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arXiv · 2112.11351

Braid stability and the Hofer metric

Marcelo R. R. Alves·Matthias Meiwes
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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 $\phi$.

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Marcelo R. R. Alves, Matthias Meiwes. 2021-12-21. Braid stability and the Hofer metric. https://arxiv.org/abs/2112.11351

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