A cat(0)-valued pointwise ergodic theorem

Tim Austin

Research output: Contribution to journalArticle

Abstract

This note proves a version of the pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space. The precise setting consists of an amenable locally compact group G with left Haar measure m G, a jointly measurable, probability-preserving action T : G (Ω, F, P) of G on a probability space, and a separable complete CAT(0)-space (X, d) with barycentre map b. In this setting we show that if (F n) n < 1 is a tempered Følner sequence of compact subsets of G and f : Ω → X is a measurable map such that for some (and hence any) fixed x ∈ X, we have ∫ Ω d(f(ω),x) 2,P(dω) < ∞, then as n → ∞ the functions of empirical barycentres ω b(1/m G(F n) ∫ Fnδ f(T gω),m G(dg)) converge pointwise for almost every ω to a T-invariant function f̄: Ω → X.

Original languageEnglish (US)
Pages (from-to)145-152
Number of pages8
JournalJournal of Topology and Analysis
Volume3
Issue number2
DOIs
StatePublished - Jun 1 2011

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Keywords

  • CAT(0)-space
  • Pointwise ergodic theorem

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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