Hofer's geometry and topological entropy

TL;DR

This paper investigates how topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces are stable under Hofer's metric, using advanced geometric and topological tools to establish new lower bounds.

Contribution

It introduces new stability results for topological entropy and orbit growth in Hamiltonian dynamics, utilizing enhanced lower bounds based on geometric intersection and homotopy invariants.

Findings

Stability of topological entropy under Hofer's metric.

Enhanced lower bounds for entropy related to geometric intersection.

Application of forcing theory to dynamical quantities.

Abstract

In this article we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong sense for a specific family of maps introduced by Polterovich and Shelukhin. A crucial ingredient comes from some enhancement of lower bounds for the topological entropy and orbit growth forced by a periodic point, formulated in terms of the geometric self-intersection number and a variant of Turaev's cobracket of the free homotopy class that it induces. Those bounds are obtained within the framework of Le Calvez and Tal's forcing theory.