A CAT(0)-valued pointwise ergodic theorem

TL;DR

This paper extends the pointwise ergodic theorem to functions valued in separable complete CAT(0)-spaces, using barycentres of empirical distributions and approximation techniques.

Contribution

It introduces a novel ergodic theorem for CAT(0)-valued functions, generalizing classical results from real-valued functions to metric space-valued functions.

Findings

Proves a pointwise ergodic theorem for CAT(0)-valued functions.

Uses barycentres of empirical distributions in the proof.

Builds on classical ergodic theorems for real-valued functions.

Abstract

In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space subject to a probability-preserving action of an amenable l.c.s.c. group, where in the CAT(0) setting the role of ergodic averages is played by the barycentres of the empirical distributions of a CAT(0)-valued map along an orbit of the group action. The proof rests on an approximation argument and an appeal to that result for real-valued maps.