A prime orbit theorem for smooth surface diffeomorphisms

Abstract

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 ${\chi }_2<0$ such that for any ${\chi }_1\in (0,h)$, \[ \lim_{\substack{l(\mathcal{H}) \mid n \\ n\to\infty}} \frac{\sharp P_{χ_1,χ_2}(n)}{e^{nh}} = l(\mathcal{H}). \] Here $P_{{\chi }_1,{\chi }_2}(n)$ stands for the set of period-$n$ saddle points in $\mathcal{H}$ with Lyapunov exponents lying outside the interval $[{\chi }_2,{\chi }_1]$, and $l(\mathcal{H})$ denotes the period associated with the homoclinic class $\mathcal{H}$.