When do two rational functions have locally biholomorphic Julia sets?

Dujardin, Romain; Favre, Charles; Gauthier, Thomas

doi:10.1090/tran/8775

When do two rational functions have locally biholomorphic Julia sets?
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by Romain Dujardin, Charles Favre and Thomas Gauthier;

Trans. Amer. Math. Soc. 376 (2023), 1601-1624

DOI: https://doi.org/10.1090/tran/8775

Published electronically: October 3, 2022

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Abstract:

In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.

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