Dynamics of asymptotically holomorphic polynomial-like maps

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.

arXiv comment47 pages

References in corpus (6)

• Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials
• Rigidity of critical circle maps
• Renormalization theory for multimodal maps
• Satellite renormalization of quadratic polynomials
• Real bounds and Lyapunov exponents
• Differential Calculus, Tensor Products and the Importance of Notation

Cited by in corpus (1)

• The boundary of chaos for interval mappings