A Central Limit Theorem for Periodic Orbits of Hyperbolic Flows

TL;DR

This paper proves a central limit theorem for the distribution of prime periodic orbits in hyperbolic flows, under certain conditions, revealing statistical regularities in their long-term behavior.

Contribution

It establishes a CLT for prime periodic orbits of hyperbolic flows, assuming an approximability condition, extending statistical understanding of hyperbolic dynamical systems.

Findings

Proves a CLT for prime periodic orbits in hyperbolic flows.

Shows the proportion of orbits satisfying an averaging condition converges to a normal distribution.

Requires an approximability condition on the flow for the CLT to hold.

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.