Braid stability for periodic orbits of area-preserving surface diffeomorphisms

TL;DR

This paper proves that the braid classes of finite sets of periodic orbits in area-preserving surface diffeomorphisms are stable under small Hamiltonian perturbations, implying lower semicontinuity of topological entropy in the Hofer metric.

Contribution

It extends previous results by showing braid stability and entropy lower semicontinuity for a broader class of surface diffeomorphisms under Hamiltonian perturbations.

Findings

Braid classes of periodic orbits are stable under small Hamiltonian perturbations.

Topological entropy is lower semicontinuous with respect to the Hofer metric.

Results extend prior work on fixed points to more general periodic orbits.

Abstract

We consider an area-preserving diffeomorphism of a compact surface, which is assumed to be an irrational rotation near each boundary component. A finite set of periodic orbits of the diffeomorphism gives rise to a braid in the mapping torus. We show that under some nondegeneracy hypotheses, the isotopy classes of braids that arise from finite sets of periodic orbits are stable under Hamiltonian perturbations that are small with respect to the Hofer metric. A corollary is that for a Hamiltonian isotopy class of such maps, the topological entropy is lower semicontinuous with respect to the Hofer metric. This extends results of Alves-Meiwes for braids arising from finite sets of fixed points of Hamiltonian surface diffeomorphisms.