Sierpinski curve Julia sets for quadratic rational maps

Devaney, Robert L.; Fagella, Núria; Garijo, Antonio; Jarque, Xavier

doi:10.5186/aasfm.2014.3903

Cited by 16 publications

(16 citation statements)

References 13 publications

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“…The first example of the Sierpiński curve as the Julia set of a rational map was given in [17,Appendix F]. For more rational maps whose Julia sets are Sierpiński curves, see [6]. For the study of non-existence of Herman rings for rational families, see [27,30] and the references therein.…”

Section: Statement Of the Resultsmentioning

confidence: 98%

Singular Perturbations with Multiple Poles of the Simple Polynomials

Xiao

Yang

2016

Qual. Theory Dyn. Syst.

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In this article, we study the dynamics of the following family of rational maps with one parameter:where n ≥ 3 and λ ∈ C * . This family of rational maps can be viewed as a singular perturbations of the simple polynomial P n (z) = z n . We give a characterization of the topological properties of the Julia sets of the family f λ according to the dynamical behaviors of the orbits of the free critical points.

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Section: Statement Of the Resultsmentioning

confidence: 98%

Singular Perturbations with Multiple Poles of the Simple Polynomials

Xiao

Yang

2016

Qual. Theory Dyn. Syst.

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“…The first example of the Sierpiński carpet as the Julia set of a rational map was given in [12,Appendix F]. For more rational maps whose Julia sets are Sierpiński carpets, see [5]. We call a compact set in C a degenerated Sierpiński carpet if it satisfies all the conditions of the Sierpiński carpet except it allows that the intersection of the boundaries of complementary domains can be non-empty.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

On the dynamics of a family of singularly perturbed rational maps

Yang

2015

Journal of Mathematical Analysis and Applications

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“…As a kind of Julia sets, the Sierpiński carpets attract much interest recently. One may refer to [4], [2], [34], [11], [25] and the references therein.…”

Section: Some Dynamical Properties In the Non-quasicircle Casementioning

confidence: 99%

Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

Wang¹,

Yang²,

Zhang³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we investigate the dynamics of the following family of rational maps f λ (z) = z 2n − λ 3n+1 z n (z 2n − λ n−1 ) with one parameter λ ∈ C * − {λ : λ 2n+2 = 1}, where n ≥ 2. This family of rational maps is viewed as a singular perturbation of the bi-critical map P −n (z) = z −n if λ = 0 is small. It is proved that the Julia set J(f λ ) is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerate Sierpiński carpet provided the free critical orbits of f λ are attracted by the super-attracting cycle 0 ↔ ∞. Furthermore, we prove that there exists suitable λ such that J(f λ ) is a Cantor set of circles but the dynamics of f λ on J(f λ ) is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of J(f λ ) is also proved if the free critical orbits are not attracted by the cycle 0 ↔ ∞. Finally we give an estimate of the Hausdorff dimension of the Julia set of f λ in some special cases.

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Sierpinski curve Julia sets for quadratic rational maps

Abstract: Abstract. We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve.

Cited by 16 publications

References 13 publications

Singular Perturbations with Multiple Poles of the Simple Polynomials

Singular Perturbations with Multiple Poles of the Simple Polynomials

On the dynamics of a family of singularly perturbed rational maps

Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

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