On the dynamics of the McMullen family 𝑅(𝑧)=𝑧^{𝑚}+𝜆/𝑧^{ℓ}

Steinmetz, Norbert

doi:10.1090/s1088-4173-06-00149-4

Cited by 32 publications

(12 citation statements)

References 23 publications

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“…It is well known that z → z 2 − 1 has two bounded periodic Fatou components whose boundaries intersect at a point. This means that the Julia set of J(f 4 ) is connected but not a Sierpiński carpet (see [Ste06,§6] and Figure 2). …”

Section: Proofs Of Theorems and Some Examplesmentioning

confidence: 99%

A criterion to generate carpet Julia sets

Yang

2017

Proc. Amer. Math. Soc.

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Abstract. It was known that the Sierpiński carpets can appear as the Julia sets in the families of some rational maps. In this article we present a criterion that guarantees the existence of the carpet Julia sets in some rational maps having exactly one fixed (super-) attracting or parabolic basin. We show that this criterion can be applied to some well known rational maps, such as McMullen maps and Morosawa-Pilgrim family. Moreover, we give also some special examples whose Julia sets are Sierpiński carpets.

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Section: Proofs Of Theorems and Some Examplesmentioning

confidence: 99%

A criterion to generate carpet Julia sets

Yang

2017

Proc. Amer. Math. Soc.

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“…It's known from [11] that in the parameter space of f p , the McMullen domain M := {p ∈ C − {0}; J(f p ) is a Cantor circle} is a deleted neighborhood of the origin. It turns out that V = M ∪ {0} is a topological disk containing 0.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps

Wang

Yang

2014

Proc Math Sci

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“…Another reason is that this family provides many examples for different purpose to understand the dynamic of rational maps. We refer the reader to [DK,DLU,DP,HP,QWY,S,R1] and the reference therein for a number of related results.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic components of McMullen maps

Qiu¹,

Roesch²,

Wang³

et al. 2015

Ann. Sci. École Norm. Sup.

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Abstract. In this article, we study the hyperbolic components of McMullen maps. We show that the boundaries of all hyperbolic components are Jordan curves. This settles a problem posed by Devaney. As a consequence, we show that cusps are dense on the boundary of the unbounded hyperbolic component. This is a dynamical analogue of McMullen's theorem that cusps are dense on the Bers' boundary of Teichmüller space.

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On the dynamics of the McMullen family 𝑅(𝑧)=𝑧^{𝑚}+𝜆/𝑧^{ℓ}

Abstract: Abstract. In this note we discuss the parameter plane and the dynamics of the rational family R(z) = z m + λ/z , with m ≥ 2, ≥ 1, and 0 < |λ| < ∞.

Cited by 32 publications

References 23 publications

A criterion to generate carpet Julia sets

A criterion to generate carpet Julia sets

Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps

Hyperbolic components of McMullen maps

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