Central limit theorem for products of toral automorphisms

Abstract

Let $({\tau }_n)$ be a sequence of toral automorphisms $τ_n : x \rightarrow A_n x \hbox{mod}\ZZ^d$ with $A_n\in \mathcal{A}$, where $\mathcal{A}$ is a finite set of matrices in $SL(d,\mathbb{Z})$. Under some conditions the method of "multiplicative systems" of Komlòs can be used to prove a Central Limit Theorem for the sums ${\sum }_{k=1}^{n}f({\tau }_k\circ {\tau }_{k-1}\cdots \circ {\tau }_{1}x)$ if $f$ is a Hölder function on ${\mathbb{T}}^d$. These conditions hold for $2\times 2$ matrices with positive coefficients. In dimension $d$ they can be applied when $A_n=A_n(\omega )$, with independent choices of $A_n(\omega )$ in a finite set of matrices $\in SL(d,\mathbb{Z})$, in order to prove a "quenched" CLT.