Local rigidity of Julia sets

We find criteria ensuring that a local (holomorphic, real analytic, $C^1$) homeomorphism between the Julia sets of two given rational functions comes from an algebraic correspondence. For example, we show that if there is a local $C^1$-symmetry between the maximal entropy measures of two rational functions, then probably up to a complex conjugation, the two rational functions are dynamically related by an algebraic correspondence. The holomorphic case of our criterion will play an important role in the authors' upcoming proof of the Dynamical Andr\'e-Oort conjecture for curves.