Dynamics of asymptotically holomorphic polynomial-like maps

Edson de Faria, Trevor Clark, Sebastian van Strien

Publication date: 17 April 2018

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.

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