When do two rational functions have locally biholomorphic Julia sets?

Abstract

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.

A brief introduction to one-dimensional complex dynamics

“…反过来, 若有理函数 f 和 g 具有相同的 Julia 集, 则在一些情形下可以推导出 f 和 g 可交 换 (参见文献 [45]). 通过最大熵测度刻画有理函数可交换性的研究还可以参见 Levin 和 Przytycki [436] 及叶和溪 [780] 和 Dujardin 等 [278] 的工作.…”