A generalization of the Brouwer plane translation theorem

TL;DR

This paper extends the Brouwer plane translation theorem by proving that any orientation-preserving homeomorphism of the plane with a topologically chain recurrent point must have a fixed point.

Contribution

It generalizes the classical theorem to a broader class of homeomorphisms based on chain recurrence.

Findings

Homeomorphisms with chain recurrent points have fixed points

Extension of Brouwer's theorem to more general conditions

New insights into plane dynamics and fixed point existence

Abstract

We show that if an orientation-preserving homeomorphism of the plane has a topologically chain recurrent point, then it has a fixed point, generalizing the Brouwer plane translation theorem.