Renormalizations and wandering Jordan curves of rational maps

2014/03/20 by Guizhen Cui, Cui, Guizhen, Wenjuan Peng +3 · 1 citation
Computer Science · Mathematics · #37F10 #37F20 #Digital Image Processing Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DS #msc:37F10 #msc:37F20

paper · pdf · doi:10.48550/arxiv.1403.5024

49 pages, 3 figures

openalex publication_date 2014/03/20 · arxiv created 2015/07/15 · arxiv updated 2015/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We realize a dynamical decomposition for a post-critically finite rational map which admits a combinatorial decomposition. We split the Riemann sphere into two completely invariant subsets. One is a subset of the Julia set consisting of uncountably many Jordan curve components. Most of them are wandering. The other consists of components that are pullbacks of finitely many renormalizations, together with possibly uncountably many points. The quotient action on the decomposed pieces is encoded by a dendrite dynamical system. We also introduce a surgery procedure to produce post-critically finite rational maps with wandering Jordan curves and prescribed renormalizations.

Citations

• Rational rays and critical portraits of complex polynomials
• A partial description of parameter space of rational maps of degree two: Part I

Cited by

• Rational maps with constant Thurston pullback mapping
• Renormalization and wandering continua of rational maps

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