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 $α\geq 0$ there is a sequence of Hamiltonian diffeomorphisms $φ<em>k$ on the two-torus $T<sup>2$ such that $$d_H(\operatorname{Ent}</em>{\leq α}(T^{2,ω),φ_k)\to} \infty \quad (k \to \infty),$$ where $$\text{Ent}_{\leq α}(T^2,ω)\subset \text{Ham}(T<sup>2,ω)$$ denotes the set of Hamiltonian diffeomorphisms with topological entropy at most $α$ 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.