On fixed point theorems and nonsensitivity

TL;DR

This paper explores the role of nonsensitivity in fixed point theory for affine dynamical systems, extending classical theorems and examining properties of function algebras related to topological groups.

Contribution

It extends Ryll-Nardzewski's fixed point theorem using nonsensitivity concepts and establishes properties of Asp(G) related to invariant means and amenability.

Findings

Extended fixed point theorem to nonsensitive systems

Proved left amenability of Asp(G)

Identified uncountably many invariant means on Asp(G)

Abstract

Sensitivity is a prominent aspect of chaotic behavior of a dynamical system. We study the relevance of nonsensitivity to fixed point theory in affine dynamical systems. We prove a fixed point theorem which extends Ryll-Nardzewski's theorem and some of its generalizations. Using the theory of hereditarily nonsensitive dynamical systems we establish left amenability of Asp(G), the algebra of Asplund functions on a topological group G (which contains the algebra WAP(G) of weakly almost periodic functions). We note that, in contrast to WAP(G), for some groups there are uncountably many invariant means on Asp(G). Finally we observe that dynamical systems in the larger class of tame G-systems need not admit an invariant probability measure.