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Almost everywhere convergence of convolution powers

Published online by Cambridge University Press:  19 September 2008

Alexandra Bellow ,
Roger Jones  and
Joseph Rosenblatt
Alexandra Bellow
Affiliation:
Mathematics Department, Northwestern University, Evanston, IL 60201, USA
Roger Jones
Affiliation:
Mathematics Department, DePaul University, 2219 N. Kenmore, Chicago, IL 60614, USA
Joseph Rosenblatt
Affiliation:
Mathematics Department, Ohio State University, Columbus, OH 43210, USA
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Abstract

Given an ergodic dynamical system (X,B,m, τ) and a probability measure μ on the integers, define for all f ∈ L1(X) The almost everywhere convergence of the convolution powers μnf(x) depends on the properties of μ. If μ has finite and then for all f ∈ Lp(X), 1< p < ∞, exists for a.e. x. However, if m2(μ) is finite and E(μ)≠0, then there exists E∈B such that a.e. and a.e. In the case when m2(μ) is infinite and E(μ)=0 we give examples for which we have divergence and other examples which show convergence is possible.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 3 , September 1994 , pp. 415 - 432
Copyright
Copyright © Cambridge University Press 1994

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