A Central Limit Theorem for Periodic Orbits of Hyperbolic Flows

Abstract

We consider a counting problem in the setting of hyperbolic dynamics. Let $\phi_t : \Lambda \to \Lambda$ be a weak mixing hyperbolic flow. We count the proportion of prime periodic orbits of $\phi_t$, with length less than $T$, that satisfy an averaging condition related to a H\"older continuous function $f: \Lambda \to \mathbb{R}$. We show, assuming an approximability condition on $\phi$, that as $T \to \infty$, we obtain a central limit theorem.