  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.
