Prime orbit theorems for expanding Thurston maps

TL;DR

This paper establishes a prime orbit theorem for expanding Thurston maps, providing asymptotic counts of primitive periodic orbits weighted by a Hölder continuous function, extending classical number theory results to complex dynamical systems.

Contribution

It introduces a prime orbit theorem for expanding Thurston maps, analyzing dynamical zeta functions and Dirichlet series without smoothness assumptions, and shows the regularity conditions are generic.

Findings

Asymptotic count of primitive periodic orbits matches logarithmic integral.

Holomorphic extension properties of zeta functions are crucial for the result.

The theorem applies to postcritically-finite rational maps with Julia set as the whole sphere.

Abstract

We obtain an analogue of the prime number theorem for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$, which are topological models of some rational maps without any smoothness or holomorphicity assumption. More precisely, by studying dynamical zeta functions and, more generally, dynamical Dirichlet series for $f$, we show that the number of primitive periodic orbits of $f$, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued H\"{o}lder continuous function $\phi$ on $S^2$ satisfying some additional regularity conditions, is asymptotically the same as the well-known logarithmic integral, with an exponential error term. Such a result, known as a Prime Orbit Theorem, follows from our quantitative study of the holomorphic extension properties of the associated dynamical zeta functions and dynamical Dirichlet series. In particular, the above result applies to postcritically-finite rational maps whose Julia set is the whole Riemann sphere. Moreover, we prove that the regularity conditions needed here are generic; and for a Latt\`{e}s map $f$ and a continuously differentiable (eventually) positive function $\phi$, such a Prime Orbit Theorem holds if and only if $\phi$ is not co-homologous to a constant.