Counting prime orbits in shrinking intervals for expanding Thurston maps

TL;DR

This paper proves a local central limit theorem for primitive periodic orbits of expanding Thurston maps, refining the Prime Orbit Theorem by counting orbits with Birkhoff sums in shrinking intervals, especially for certain rational maps.

Contribution

It introduces a local limit theorem for primitive periodic orbits in expanding Thurston maps, extending prime orbit counting to fine-scale intervals under specific conditions.

Findings

Established a local central limit theorem for primitive periodic orbits.

Derived precise asymptotics for Birkhoff sums in shrinking intervals.

Applicable to postcritically-finite rational maps with Julia set equal to the sphere.

Abstract

We establish a local central limit theorem for primitive periodic orbits of expanding Thurston maps, providing a fine-scale refinement of the Prime Orbit Theorem in the context of non-uniformly expanding dynamics. Specifically, we count the number of primitive periodic orbits whose Birkhoff sums for a given potential lie within a family of shrinking intervals. For eventually positive, real-valued \holder continuous potentials that satisfy the strong non-integrability condition, we derive precise asymptotic estimates. In particular, our results apply to postcritically-finite rational maps whose Julia set is the whole Riemann sphere.