A symplectic proof of a theorem of Franks

TL;DR

This paper provides a new symplectic topology-based proof of Franks' theorem, which states that area-preserving sphere homeomorphisms have either two or infinitely many periodic points, assuming smoothness.

Contribution

It offers the first proof of Franks' theorem using symplectic topology tools, differing from previous methods and relying on results by Ginzburg and Kerman.

Findings

Reproves Franks' theorem under smoothness assumption

Uses symplectic topology tools instead of classical methods

Highlights the role of resonance relations in Hamiltonian dynamics

Abstract

A celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this work we reprove Franks' theorem under the additional assumption that the map is smooth. Our proof uses only tools from symplectic topology and thus differs significantly from all previous proofs. A crucial role is played by the results of Ginzburg and Kerman concerning resonance relations for Hamiltonian diffeomorpisms.