Braid stability and the Hofer metric

Alves, Marcelo R.R.; Meiwes, Matthias

doi:10.5802/ahl.205

Braid stability and the Hofer metric
[Stabilité des tresses et métrique de Hofer]

Alves, Marcelo R.R.¹ ; Meiwes, Matthias ²

¹Faculty of Science, University of Antwerp, Campus Middelheim, Middelheimlaan 1, BE-2020 Antwerpen (Belgium)
²School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978 (Israel)

Annales Henri Lebesgue, Tome 7 (2024), pp. 521-581

Abstract (VO)
Résumé

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 $φ$ .

Dans cet article, nous montrons que le type de tresse d’un ensemble de points fixes d’un difféomorphisme hamiltonien non-dégénéré d’une surface est stable sous des perturbations suffisamment petites par rapport à la métrique de Hofer $d_{Hofer}$ . Nous appelons ce nouveau phénomène stabilité des tresses pour la métrique de Hofer.

Nous appliquons la stabilité des tresses pour étudier la stabilité de l’entropie topologique $h_{top}$ des difféomorphismes hamiltoniens des surfaces par rapport à de petites perturbations pour $d_{Hofer}$ . Nous montrons que $h_{top}$ est semi-continue inférieurement sur l’espace des difféomorphismes hamiltoniens d’une surface fermée, muni de la métrique de Hofer, et sur l’espace des difféomorphismes à support compact du disque bidimensionnel $𝔻$ muni de la métrique Hofer. Cela répond au cas bidimensionnel d’une question de Polterovitch.

Afin de prouver la semi-continuité inférieure de $h_{top}$ par rapport à $d_{Hofer}$ , nous montrons que l’entropie topologique d’un difféomorphisme $φ$ d’une surface compacte peut être reconstituée à partir des types de tresses réalisés par les orbites périodiques de $φ$ .

Reçu le : 2022-04-18
Révisé le : 2023-10-31
Accepté le : 2023-11-05
Publié le : 2024-09-05

MR Zbl

DOI : 10.5802/ahl.205

Classification : 37E30, 37B40, 37J10, 53D40
Keywords: Low-dimensional dynamical systems, Topological entropy, Hamiltonian systems, Floer homology

Affiliations des auteurs :

Alves, Marcelo R.R. ¹ ; Meiwes, Matthias ²

¹ Faculty of Science, University of Antwerp, Campus Middelheim, Middelheimlaan 1, BE-2020 Antwerpen (Belgium)
² School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978 (Israel)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Braid stability and the {Hofer} metric},
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Alves, Marcelo R.R.; Meiwes, Matthias. Braid stability and the Hofer metric. Annales Henri Lebesgue, Tome 7 (2024), pp. 521-581. doi: 10.5802/ahl.205

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