Counting prime orbits in shrinking intervals for expanding Thurston maps

Zhiqiang Li; Xianghui Shi

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arXiv · 2511.22601

Counting prime orbits in shrinking intervals for expanding Thurston maps

Zhiqiang Li·Xianghui Shi

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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.

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BibTeX RIS

Zhiqiang Li, Xianghui Shi. 2025-11-27. Counting prime orbits in shrinking intervals for expanding Thurston maps. https://arxiv.org/abs/2511.22601

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