Braid stability and the Hofer metric
Year: 2024
Type: article
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 .
Source: Annales Henri Lebesgue
DOI: 10.5802/ahl.205
Authors: Marcelo R.R. Alves, Matthias Meiwes
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4402269993