Zhuchao Ji, Junyi Xie
Published 2023-02-06Version 1
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.
Real-analyticity of Hausdorff Dimension of Julia Sets along the Parabolic Arcs of the Multicorns
Meromorphic functions whose action on their Julia sets is non-ergodic
Hausdorff measure of Julia sets in the exponential family
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