  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.
