Central limit theorem for commutative semigroups of toral endomorphisms

2013/04/16 by Guy Cohen, Cohen, Guy, Jean-Pierre Conze +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1304.4556

arxiv created 2013/05/16 · arxiv updated 2013/05/17

Abstract

Let \Cal S be an abelian finitely generated semigroup of endomorphisms of a probability space (Ω, \Cal A, μ), with (T₁, ..., T_d) a system of generators in \Cal S. Given an increasing sequence of domains (D_n) ⊂ \N^d, a question is the convergence in distribution of the normalized sequence |D_n|^-\frac12 ∑_\k ∈ D_n f ∘ T^ \k, for f ∈ L²₀(μ), where T^\k= T₁^k₁ ... T_d^k_d, \k= (k₁, ..., k_d) ∈ \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^ρ, ρ≥ 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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