AI 中文总结
该研究引入接触同胚的平坦平移链等概念，将维特博定理推广，证明\(\mathbb{R}^{2n + 1}\)或\(\mathbb{R}^{2n} \times S^1\)中特定接触同胚在支集内部有无限多几何不同的平坦平移链，非负时增长率至少线性，周期链也有类似结论。
AI 中文摘要
我们引入了接触同胚的平坦平移链和有限接触同胚序列的周期平坦平移链的概念，并将维特博（1992）关于\(\mathbb{R}^{2n}\)的紧支哈密顿微分同胚的周期点重数的定理推广到这些概念。具体而言，我们表明，对于\(\mathbb{R}^{2n + 1}\)或\(\mathbb{R}^{2n} \times S^1\)中每一个与恒等同痕的非平凡紧支接触同胚，相对于标准接触形式，在其支集内部有无限多个几何上不同的平坦平移链；如果接触同胚是非负的，这种平移链的增长率至少是线性的，对于有限接触同胚序列的周期平坦平移链也有类似的陈述。
英文摘要
We introduce the notions of flat translated chains of contactomorphisms and periodic flat translated chains of finite sequences of contactomorphisms, and extend to these notions the theorem of Viterbo (1992) on the multiplicity of periodic points of compactly supported Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$. More precisely, we show that every non-trivial compactly supported contactomorphism of either $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$ that is contact isotopic to the identity has infinitely many geometrically distinct flat translated chains in the interior of the support with respect to the standard contact form, that the growth-rate of such translated chains is at least linear if the contactomorphism is non-negative, as well as similar statements for periodic flat translated chains of finite sequences of contactomorphisms.
Comments23 pages
