A Central Limit Theorem for Periodic Orbits of Hyperbolic Flows

Abstract

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

9 pages