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arXiv · 1809.02586

Location of Siegel capture polynomials in parameter spaces

Alexander Blokh·Arnaud Cheritat·Lex OVersteegen·Vladlen Timorin
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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.

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Alexander Blokh, Arnaud Cheritat, Lex OVersteegen, Vladlen Timorin. 2018-09-07. Location of Siegel capture polynomials in parameter spaces. https://doi.org/10.1088/1361-6544/abb9f9

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