A note on an ergodic theorem in weakly uniformly convex geodesic spaces

TL;DR

This paper extends an ergodic theorem originally proven for uniformly convex geodesic spaces with nonpositive curvature to a broader class of spaces with weaker convexity assumptions, broadening its applicability.

Contribution

It demonstrates that the ergodic theorem by Karlsson and Margulis applies under weaker uniform convexity conditions, not requiring nonpositive curvature.

Findings

The ergodic theorem holds in weakly uniformly convex geodesic spaces.

The result broadens the class of spaces where the ergodic theorem applies.

It maintains the asymptotic behavior analysis of integrable cocycles.

Abstract

Karlsson and Margulis proved in the setting of uniformly convex geodesic spaces, which additionally satisfy a nonpositive curvature condition, an ergodic theorem that focuses on the asymptotic behavior of integrable cocycles of nonexpansive mappings over an ergodic measure-preserving transformation. In this note we show that this result holds true when assuming a weaker notion of uniform convexity.