  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.
