Central limit theorem for commutative semigroups of toral endomorphisms

Let $\Cal S$ be an abelian finitely generated semigroup of endomorphisms of a probability space $(\Omega, {\Cal A}, \mu)$, with $(T_1, ..., T_d)$ a system of generators in ${\Cal S}$. Given an increasing sequence of domains $(D_n) \subset \N^d$, a question is the convergence in distribution of the normalized sequence $|D_n|^{-\frac12} \sum_{{\k} \, \in D_n} \, f \circ T^{\,{\k}}$, for $f \in L^2_0(\mu)$, 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 $\N^d$- or $\Z^d$-actions given by commuting toral automorphisms or endomorphisms on $\T^\rho$, $\rho \geq 1$. 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.