A CAT(0)-VALUED POINTWISE ERGODIC THEOREM

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 mG, a jointly measurable, probability-preserving action [Formula: see text] 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 (Fn)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 [Formula: see text] then as n → ∞ the functions of empirical barycentres [Formula: see text] converge pointwise for almost every ω to a T-invariant function [Formula: see text].