Random ergodic theorems and real cocycles

Abstract

We study mean convergence of ergodic averages 1/N σn=0N-1 f ο τk(n)(ω) (*) associated to a measure-preserving transformation or flow τ along the random sequence of times kn(ω) = σj=0n-1 F(Tjω) given by the Birkhoff sums of a measurable function F for an ergodic measure-preserving transformation T. We prove that the sequence (kn(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every ω, the averages (*) converge for every choice of τ, if and only if the "cocycle" F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the "associated flow" of F is countable. We show that this condition holds in many natural situations. When no assumption is made on F, the random sequence (kn(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However, for any aperiodic transformation T, we are able to construct an integrable function F for which the sequence (kn(ω)) is not almost surely universally good for the class of weakly mixing transformations.

Publication Title

Israel Journal of Mathematics

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