Effective results on nonlinear ergodic averages in CAT$(\kappa)$ spaces

TL;DR

This paper uses proof mining techniques to derive explicit, uniform rates of convergence for nonlinear ergodic averages in CAT(0) spaces, extending the classical ergodic theorem.

Contribution

It provides the first effective, quantitative bounds for nonlinear ergodic averages in CAT(0) spaces, generalizing the von Neumann theorem.

Findings

Derived explicit rates of asymptotic regularity

Established uniform metastability bounds

Extended classical ergodic results to nonlinear settings

Abstract

In this paper we apply proof mining techniques to compute, in the setting of CAT$(\kappa)$ spaces (with $\kappa >0$), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.