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Asymptotically holomorphic methods for infinitely renormalizable $C^r$ unimodal maps

Part of: Complex dynamical systems Geometric function theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  29 November 2022

TREVOR CLARK ,
EDSON DE FARIA Open the ORCID record for EDSON DE FARIA [Opens in a new window]  and
SEBASTIAN VAN STRIEN Open the ORCID record for SEBASTIAN VAN STRIEN [Opens in a new window]
TREVOR CLARK
Affiliation:
Imperial College London, London, UK (e-mail: t.clark@imperial.ac.uk)
EDSON DE FARIA
Affiliation:
Instituto de Matemática e Estatística, USP, São Paulo, SP, Brazil (e-mail: edson@ime.usp.br)
SEBASTIAN VAN STRIEN*
Affiliation:
Imperial College London, London, UK
*
e-mail: s.van-strien@imperial.ac.uk
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Abstract

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.

Keywords

renormalizationunimodal mapsasymptotically holomorphic mapsFatou–Julia–Sullivan theory

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions 37E20: Universality, renormalization
Secondary: 30C62: Quasiconformal mappings in the plane
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 11 , November 2023 , pp. 3636 - 3684
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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