Counting prime orbits in shrinking intervals for expanding Thurston maps
Zhiqiang Li, Xianghui Shi
TL;DR
The paper proves a local central limit theorem for primitive periodic orbits of expanding Thurston maps by counting orbits whose Birkhoff sums with respect to a Hölder potential lie in shrinking intervals. The authors develop a refined thermodynamic framework using split Ruelle operators to handle non-uniform expansion and critical points, obtaining precise decay estimates for twisted partition functions and analytic properties of the pressure function. Central contributions include the existence of a unique rootpressure, explicit expressions for the limiting Gaussian variance, and a sharp n^{-3/2} prefactor in the shrinking-interval asymptotics, with a corollary applying to postcritically-finite rational maps without periodic critical points. The work bridges non-uniform dynamics and fine-scale orbit distribution, extending local limit phenomena beyond uniformly hyperbolic settings and enriching the Prime Orbit Theorem in complex dynamics with a local-statistical perspective.
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.