A fixed point theorem for twist maps

Zhihong Xia; Peizheng Yu

doi:10.3934/dcds.2022045

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2022, Volume 42, Issue 8: 4051-4059. Doi: 10.3934/dcds.2022045

This issue Previous Article Local behavior of solutions to a fractional equation with isolated singularity and critical Serrin exponent Next Article Concentrated solutions for a critical elliptic equation

A fixed point theorem for twist maps

Zhihong Xia^1, and
Peizheng Yu^2,

1.
Department of Mathematics, Northwestern University, Evanston, IL 60208 USA
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

Received: November 2021

Revised: March 2022

Early access: April 2022

Published: August 2022

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Abstract

Poincaré's last geometric theorem (Poincaré-Birkhoff Theorem [2]) 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 Poincaré's geometric theorem, our result also has some applications to reversible systems.

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Mathematics Subject Classification: Primary: 37E40, 37E30, 37J12.

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References

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