Modeling core parts of Zakeri slices I

TL;DR

This paper models the central part of Zakeri slices of cubic polynomials with connected Julia sets, providing a continuous dynamical projection that aligns with known parameterizations of hyperbolic components.

Contribution

It introduces a model for the central part of Zakeri slices and constructs a continuous dynamical projection consistent with existing hyperbolic domain parameterizations.

Findings

A model for the central part of Zakeri slices is established.

A continuous projection from the model to the slice is constructed.

The projection aligns with the dynamical-analytic parameterization of the Principal Hyperbolic Domain.

Abstract

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.