Periodic points of rational area-preserving homeomorphisms

Abstract

An area-preserving homeomorphism isotopic to the identity is said to have rational rotation direction if its rotation vector is a real multiple of a rational class. We give a short proof that any area-preserving homeomorphism of a compact surface of genus at least two, which is isotopic to the identity and has rational rotation direction, is either the identity or has periodic points of unbounded minimal period. This answers a question of Seyfaddini and can be regarded as a Conley conjecture-type result for symplectic homeomorphisms of surfaces beyond the Hamiltonian case. We also discuss several variations, such as maps preserving arbitrary Borel probability measures with full support, maps not isotopic to the identity, and maps on lower genus surfaces. The proofs of the main results combine topological arguments with periodic Floer homology.

arXiv comment14 pages + references. v2: Expository improvements

References in corpus (13)

• Periodic points of Hamiltonian surface diffeomorphisms
• Action and Index Spectra and Periodic Orbits in Hamiltonian Dynamics
• On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms
• Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces
• Hyperbolic Fixed Points and Periodic Orbits of Hamiltonian Diffeomorphisms
• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• On hyperbolic points and periodic orbits of symplectomorphisms
• Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms
• Conley conjecture and local Floer homology
• PFH spectral invariants and $C^{\infty }$ closing lemmas
• On Periodic Points of Symplectomorphisms on Surfaces
• Homotopic rotation sets for higher genus surfaces
• Non contractible periodic orbits for generic hamiltonian diffeomorphisms of surfaces