Braid stability and the Hofer metric

TL;DR

This paper introduces braid stability under the Hofer metric for Hamiltonian diffeomorphisms on surfaces, demonstrating lower semicontinuity of topological entropy with respect to this metric, and connecting braid types to entropy.

Contribution

It establishes the concept of braid stability under the Hofer metric and proves the lower semicontinuity of topological entropy for surface diffeomorphisms in this metric.

Findings

Braid type of periodic orbits is stable under small Hofer perturbations.

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

Entropy can be recovered from braid types of periodic orbits.

Abstract

In this article we show that the braid type of a set of $1$-periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations which are sufficiently small with respect to the Hofer metric $d_{\rm Hofer}$. We call this new phenomenon braid stability for the Hofer metric. We apply braid stability to study the stability of the topological entropy $h_{\rm top}$ of Hamiltonian diffeomorphisms on surfaces with respect to small perturbations with respect to $d_{\rm Hofer}$. We show that $h_{\rm top}$ is lower semicontinuous on the space of Hamiltonian diffeomorphisms of a closed surface endowed with the Hofer metric, and on the space of compactly supported diffeormophisms of the two-dimensional disk $\mathbb{D}$ endowed with the Hofer metric. This answers the two-dimensional case of a question of Polterovich. En route to proving the lower semicontinuity of $h_{\rm top}$ with respect to $d_{\rm Hofer}$, we prove that the topological entropy of a diffeomorphism $\phi$ on a compact surface can be recovered from the topological entropy of the braid types realised by the periodic orbits of $\phi$.