Central limit theorem for products of toral automorphisms

Let $({\tau }_n)$ be a sequence of toral automorphisms $\tau_n : x \rightarrow A_n x \hbox{mod}\ZZ^d$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 Koml`os 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$f$isaH\ddot{o}lderfunctionon$$\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.