A Central Limit Theorem for Periodic Orbits of Hyperbolic Flows

We consider a counting problem in the setting of hyperbolic dynamics. Let ${\varphi }_t:\mathrm{\Lambda }\to \mathrm{\Lambda }$ 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 H"older continuous function $f:\mathrm{\Lambda }\to \mathbb{R}$. We show, assuming an approximability condition on $\varphi$, that as $T\to \mathrm{\infty }$, we obtain a central limit theorem.