When do Two Rational Functions have Locally Biholomorphic Julia Sets?

We address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics and goes back to classical work of Fatou and Julia: 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 is joint work with Charles Favre and Thomas Gauthier.

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