Models for spaces of dendritic polynomials

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of...

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Bibliographic Details
Published in:arXiv.org 2018-02
Main Authors: Blokh, Alexander, Oversteegen, Lex, Ptacek, Ross, Timorin, Vladlen
Format: Article
Language:English
Subjects:
Dendritic structure
Polynomials
Topology
Online Access:Get full text
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Summary:Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the "pinched disk" model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.
ISSN:2331-8422