A symplectic proof of a theorem of Franks

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.

15 pages. Minor changes. Final version to appear in Compositio Mathematica

Cited by in corpus (13)

• Hamiltonian Pseudo-rotations of Projective Spaces
• On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms
• Hyperbolic Fixed Points and Periodic Orbits of Hamiltonian Diffeomorphisms
• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• Periodic Orbits of Hamiltonian Systems Linear and Hyperbolic at Infinity
• Contact three-manifolds with exactly two simple Reeb orbits
• On hyperbolic points and periodic orbits of symplectomorphisms
• On periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}{\text{P}}^d$ via generating functions
• On Periodic Points of Symplectomorphisms on Surfaces
• On the Hofer-Zehnder conjecture on $\mathbb{C}{\text{P}}^d$ via generating functions (with an appendix by Egor Shelukhin)
• On reversible maps and symmetric periodic points
• On the Hofer-Zehnder conjecture on weighted projective spaces
• Perfect Reeb flows and action-index relations