In my talk I will address some aspects of a question of Polterovich on the stability of topological entropy for Hamiltonian diffeomorphisms with respect to Hofer's metric. I will focus on some results in dimension two. First I discuss examples for which positive entropy persists under large perturbations. Then I present a braid stability result in the context of Hofer's geometry, and explain what it implies for the question of entropy stability. All relevant definitions will be explained. This is based on joint works with Arnon Chor, and Marcelo Alves.

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