Recurrence in mean
type: Item
instance of: scholarly article
title: Recurrence in mean
zbMATH Open document ID: 1029.37008
DOI: 10.1023/A:1020548815535
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: Let \((X,d)\) be a metric space. Using a generalization \(H_h\) of the standard Hausdorff measure \(H_\alpha\) (coinciding with \(H_\alpha\) for \(h(t)=t^\alpha\)), the author shows that for a discrete time dynamical system \((T,X,\mu)\), such that \(H_h(X)<\infty\) and any \(\mu\)-measurable set is \(H_h\)-measurable, the \(\mu\)-average value of the recurrence constant \(C(x):=\liminf_{n\to\infty} nh(d(T^nx,x))\) over any measurable set \(A\) cannot exceed \(H_h(A)\). This estimate generalizes results by \textit{M. D. Boshernitzan} [Invent. Math. 113, 617-631 (1993; Zbl 0839.28008)] obtained in the case of \(H_\alpha\).
reviewed by: Michael L. Blank
Mathematics Subject Classification ID: 37B20
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 1928319
zbMATH Keywords: recurrence
zbMATH Keywords: invariant measure
author: Ilya D. Shkredov
MaRDI profile type: Publication
full work available at URL: a:1020548815535
OpenAlex ID: W128090924
description: scientific article; zbMATH DE number 1928319