Weak mixing for nonsingular Bernoulli actions of countable amenable\n groups

2018/07/16 by Alexandre I. Danilenko, Danilenko, Alexandre I.
Mathematics · #37A20 #37A40 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1807.05905

openalex publication_date 2018/07/16 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

Let G be an amenable discrete countable infinite group,\n A a finite set, and (\μ_g)_{g\∈ G} a family of probability measures on\nA such that \inf_{g\∈ G}\min_{a\∈ A}\μ_g(a)>0. It is shown (among other\nresults) that if the Bernoulli shiftwise action of G on the infinite product\nspace bigotimes_{g\∈ G}(A,\μ_g) is nonsingular and conservative then it is\nweakly mixing. This answers in positive a question by Z.~Kosloff who proved\nrecently that the conservative Bernoulli Bbb Z^d-actions are ergodic. As a\nbyproduct, we prove a weak version of the pointwise ratio ergodic theorem for\nnonsingular actions of G.\n

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