Location of Siegel capture polynomials in parameter spaces

TL;DR

This paper investigates the placement of IS-capture polynomials within the parameter space of cubic polynomials, revealing their boundary location relative to hyperbolic components and their relation to the cubic Principal Hyperbolic Domain.

Contribution

It characterizes the boundary placement of IS-capture polynomials and connects them to the structure of the cubic Principal Hyperbolic Domain and its boundary in parameter space.

Findings

IS-captures lie on the boundary of unique hyperbolic components.

The closure of the cubic Principal Hyperbolic Domain may have bounded complementary domains with specific properties.

These properties include critical points in the Julia set and Julia sets with positive Lebesgue measure and invariant line fields.

Abstract

A cubic polynomial $f$ with a periodic Siegel disk containing an eventual image of a critical point is said to be a \emph{Siegel capture polynomial}. If the Siegel disk is invariant, we call $f$ a \emph{IS-capture polynomial} (or just an IS-capture; IS stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials and show that any IS-capture is on the boundary of a unique hyperbolic component determined by the rational lamination of the map. We also relate IS-captures to the cubic Principal Hyperbolic Domain and its closure (by definition, the \emph{cubic Principal Hyperbolic Domain} consists of cubic hyperbolic polynomials with Jordan curve Julia sets) and prove that, in the slice of cubic polynomials given by a fixed multiplier at one of the fixed points, the closure of the cubic principal hyperbolic domain might possibly only have bounded complementary domains $U$ such that (1) critical points of $f\in U$ are distinct and belong to $J(f)$, and (2) $J(f)$ has positive Lebesgue measure and carries an invariant line field.