Braid Stability for Periodic Orbits of Area-preserving Surface Diffeomorphisms

Given an area-preserving surface diffeomorphism, what can one say about the topological properties of its periodic orbits? In particular, a finite set of periodic orbits gives rise to a braid in the mapping torus, and one can ask which isotopy classes of braids arise this way. 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 within a Hamiltonian isotopy class, the topological entropy is lower semicontinuous with respect to the Hofer metric. It is an open question whether analogous statements hold for Reeb orbits on contact three-manifolds.