The central limit theorem for function systems on

Theorem 5. If Γ = {g₁,...,g_k} acts minimally, then for any probability vector p = (p₁,...p_k) the iterated function system (Γ,p) admits a unique invariant measure.
Proof The iterated function system (Γ⁻¹,p) satisfies the e–property. Denote by P˜ and U˜ the Markov operator and the dual operator corresponding to (Γ⁻¹,p), respectively. From the proof of the previous proposition it follows that the hypothesis holds provided the unique invariant measure ˜µ for (Γ⁻¹,p) satisfies the condition supp ˜µ = S¹. Now assume that S¹ \ supp ˜µ =6 ∅. Let (a,b) ⊂ S¹ \ supp ˜µ. Set

and observe that S₀ is open and dense in S¹, by the minimal action of Γ. Let µ_∗ ∈ M1(S¹) be an ergodic invariant measure for (Γ,p). Since suppµ_∗ = S¹ and g_i(S₀) ⊂ S₀ for any i = 1,...,k we have µ_∗(S₀) > 0 and U1_S0 = 1_S0. Thus µ_∗(S₀) = 1, by ergodicity of µ_∗. We are going to apply Proposition 2 therefore we have to check that the Cesa´ro e-property holds at any x ∈ S₀. To do this fix x ∈ S₀ and ε > 0. Let I ⊂ S₀ be an open neighbourhood of x. Let f : S¹ → R be a Lipschitz function with the Lipschitz constant L. Choose a finite set {x₀,...,x_N} ⊂ S¹ such that |[x_i−1,x_i]| < ε/L. Since converges weakly to µ˜ for any x ∈ S¹ and ˜µ(I) = 0 we have
0 as n → ∞.
On the other hand, we have
1 n U˜k1I(x) = 1 Xn 1 Xk pi1 ···pik1I(gi−11 ◦ ··· ◦ gi−k1(x)) nn

k=1 k=1 (i ,...,i )∈Σ_k n 1 ⁿ X 1 Xk i₁ ···pi_k1g_i1,...,i_k(I)(x)
=p
k=1 (i ,...,i )∈Σ_k
1 n ⁿ X 1 Xk i₁ ···pi_k1_{x}(gi₁,...,i_k(I)).
=p
k=1 (i ,...,i )∈Σ_k
Consequently, we have
0 as n → ∞.
Thus, for any x,y ∈ I, the interval g_i1···ik([x,y]) typically will be located between some points x_i−1 and x_i, so that its length will be less than ε/L. Hence

Since ε > 0 was arbitrary, the operator P satisfies the Cesa´ro e–property. This completes the proof. •

5. Central Limit Theorem

Let Γ = {g₁,...,g_k} be a family of homeomorphisms on S¹ and let p = (p₁,...p_k) be a probability vector. Let µ_∗ ∈ M1(S¹) be an invariant measure for the iterated function system (Γ,p). By (X_n)_n≥0 we shall denote the stationary Markov chain corresponding to the iterated function system (Γ,p). Let ϕ : S¹ → R be a H¨older continuous function satisfying RS₁ ϕdµ_∗ = 0. Set
S_n := S_n(ϕ) = ϕ(X₀) + ... + ϕ(X_n)
and for n ≥ 1.
Our main purpose in this section is to prove that is asymptotically normal (the CLT theorem). Maxwell and Woodroofe in [9] studied general Markov chains and formulated a simple sufficient condition for the CLT which in our case takes the form
(2) ,
where k · kL2_(µ∗₎ denotes the L²(µ_∗) norm. More precisely, the result proved in [9] says that if (2) holds, then the limit ) exists and is finite, and then the distribution of tends to N(0,σ).
We start with recalling some properties of iterated function systems obtained by D. Malicet (see Theorem A and Corollary 2.6 in [8]):