A prime orbit theorem for smooth surface diffeomorphisms

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 $\chi_1 \in (0, h)$, $$\lim_{\substack{l(\mathcal{H}) \mid n \\ n\to\infty}} \frac{\sharp P_{\chi_1,\chi_2}(n)}{e^{nh}} = l(\mathcal{H}).$$ 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 $\mathcal{H}$.