辫子稳定性与辫子同位素的Floer理论
Braid Stability and a Floer theory of Braid Isotopies
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中文总结 AI 辅助
本文在闭辛曲面哈密顿流周期轨道辫子空间上建立Floer理论，证明定量辫子稳定性，并应用于二维环面，仅用可缩轨道证明拓扑熵约束集的Hofer距离趋于无穷。
中文摘要 AI 辅助
我们在闭辛曲面上的哈密顿流的周期轨道辫子空间中构建了一个Floer理论。微分映射和延续映射通过计数辫子的Floer同位素来定义：即具有两两不相交图的Floer圆柱元组。借助这一新视角，我们证明了一个定量辫子稳定性结果，该结果表明辫子在可能较大的哈密顿扰动下仍然持续存在。作为应用，我们证明对于每个$\alpha\geq 0$，在二维环面$T^2$上存在一列哈密顿微分同胚$\phi_k$，使得$$d_H(\operatorname{Ent}_{\leq \alpha}(T^2,\omega),\phi_k)\to \infty \quad (k \to \infty),$$其中$\text{Ent}_{\leq \alpha}(T^2,\omega)\subset \text{Ham}(T^2,\omega)$表示拓扑熵至多为$\alpha$的哈密顿微分同胚的集合，$d_H$为Hofer度量。我们仅通过研究可缩周期轨道来证明这一结果，而先前类似的高亏格结论是通过使用不可缩轨道获得的。
英文摘要
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 $ϕ_k$ on the two-torus $T^2$ such that $$d_H(\operatorname{Ent}_{\leq α}(T^2,ω),ϕ_k)\to \infty \quad (k \to \infty),$$ where $\text{Ent}_{\leq α}(T^2,ω)\subset \text{Ham}(T^2,ω)$ 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.
