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. 2020 Mathematics Subject Classification. 37E30, 37J46. A. Chor was partially supported by the Israel Science Foundation grant 667/18. M. Meiwes was partially supported by the Israel Science Foundation grant 2026/17. 1 In fact a similar result holds for C 0 -perturbations of T , see [45].1 * (φ) α for r ∈ Ê, and whose linear maps π r,s : HF r * (φ) α → HF s * (φ) α are induced by the inclusion maps CF r * (M, H) α → CF s * (M, H) α , where H is a Hamiltonian that generates φ.
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