The main cubioid

Blokh, Alexander; Oversteegen, Lex G.; Ptacek, Ross; Timorin, Vladlen

doi:10.1088/0951-7715/27/8/1879

Cited by 18 publications

(41 citation statements)

References 24 publications

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“…Generally, the behavior of critical orbits to a large extent determines the dynamics of other orbits. For example, by a classical theorem of Actually, in [BOPT14] we claim that all non-repelling periodic cutpoints in the Julia set J(f ), except perhaps 0, have multiplier 1; still, literally repeating the same arguments one proves the version of [BOPT14, Theorem A] given by Theorem 1.1. This motivates Definition 1.2, in which we define a special set CU such that PHD 3 ⊂ CU.…”

Section: Introductionmentioning

confidence: 93%

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Quadratic-Like Dynamics of Cubic Polynomials

Blokh

Oversteegen

Ptacek

et al. 2016

Commun. Math. Phys.

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Abstract. A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.

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Section: Introductionmentioning

confidence: 93%

“…Definition 1.2 ( [BOPT14]). Let CU be the family of classes of cubic polynomials f with connected J(f ) such that f has a non-repelling fixed point, no repelling periodic cutpoints in J(f ), and all its nonrepelling periodic points, except at most one fixed point, have multiplier 1.…”

Section: Introductionmentioning

confidence: 99%

Quadratic-Like Dynamics of Cubic Polynomials

Blokh

Oversteegen

Ptacek

et al. 2016

Commun. Math. Phys.

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“…We want to improve the description of the dynamics of maps in a bounded component of F λ \ P λ given in [BOPT14] (see Theorem 1.4). Let f ∈ F λ , |λ| 1, belong to a bounded component W f of F λ \ P λ .…”

Section: Bounded Components Of F λ \ P λ Must Be Of Siegel Capture Tymentioning

confidence: 99%

“…Equivalently, c ∈ PHD 2 if and only if z 2 + c has an attracting fixed point. The closure of PHD 2 consists of all parameter values c such that z 2 + c Observe that, strictly speaking, in [BOPT14] we claim that all non-repelling periodic cutpoints in the Julia set J(f ), except perhaps one, have multiplier 1; however, literally repeating the same arguments one can prove the version of the results of [BOPT14] given by Theorem 1.1 (i.e., we can talk about all non-repelling periodic points of f , not only its periodic cutpoints). Theorem 1.1 motivates the following definition; notice that from now on in the paper we concentrate upon the cubic case (thus, unlike Theorem 1.1, Definition 1.2 deals with cubic polynomials).…”

Section: Introductionmentioning

confidence: 99%

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Complementary components to the cubic principal hyperbolic domain

Blokh

Oversteegen

Ptacek

et al. 2018

Proc. Amer. Math. Soc.

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We study the closure of the cubic Principal Hyperbolic Domain and its intersection P λ with the slice F λ of the space of all cubic polynomials with fixed point 0 defined by the multiplier λ at 0. We show that any bounded domain W of F λ \ P λ consists of J-stable polynomials f with connected Julia sets J(f ) and is either of Siegel capture type (then f ∈ W has an invariant Siegel domain U around 0 and another Fatou domain V such that f | V is twoto-one and f k (V ) = U for some k > 0) or of queer type (then a specially chosen critical point of f ∈ W belongs to J(f ), the set J(f ) has positive Lebesgue measure, and carries an invariant line field). 1 2 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORINhas a non-repelling fixed point. As follows from the Douady-Hubbard parameter landing theorem [DH8485, Hub93], the Mandelbrot set itself can be thought of as the union of the main cardioid and limbs (connected components of M 2 \ PHD 2 ) parameterized by reduced rational fractions p/q ∈ (0, 1).This motivates our study of higher degree analogs of PHD 2 started in [BOPT14]. More precisely, complex numbers c are in one-to-one correspondence with affine conjugacy classes of quadratic polynomials (throughout we call affine conjugacy classes of polynomials classes of polynomials). Thus a natural higher-degree analog of the set M 2 is the degree d connectedness locus M d defined as the set of classes of degree d polynomials f , all of whose critical points do not escape, or, equivalently, whose Julia set J(f ) is connected. The Principal Hyperbolic Domain PHD d of M d is defined as the set of classes of hyperbolic degree d polynomials with Jordan curve Julia sets. Equivalently, the class [f ] of a degree d polynomial f belongs to PHD d if all critical points of f are in the immediate attracting basin of the same attracting (or super-attracting) fixed point. In [BOPT14] we describe properties of cubic polynomials f such that [f ] ∈ PHD d ; notice that Theorem 1.1 holds for any d 2.Theorem 1.1 ([BOPT14]). If [f ] ∈ PHD d , then f has a non-repelling fixed point, no repelling periodic cutpoints in J(f ), and all its non-repelling periodic points, except at most one fixed point, have multiplier 1.

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Location of Siegel capture polynomials in parameter spaces

et al. 2021

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A cubic polynomial with a marked fixed point 0 is called an IS-capture polynomial if it has a Siegel disk D around 0 and if D contains an eventual image of a critical point. We show that any IS-capture polynomial is on the boundary of a unique bounded hyperbolic component of the polynomial parameter space determined by the rational lamination of the map and relate IS-capture polynomials to the cubic principal hyperbolic domain and its closure.

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The main cubioid

Abstract: Abstract. We discuss different analogs of the main cardioid in the parameter space of cubic polynomials, and establish relationships between them.

Cited by 18 publications

References 24 publications

Quadratic-Like Dynamics of Cubic Polynomials

Quadratic-Like Dynamics of Cubic Polynomials

Complementary components to the cubic principal hyperbolic domain

Location of Siegel capture polynomials in parameter spaces

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