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

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 t...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2023-11, Vol.43 (11), p.3636-3684
Main Authors: CLARK, TREVOR, DE FARIA, EDSON, VAN STRIEN, SEBASTIAN
Format: Article
Language:English
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Summary: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2022.72