Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions by toral endomorphisms

Abstract

In this paper we prove the central limit theorem for the following multisequence $$\sum_{n_1=1}^{N_1} ... \sum_{n_d=1}^{N_d} f(A_1^{n_1}...A_d^{n_d} {\bf x} )$$ where $f$ is a Hölder's continue function, $A_1,\ldots,A_d$ are $s\times s$ partially hyperbolic commuting integer matrices, and $\bf x$ is a uniformly distributed random variable in $[0,1]^s$. Next we prove the functional central limit theorem, and the almost sure central limit theorem. The main tool is the $S$-unit theorem.