Periodic and fixed points of multivalued maps on Euclidean spaces

TL;DR

This paper investigates the existence of periodic points for multivalued maps on Euclidean spaces and their extensions, establishing conditions under which such points exist or are fixed points, with implications for topological dynamics.

Contribution

It provides new characterizations of periodic points for multivalued maps via their extensions to Stone-Čech compactifications, extending previous results to locally compact Lindelöf spaces.

Findings

A multivalued map has a point of period M iff its extension has such a point.

Extensions of certain maps are fixed-point free under specific conditions.

Results apply to maps on Euclidean and locally compact Lindelöf spaces.

Abstract

We show, in particular, that a multivalued map $f$ from a closed subspace $X$ of $\mathbb R^n$ to ${\rm exp}_k(\mathbb R^n)$ has a point of period exactly $M$ if and only if its continuous extension $\tilde f: \beta X\to {\rm exp}_k(\beta \mathbb R^n)$ has such a point. The result also holds if one repace $\mathbb R^n$ by a locally compact Lindel\"of space of finite dimension. We also show that if $f$ is a colorable map froma normal space $X$ to the space ${\mathcal K}(X)$ of all compact subsets of $X$ then its extension $\tilde f:\beta X\to {\mathcal K}(\beta X)$ is fixed-point free.