A fixed point theorem for twist maps

TL;DR

This paper generalizes Poincare's last geometric theorem by replacing the area-preserving condition with a weaker intersection property, proving the existence of fixed points for a broader class of twist maps and exploring applications to reversible systems.

Contribution

It introduces a new fixed point theorem for twist maps under a weaker intersection condition, extending classical results and providing a new proof of Poincare's theorem.

Findings

Any twist map with the intersection property has at least one fixed point.

The result applies to reversible systems, broadening the scope of fixed point theorems.

Provides a new proof of Poincare's geometric theorem.

Abstract

Poincare's last geometric theorem (Poincare-Birkhoff Theorem) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under $f$ at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincare's geometric theorem, our result also has some applications to reversible systems.