Braid Stability and a Floer theory of Braid Isotopies

Abstract

We build a Floer theory on the space of braids of periodic orbits of a Hamiltonian flow on a closed symplectic surface. The differential and continuation maps are defined by counting Floer isotopies of braids: tuples of Floer cylinders with pairwise disjoint graphs. With this new perspective, we prove a quantitative braid stability result that shows the persistence of braids under possibly large Hamiltonian perturbations. As an application, we show that for every $\alpha \ge 0$ there is a sequence of Hamiltonian diffeomorphisms ${\varphi }_k$ on the two-torus $T^2$ such that $$d_H({\mathrm{Ent}}_{\le \alpha }(T^2,\omega ),{\varphi }_k)\to \infty (k\to \infty ),$$ where ${\text{Ent}}_{\le \alpha }(T^2,\omega )\subset \text{Ham}(T^2,\omega )$ denotes the set of Hamiltonian diffeomorphisms with topological entropy at most $\alpha$ and $d_H$ the Hofer metric. We prove this result by studying only contractible periodic orbits, whereas analogous higher-genus statements were previously obtained using non-contractible orbits.

arXiv comment67 pages, 15 figures, v2: corrected typos and added more citations and new remarks, comments are welcome