When do two rational functions have locally biholomorphic Julia sets?

TL;DR

This paper investigates conditions under which two rational functions have locally biholomorphic Julia sets, providing criteria that connect local biholomorphisms to algebraic correspondences, extending classical results in complex dynamics.

Contribution

It establishes new criteria linking local biholomorphisms of Julia sets to algebraic correspondences, unifying and extending previous classical results.

Findings

Criteria for local biholomorphisms between Julia sets

Connection between local isomorphisms and algebraic correspondences

Use of entire curves and positive currents in proofs

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.