Location of Siegel capture polynomials in parameter spaces

Abstract

We study the set of cubic polynomials f with a marked fixed point. If f has a Siegel disk at the marked fixed point, and if this disk contains an eventual image of a critical point, we call f a IS-capture polynomial ("IS" stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all marked cubic polynomials modulo affine conjugacy. In particular, we show that any IS-capture polynomial is on the boundary of a unique bounded hyperbolic component determined by the rational lamination of the map. We also relate IS-capture polynomials to the cubic Principal Hyperbolic Domain and its closure (by definition, the cubic Principal Hyperbolic Domain consists of cubic hyperbolic polynomials with Jordan curve Julia sets).

Quadratic rational maps with a $2$-cycle of Siegel disks

“…For the study of the parameter spaces of the holomorphic families containing a fixed Siegel disk, one may also refer to [BH01], [BF10], [BCOT18], [Zak18] and [Ché20].…”