Convergence of polynomial ergodic averages
type: Item
instance of: scholarly article
title: Convergence of polynomial ergodic averages
zbMATH Open document ID: 1085.28009
DOI: 10.1007/BF02772534
author: Bryna Kra
author: Bernard Host
published in: Israel Journal of Mathematics
publication date: 2006-02-06T00:00:00Z
review text: The main result of the paper is the proof of \(L^2\)-convergence of a product of measurable functions evaluated along polynomial times. Let \((X,{\mathcal B}, \mu,T)\) be an invertible dynamical system, let \(\{p_i(n)\}_{i=1}^\ell\) be integer polynomials, and let at least one of the following conditions hold: (i) the system \((X,{\mathcal B}, \mu,T)\) is totally ergodic; (ii) the polynomials are all of degree greater than one; (iii) the polynomials are all of degree one. Then for any collection of \(L^\infty(\mu)\) functions \(\{f_i\}_{i=1}^\ell\) and any two sequences \(\{M_i\}, \{N_i\}\) of integers with \(N_i\to\infty\) the averages \[ \frac1{N_i} \sum_{n=M_i}^{M_i+N_i-1} \prod_{k=1}^l f_k\circ T^{p_k} ~d\mu \] converge as \(i\to\infty\).
reviewed by: Michael L. Blank
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A25
zbMATH DE Number: 5003388
zbMATH Keywords: ergodicity
zbMATH Keywords: mixing
zbMATH Keywords: ergodic average
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5003388