Fixed Point Theorem for Non-Self Maps of Regions in the Plane

TL;DR

This paper generalizes Brouwer's plane translation theorem to non-self maps between certain planar regions, showing that such maps without fixed points also lack recurrent points.

Contribution

It extends classical fixed point results to non-self maps of planar regions, broadening the scope of fixed point theory in topology.

Findings

Maps with no fixed points have no recurrent points

Generalization of Brouwer's theorem to non-self maps

Applicable to compact, simply connected, locally connected regions in R^2

Abstract

Let X and Y be compact, simply connected and locally connected subsets of R^2, and let f : X -> Y be a homeomorphism isotopic to the identity on X. Generalizing Brouwer's plane translation theorem for self-maps of the plane, we prove that f has no recurrent (in particular, no periodic) points if it has no fixed points.