A Mean Ergodic Theorem for Nonexpansive Mappings in Hadamard spaces

TL;DR

This paper extends the Baillon nonlinear ergodic theorem to Hadamard spaces, showing that Karcher means of nonexpansive mappings converge weakly to fixed points, including for continuous semigroups.

Contribution

It establishes a mean ergodic theorem for nonexpansive mappings in Hadamard spaces, broadening the scope of ergodic results to nonpositive curvature metric spaces.

Findings

Karcher means converge weakly to fixed points

Results apply to 1-parameter continuous semigroups

Extends Baillon nonlinear ergodic theorem

Abstract

In this paper, we prove a mean ergodic theorem for nonexpansive mappings in Hadamard (nonpositive curvature metric) spaces, which extends the Baillon nonlinear ergodic theorem. The main result shows that the sequence given by the Karcher means of iterations of a nonexpansive mapping with a nonempty fixed point set converges weakly to a fixed point of the mapping. This result also remains true for a 1-parameter continuous semigroup of contractions.