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fatou–julia–sullivan theory

Ergodic Theory and Dynamical Systems (1)

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Asymptotically holomorphic methods for infinitely renormalizable $C^r$ unimodal maps
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TREVOR CLARK, EDSON DE FARIA, SEBASTIAN VAN STRIEN
Journal:

Ergodic Theory and Dynamical Systems / Volume 43 / Issue 11 / November 2023

Published online by Cambridge University Press:

29 November 2022, pp. 3636-3684

Print publication:

November 2023
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The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.