Central Limit Theorem for Commutative Semigroups of Toral Endomorphisms

Abstract. Let S be an abelian finitely generated semigroup of endomorphisms of a probability space (Ω,A, μ), with (T1, ..., Td) a system of generators in S. Given an increasing sequence of domains (Dn) ⊂ N, a question is the convergence in distribution of the normalized sequence |Dn| 12 ∑ k∈Dn f ◦ T , or normalized sequences of iterates of barycenters Pf = ∑ j pjf ◦ Tj, where T k = T k1 1 ...T kd d , k = (k1, ..., kd) ∈ N. After a preliminary spectral study when the action of S has a Lebesgue spectrum, we consider totally ergodic d-dimensional actions given by commuting endomorphisms on a compact abelian connected group G and we show a CLT, when f is regular on G. When G is the torus, a criterion of non-degeneracy of the variance is given.