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Prime number theorems and holonomies for hyperbolic rational maps

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Abstract

For a hyperbolic rational map f of degree at least two on the Riemann sphere, we obtain estimates for the number of primitive periodic orbits of f ordered by their multiplier, and establish equidistribution of the associated holonomies, both with power saving error terms.

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Acknowledgments

We would like to thank Dennis Sullivan for bringing our attention to this problem, and Curt McMullen for telling us about Zdunik’s work. We would also like to thank Ralf Spatzier and Mark Pollicott for helpful discussions.

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Authors and Affiliations

  1. Mathematics Department, Yale University, New Haven, CT, 06511, USA

    Hee Oh

  2. Korea Institute for Advanced Study, Seoul, Korea

    Hee Oh

  3. IAS, Princeton, NJ, 08540, USA

    Dale Winter

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  1. Hee Oh
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  2. Dale Winter
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Correspondence to Hee Oh.

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Supported in parts by the NSF.

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Oh, H., Winter, D. Prime number theorems and holonomies for hyperbolic rational maps. Invent. math. 208, 401–440 (2017). https://doi.org/10.1007/s00222-016-0693-1

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  • DOI: https://doi.org/10.1007/s00222-016-0693-1

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