光滑曲面微分同胚的素轨道定理
A prime orbit theorem for smooth surface diffeomorphisms
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中文总结 AI 辅助
该研究针对闭曲面上具正拓扑熵的$C^\infty$微分同胚的每个同宿类，建立了精确的素轨道定理，给出了满足李雅普诺夫指数区间条件的周期鞍点计数的渐近极限公式。
中文摘要 AI 辅助
我们针对闭曲面上具有正拓扑熵的$C^\infty$微分同胚的每一个同宿类，建立了精确的素轨道定理。设$\mathcal{H}$是拓扑熵为$h>0$的同宿类，则存在常数$\chi_2<0$，使得对任意$\chi_1\in(0,h)$，有\\[ \lim_{\substack{l(\mathcal{H}) \mid n \\\\ n\to\infty}} \frac{\sharp P_{\chi_1,\chi_2}(n)}{e^{nh}} = l(\mathcal{H}). \\] 其中$P_{\chi_1,\chi_2}(n)$表示$\mathcal{H}$中李雅普诺夫指数落在区间$[\chi_2,\chi_1]$之外的周期为$n$的鞍点集，$l(\mathcal{H})$表示与同宿类$\mathcal{H}$相关的周期。
英文摘要
We establish a sharp prime orbit theorem for every homoclinic class of a $C^\infty$ diffeomorphism on a closed surface with positive topological entropy. Let $\mathcal{H}$ be a homoclinic class with topological entropy $h > 0$. Then there exists a constant $χ_2 < 0$ such that for any $χ_1 \in (0, h)$, \[ \lim_{\substack{l(\mathcal{H}) \mid n \\ n\to\infty}} \frac{\sharp P_{χ_1,χ_2}(n)}{e^{nh}} = l(\mathcal{H}). \] Here $P_{χ_1,χ_2}(n)$ stands for the set of period-$n$ saddle points in $\mathcal{H}$ with Lyapunov exponents lying outside the interval $[χ_2,χ_1]$, and $l(\mathcal{H})$ denotes the period associated with the homoclinic class $\mathcal{H}$.
