Universiteit Utrecht

Department of Mathematics

Central limit theorems in ergodic theory

Attila Herczegh

30 June 2008

Let I=[0,1) and B be the Borel σ-algebra and let T and S be ergodic and measure preserving transformations with μ_1 and μ_2 probability measures on the space (I, B). Take two partitions on I, P and Q. Then we look at the sequences of partitions P_n and Q_n generated by T and S, respectively. Define K_n(x)= min {j ≥ 1: T^j x ∈ Q_n(x)}, where Q_n(x) is the element of Q_n containing x. We give conditions on T and S such that a central limit theorem holds for log(K_n). Then we also look at the quantity m(n,x)= max{ m ≥ 1 : P_n(x) ⊆ Q_m(x)} and focus on the speed of convergence in m(n,x)/n → h(T)/h(S), i.e. we give conditions on T and S such that the central limit theorem holds for m(n,x) or that the convergence is even faster than 1/√n. In the end, we give some examples that satisfy the conditions made on T and S.

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