Existence of group nonexpansive retractions and ergodic theorems in topological groups

TL;DR

This paper investigates the existence of group nonexpansive retractions and establishes ergodic theorems within the context of topological groups, focusing on compact subsets and fixed point properties.

Contribution

It introduces the concept of group nonexpansive mappings and proves the existence of retractions onto fixed point sets in topological groups.

Findings

Existence of group nonexpansive retractions onto fixed point sets

Development of ergodic theorems for these mappings

Extension of fixed point theory in topological groups

Abstract

Suppose that $G$ is a topological group and $ C $ a compact subset of $G$. In this paper we define group nonexpansive mappings and then we consider $\sc = \{T_{i} : i \in I \}$ as a family of the group nonexpansive mappings on $C$. Also we study the existence of group nonexpansive retractions $P_{i}$ from $C$ onto $\text{Fix}(\sc)$ such that $P_{i}T_{i} = T_{i}P_{i} = P_{i}$.