Entropy, braids, and Hofer's metric

Abstract: Topological entropy captures the orbit complexity of a dynamical system with the help of a single non-negative number. Detecting robustness of this number under perturbation is a way to understand stability features of a chaotic system. In my talk, I will address the problem of robustness of entropy for Hamiltonian diffeomorphisms in terms of Hofer's metric. Our main focus lies on dimension 2, where there is a strong connection between topological entropy and the existence of specific braid types of periodic orbits. I explain that the construction of eggbeater maps of Polterovich-Shelukhin and their generalizations by Chor provide robustness even under large perturbation: the entropy will not drop much when perturbing the specific diffeomorphism in some ball of large Hofer-radius. I furthermore discuss a result that any braid of non-degenerate one-periodic orbits with pairwise homotopic strands persists under generic Hofer-small perturbations, which yields a local entropy robustness result for surfaces. This talk is based on joint works with Arnon Chor, and Marcelo Alves.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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