Moscow Mathematical Journal

Volume 22, Issue 2, April–June 2022 pp. 265–294.

Modeling Core Parts of Zakeri Slices I

Authors: Alexander Blokh (1), Lex Oversteegen (1), Anastasia Shepelevtseva (2), and Vladlen Timorin (3)
Author institution:(1) Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170
(2) Faculty of Mathematics, HSE University, Russian Federation, 6 Usacheva St., 119048 Moscow
Scuola Normale Superiore, 7 Piazza dei Cavalieri, 56126 Pisa, Italy
(3) Faculty of Mathematics, HSE University, Russian Federation, 6 Usacheva St., 119048 Moscow
Independent University of Moscow, Bolshoy Vlasyevskiy Per. 11, 119002 Moscow, Russia

Summary:

The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them Zakeri slices. We give a model of the central part of a slice (the subset of the slice that can be approximated by hyperbolic polynomials with Jordan curve Julia sets), and a continuous projection from the central part to the model. The projection is defined dynamically and agrees with the dynamical-analytic parameterization of the Principal Hyperbolic Domain by Petersen and Tan Lei.

2020 Math. Subj. Class. Primary: 37F46, 37F20; Secondary: 37F10, 37F50.

Keywords: Complex dynamics, Julia set, cubic polynomial, Siegel disk, connectedness locus, external rays.

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