Central limit theorem for products of toral automorphisms

2010/06/21 by Jean-Pierre Conze, Stéphane Le Borgne, Conze, Jean-Pierre +4
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1006.4051

arxiv created 2010/06/21 · arxiv updated 2010/06/22

Abstract

Let (τ_n) be a sequence of toral automorphisms τ_n : x → A_n x \hboxmod\ZZ^d with A_n ∈ \cal A, where \cal A is a finite set of matrices in SL(d, ℤ). Under some conditions the method of "multiplicative systems" of Komlòs can be used to prove a Central Limit Theorem for the sums ∑_k=1ⁿ f(τ_k ∘ τ_k-1 ⋯ ∘ τ₁ x) if f is a Hölder function on \mathbbT^d. These conditions hold for 2× 2 matrices with positive coefficients. In dimension d they can be applied when A_n= A_n(ω), with independent choices of A_n(ω) in a finite set of matrices ∈ SL(d, ℤ), in order to prove a "quenched" CLT.

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