ → R, x ∈ S¹, ω =
( i₁,i₂,...) ∈ Σ or) we shall denote the sum:
.
Lemma 8. Let Γ = {g₁,...,g_k} act minimally and there is no measure invariant by Γ. Assume that p₁ = ··· = p_k = 1/k and let ϕ : S¹ → R be an arbitrary Lipschitz continous function with Lipschitz constant 1. Then there exist α ∈ (0,1),m ∈ N and γ > 0 such that for an arbitrary pair of points x,y ∈ S¹ there exist two collections of measurable pairwise disjoint sets P₁,P₂ ... ⊂ Σ and such that: for every l ≥ 1
Σ = P1 ∪ P2 ∪ ··· ∪ Pl ∪ Bl, ,
where the sets B_l,B_l^′ are unions (possibly infinite) of some disjoint cylinders and a (measure preserving) bijection b_l : Σ → Σ satisfying
for every j = 1,...l;