小面积形变下拓扑熵的鲁棒性
Robustness of topological entropy under small area deformations
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中文总结 AI 辅助
本文证明闭曲面上拓扑熵的鲁棒性：小面积圆盘支撑的保面积形变不会使正拓扑熵的哈密顿微分同胚熵归零，其结论基于针对谱距离的新型辫子稳定性结果。
中文摘要 AI 辅助
本文针对闭曲面上的哈密顿微分同胚的拓扑熵，建立了一种新型稳定性现象。对带有面积形式的闭曲面(Σ,ω)及其哈密顿微分同胚φ，证明了对任意ε>0，存在A=A(φ,ε)>0，使得由φ经支撑在若干不交圆盘（每个圆盘面积小于A）上的形变得到的任意哈密顿微分同胚φ'，都满足h_top(φ')>h_top(φ)-ε。特别地，若h_top(φ)>0，则无法通过支撑在小面积圆盘上的保面积形变使φ的熵变为零。这一结果源于本文针对Connery-Grigg新近引入的谱距离建立的新型辫子稳定性结论。
英文摘要
In this paper, we establish a new type of stability phenomenon for the topological entropy of Hamiltonian diffeomorphisms of closed surfaces. For a closed surface endowed with an area form $(Σ,ω)$ and a Hamiltonian diffeomorphism $ϕ$ of $(Σ,ω)$, we show that for every $\varepsilon>0$ there exists $A=A(ϕ,\varepsilon)>0$ such that \[ h_{\mathrm{top}}(ϕ') > h_{\mathrm{top}}(ϕ)-\varepsilon \] for every Hamiltonian diffeomorphism $ϕ'$ obtained from $ϕ$ by a deformation supported in a disjoint union of disks, each of area less than $A$. In particular, if $h_{\mathrm{top}}(ϕ)>0$, then $ϕ$ cannot be made to have zero entropy by an area-preserving deformation supported in disks of small area. This follows from the new braid stability result established in this paper with respect to the spectral distance recently introduced by Connery-Grigg.
发表机构
- University of Augsburg（奥格斯堡大学）
- Tel Aviv University（特拉维夫大学）
- RWTH Aachen University（亚琛工业大学）
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