Braid stability and the Hofer metric

Abstract

In this article we show that the braid type of a set of 1periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations which are sufficiently small with respect to the Hofer metric d Hofer . We call this new phenomenon braid stability for the Hofer metric.We apply braid stability to study the stability of the topological entropy h top of Hamiltonian diffeomorphisms on surfaces with respect to small perturbations with respect to d Hofer . We show that h top 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 diffeormophisms of the two-dimensional disk D endowed with the Hofer metric. This answers the two-dimensional case of a question of Polterovich.En route to proving the lower semicontinuity of h top with respect to d Hofer , we prove that the topological entropy of a diffeomorphism φ on a compact surface can be recovered from the topological entropy of the braid types realised by the periodic orbits of φ.

Non-realizability of the pure braid group as area-preserving homeomorphisms

“…The braid group is a classical object in low dimensional topology and it has been studied by mathematicians for decades. Recently, some progress has been made to study the braid group from view points of symplectic geometry [1,11,10,8]. To describe their results, we first review some definitions in symplectic geometry quickly.…”

“…We can associate it to a vector field X H called the Hamiltonian vector field by the relation ω(X H , •) = d Σ H. Let ϕ t H be the flow generated by X H . A diffeomorphism ϕ of Σ is called a Hamiltonian symplecticmorphism if ϕ = ϕ 1 H for some H. Let Ham(Σ, ω) be the set of all Hamiltonian symplecticmorphisms. In fact, Ham(Σ, ω) is a group.…”

“…H with respect to the Hofer metric, then there is a union of distinct 1-periodic orbits α ′ of ϕ 1 H ′ such that b(α) = b(α ′ ) [1]. Recently, M. Hutchings generalizes this result to the general case (without the Hamiltonian assumption and α can contain periodic orbits with arbitrary periods) by using holomorphic curve methods [8].…”

Hofer's geometry and topological entropy

“…Also recently, motivated by the present work, stability properties with respect to d Hofer on the braid types of periodic orbits of Hamiltonian surface diffeomorphisms have been studied by Alves and the second author (see [AM21]). One dynamical consequence is that h top is lower semi-continuous on (Ham(Σ, ω), d Hofer ) for closed surfaces Σ.…”