Prime orbit theorems for expanding Thurston maps: Dirichlet series and orbifolds

Abstract

We obtain an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumptions. More precisely, we show that the number of primitive periodic orbits, ordered by a weight on each point induced by an (eventually) positive real-valued Hölder continuous function on $S^2$ that is not cohomologous to a constant, is asymptotically the same as the well-known logarithmic integral. In particular, our results apply to postcritically-finite rational maps for which the Julia set is the whole Riemann sphere.

arXiv comment67 pages. This is the first of a series of 3 papers (together with arXiv:2312.06688 and arXiv:2312.06687), replacing arXiv:1804.08221

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• Fourier dimension and spectral gaps for hyperbolic surfaces
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Cited by in corpus (3)

• Ground states and periodic orbits for expanding Thurston maps
• Prime orbit theorems for expanding Thurston maps: Lattès maps and split Ruelle operators
• Prime orbit theorems for expanding Thurston maps: Genericity of strong non-integrability condition