Volume 7 (2024)
Alves, Marcelo R.R.; Meiwes, Matthias
Braid stability and the Hofer metric
Annales Henri Lebesgue, Volume 7 (2024), pp. 521-581.

Metadata

DOI10.5802/ahl.205
Keywords Low-dimensional dynamical systems, Topological entropy, Hamiltonian systems, Floer homology

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 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 top of Hamiltonian diffeomorphisms on surfaces under small perturbations with respect to d Hofer . We show that h 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 diffeomorphisms of the two-dimensional disk 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 top with respect to d Hofer , we prove that the topological entropy of a diffeomorphism φ on a compact surface can be recovered from the braid types realized by the periodic orbits of φ.


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