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arXiv · 2106.06374

A fixed point theorem for twist maps

Peizheng Yu·Zhihong Xia
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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.

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Peizheng Yu, Zhihong Xia. 2021-06-11. A fixed point theorem for twist maps. https://arxiv.org/abs/2106.06374

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