Central limit theorem for products of toral automorphisms

Let $(τ_n)$ be a sequence of toral automorphisms $τ_n : x \rightarrow A_n x \hbox{mod}\ZZ^d$ with $A_n \in {\cal A}$, where ${\cal A}$ 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 $\sum_{k=1}^n f(τ_k \circ τ_{k-1} \cdots \circ τ_1 x)$ 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 $A_n= A_n(ω)$, with independent choices of $A_n(ω)$ in a finite set of matrices $\in SL(d, \mathbb{Z})$, in order to prove a "quenched" CLT.

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