Bounded homeomorphisms of the open annulus

TL;DR

This paper generalizes the Poincaré-Birkhoff theorem for the open annulus, establishing fixed points for certain homeomorphisms and exploring their periodic orbits through rotation numbers.

Contribution

It introduces a fixed point theorem for bounded homeomorphisms of the open annulus and analyzes cases with at most one fixed point, extending classical results.

Findings

Fixed point theorem for bounded homeomorphisms

Existence of periodic orbits via rational rotation numbers

Special case analysis for homeomorphisms with at most one fixed point

Abstract

We prove a generalization of the Poincar\'e-Birkhoff theorem for the open annulus showing that if a homeomorphism satisfies a certain twist condition and the nonwandering set is connected, then there is a fixed point. Our main focus is the study of bounded homeomorphisms of the open annulus. We prove a fixed point theorem for bounded homeomorphisms and study the special case of those homeomorphisms possessing at most one fixed point. Lastly we use the existence of rational rotation numbers to prove the existence of periodic orbits.