An L^1 ergodic theorem with values in a nonpositively curved space via a canonical barycenter map

TL;DR

This paper generalizes the Birkhoff ergodic theorem to functions valued in nonpositively curved spaces using a barycenter map, removing the need for local compactness and establishing a fixed point theorem for isometric actions.

Contribution

It introduces a new barycenter map construction that extends ergodic theorems to nonpositively curved spaces without requiring local compactness.

Findings

Extended Birkhoff ergodic theorem to nonpositively curved spaces

Constructed a barycenter map applicable without local compactness

Proved a fixed point theorem for isometries on Buseman spaces

Abstract

We extend a recent result of Tim Austin (see arXiv:0905.0515) to the L^1 setting, thus providing a general version of the Birkhoff ergodic theorem for functions taking values in nonpositively curved spaces. In this setting, the notion of a Birkhoff sum is replaced by that of a barycenter along the orbit. The construction of an appropriate barycenter map is the core of this note. In particular, we solve a problem raised by K.-T. Sturm showing that local compactness for the underlying space is superfluous for the construction (this extends a result of A. Es-Sahib and H. Heinich). As a byproduct of our construction, we prove a fixed point theorem for actions by isometries on a Buseman space.