A topological version of the Poincar\'e-Birkhoff theorem with two fixed points

TL;DR

This paper establishes a topological property for certain annulus homeomorphisms, extending the Poincaré-Birkhoff theorem to cases with at most one fixed point, and provides new insights into fixed point existence under weaker conditions.

Contribution

It introduces a topological generalization of the Poincaré-Birkhoff theorem applicable to homeomorphisms with at most one fixed point, relaxing traditional boundary twist conditions.

Findings

Identifies a topological property for annulus homeomorphisms with limited fixed points.

Shows that boundary twist conditions can be replaced by orbit unboundedness.

Provides a new proof for a version of the Conley-Zehnder theorem in the annulus.

Abstract

The main result of this paper gives a topological property satisfied by any homeomorphism of the annulus $\mathbb{A}=\mathbb{S}^1 \times [-1,1]$ isotopic to the identity and with at most one fixed point. This generalizes the classical Poincar\'e-Birkhoff theorem because this property certainly does not hold for an area preserving homeomorphism of $\mathbb{A}$ with the usual boundary twist condition. We also have two corollaries of this result. The first one shows in particular that the boundary twist assumption may be weakened by demanding that the homeomorphism $h$ has a lift $H$ to the strip $\widetilde{\mathbb{A}} = \mathbb{R} \times [-1,1]$ possessing both a forward orbit unbounded on the right and a forward orbit unbounded on the the left. As a second corollary we get a new proof of a version of the Conley-Zehnder theorem in $\mathbb{A}$: if a homeomorphism of $\mathbb{A}$ isotopic to the identity preserves the area and has mean rotation zero, then it possesses two fixed points.