Central limit theorem for commutative semigroups of toral endomorphisms

Abstract

Let $\Cal S$ be an abelian finitely generated semigroup of endomorphisms of a probability space $(Ω, {\Cal A}, μ)$, with $(T_1,...,T_d)$ a system of generators in ${\Cal S}$. Given an increasing sequence of domains $(D_n)\subset {\mathbb{N}}^d$, a question is the convergence in distribution of the normalized sequence $|D_n|^{-\frac12} \sum_{{\k} \, \in D_n} \, f \circ T^{\,{\k}}$, for $f\in L_0^2(\mu )$, where $T^{\k}= T_1^{k_1} ... T_d^{k_d}$, ${\k}= (k_1, ..., k_d) \in {\N}^d$. After a preliminary spectral study when the action of $\Cal S$ has a Lebesgue spectrum, we consider ${\mathbb{N}}^d$- or ${\mathbb{Z}}^d$-actions given by commuting toral automorphisms or endomorphisms on $\T^ρ$, $\rho \ge 1$. For a totally ergodic action by automorphisms, we show a CLT for the above normalized sequence or other summation methods like barycenters, as well as a criterion of non-degeneracy of the variance, when $f$ is regular on the torus. A CLT is also proved for some semigroups of endomorphisms. Classical results on the existence and the construction of such actions by automorphisms are recalled.

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