The solar Julia sets of basic quadratic Cremer polynomials

Blokh, Alexander; Buff, Xavier; Chéritat, Arnaud; Oversteegen, Lex G.

doi:10.1017/s0143385708000990

Cited by 10 publications

(7 citation statements)

References 21 publications

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“…The remaining basic uniCremer Julia sets are called red dwarf Julia sets. They can be defined as basic uniCremer Julia sets such that all impressions contain p. These notions have been introduced in [BO06a] and were further studied in [BBCO07] where it was shown that solar Julia sets of degree two, with positive Lebesgue measure, exist. The following lemma describes red dwarf Julia sets and complements Theorem 5.10.…”

Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

The Julia sets of basic uniCremer polynomials of arbitrary degree

Blokh

Oversteegen

2009

Conform. Geom. Dyn.

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Abstract. Let P be a polynomial of degree d with a Cremer point p and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets J P . The red dwarf J P are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing p and the orbits of all critical images. The solar J P are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and J P is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of J P and show that such sets J P appear through polynomial-like maps for generic polynomials with Cremer points. Since known tools break down for d > 2 (if d > 2, it is not known if there are small cycles near p, while if d = 2, this result is due to Yoccoz), we introduce wandering ray continua in J P and provide a new application of Thurston laminations.

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Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

The Julia sets of basic uniCremer polynomials of arbitrary degree

Blokh

Oversteegen

2009

Conform. Geom. Dyn.

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“…However, the difference with our setup is that bubbles in the sense of Luo are Fatou components that are eventually mapped to a superattracting rather than a Siegel domain. Also, similar ideas are used in [BBCO10] where some quadratic Cremer Julia sets were studied by approximating them with Siegel Julia sets with specific properties.…”

Section: Polar Coordinates and Bubblesmentioning

confidence: 99%

Modeling core parts of Zakeri slices I

Blokh¹,

Oversteegen²,

Shepelevtseva³

et al. 2021

Preprint

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The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them Zakeri slices. We give a model of the central part of a slice (the subset of the slice that can be approximated by hyperbolic polynomials with Jordan curve Julia sets), and a continuous projection from the central part to the model. The projection is defined dynamically and is coherent with the dynamical-analytic parameterization of the Principal Hyperbolic Domain by Petersen and Tan Lei.

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“…2 is equal to the cross ratio of α 0 , α n , β 0 , β n . Then (9) z n = (1 − cosh(τ )) cosh δ n . 5 If the group generated by γ 0 and γ 1 is discrete then the set {cosh(δ n ) : n ∈ Z ≥2 } is discrete in C. The theorem follows from the following inductive relation on the sequence of δ n [12, p.67]: (10) cosh…”

Section: Complex Dynamics and Kleinian Groupsmentioning

confidence: 99%

“…Because of the rigidity of holomorphic functions, the theory of complex dynamics is rich in deep results: a complete combinatorial description of the structure of the Julia set of polynomials and (conjecturally) of the Mandelbrot set [53], application of the thermodynamical formalism to Julia sets (see for example the survey [55]) which allows to compute their Hausdorff dimensions, interplay with circle map dynamics and the theory of small divisors (for example [69]), as examples of realizations of unusual topologies as Julia sets (e.g. [9,57]) or of pathological dynamical systems [13].…”

Section: Introductionmentioning

confidence: 99%