Braid stability and the Hofer metric

In this article we show that the braid type of a set of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> -periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations which are sufficiently small with respect to the Hofer metric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mi mathvariant="normal">Hofer</mml:mi> </mml:msub> </mml:math> . We call this new phenomenon braid stability for the Hofer metric. We apply braid stability to study the stability of the topological entropy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi mathvariant="normal">top</mml:mi> </mml:msub> </mml:math> of Hamiltonian diffeomorphisms on surfaces under small perturbations with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mi mathvariant="normal">Hofer</mml:mi> </mml:msub> </mml:math> . We show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi mathvariant="normal">top</mml:mi> </mml:msub> </mml:math> is lower semicontinuous on the space of Hamiltonian diffeomorphisms of a closed surface endowed with the Hofer metric, and on the space of compactly supported diffeomorphisms of the two-dimensional disk <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔻</mml:mi> </mml:math> endowed with the Hofer metric. This answers the two-dimensional case of a question of Polterovich. En route to proving the lower semicontinuity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi mathvariant="normal">top</mml:mi> </mml:msub> </mml:math> with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mi mathvariant="normal">Hofer</mml:mi> </mml:msub> </mml:math> , we prove that the topological entropy of a diffeomorphism <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> on a compact surface can be recovered from the braid types realized by the periodic orbits of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> .