Pré-Publication, Document De Travail Année : 2018

Location of Siegel capture polynomials in parameter spaces

Blokh Alexander
  • Fonction : Auteur
Oversteegen Lex
  • Fonction : Auteur
Timorin Vladlen
  • Fonction : Auteur

Résumé

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. In particular, we 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). We 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 cannot have bounded complementary domains containing IS-captures.

Dates et versions

hal-01940037 , version 1 (29-11-2018)

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Arnaud Chéritat, Blokh Alexander, Oversteegen Lex, Timorin Vladlen. Location of Siegel capture polynomials in parameter spaces. 2018. ⟨hal-01940037⟩
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