# Almost Periodicity and Ergodic Theorems for Nonexpansive Mappings and Semigroups in Hadamard Spaces

## TL;DR

This paper establishes mean ergodic theorems for nonexpansive mappings and semigroups in Hadamard spaces, utilizing almost periodicity of orbits, with applications to evolution equations on Hadamard manifolds.

## Contribution

It proves the mean ergodic theorem for nonexpansive mappings in Hadamard spaces, extending ergodic theory to non-linear geometric contexts.

## Key findings

- Proved mean ergodic theorem for nonexpansive mappings in Hadamard spaces
- Established almost periodicity of orbits in metric and Hadamard spaces
- Applied results to asymptotic analysis of evolution equations on Hadamard manifolds

## Abstract

The main purpose of this paper is to prove the mean ergodic theorem for nonexpansive mappings and semigroups in locally compact Hadamard spaces, including finite dimensional Hadamard manifolds. The main tool for proving ergodic convergence is the almost periodicity of orbits of a nonexpansive mapping. Therefore, in the first part of the paper, we study almost periodicity (and as a special case, periodicity) in metric and Hadamard spaces. Then, we prove a mean ergodic theorem for nonexpansive mappings and continuous semigroups of contractions in locally compact Hadamard spaces. Finally, an application to the asymptotic behavior of the first order evolution equation associated to the monotone vector field on Hadamard manifolds is presented.