Local rigidity of Julia sets

TL;DR

This paper establishes criteria under which local symmetries between Julia sets imply a global algebraic relation between the underlying rational functions, with implications for the Dynamical André-Oort conjecture.

Contribution

It introduces new conditions linking local symmetries of Julia sets to algebraic correspondences between rational functions, advancing understanding of their global dynamics.

Findings

Local $C^1$-symmetries imply algebraic relations between rational functions

Criteria connect measures on Julia sets to algebraic correspondences

Results are relevant for the proof of the Dynamical André-Oort conjecture

Abstract

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.