A criterion to generate carpet Julia sets

Yang, Fei

doi:10.1090/proc/13924

Cited by 6 publications

(5 citation statements)

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“…第一个 Sierpiński 地毯型 Julia 集是由 Milnor 和谭蕾 [513] 找到的. 更多的 Sierpiński 地毯型 Julia 集可参见文献 [240,246,316,570,762,771]. 对于这类特殊的 Julia 集, McMullen 提出了以下问题 (参见文献 [498, 问题 5.3]):…”

Section: 如果复球面上的一个紧子集内部为空并可以写成可数多个球面直径趋向于零且闭包两两不交的unclassified

A brief introduction to one-dimensional complex dynamics

Fei

2024

Sci. Sin.-Math.

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. 1879 年, Cayley [168] 建议用该方法求解复多项式的根. 但他发现这个问题即使对于三次复多项式也已经非常困难: "The case of the cubic equation appears to present considerable difficulty." 从今天的视角来看, Cayley 在那时遇到困难是不可避免的, 因为他要解决的问题是计算一个三次有理函数的 Julia 集 (见图 1).Cayley 并不是第一个在 19 世纪 70 年代研究迭代的人. Schröder 在 1870-1871 年也考虑了用迭代法解方程, 同时还对下面的 Schröder 方程做了研究 3) :1884 年, Koenigs [416] 证明了如果导数 (或称 "乘子") λ = f ′ (0) 满足 |λ| ̸ =

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Section: 如果复球面上的一个紧子集内部为空并可以写成可数多个球面直径趋向于零且闭包两两不交的unclassified

A brief introduction to one-dimensional complex dynamics

Fei

2024

Sci. Sin.-Math.

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“…Further, it is found that the existence of polynomial semigroups such that the associated Julia set contains no interior points as well as Hausdorff dimension arbitrarily close to 2. A criterion studies in [61] for some rational maps having exactly one fixed (super-) attracting or parabolic basin that provides guarantee the existence of the carpet Julia sets.…”

Section: Different Approaches To Studying Dynamics Of One Variable Co...mentioning

confidence: 99%

On Some Advances in Dynamics of One Variable Complex Functions

Sajid¹

2022

Int. J. of Appl. Math.

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In this article, we present some recent advances in the dynamics of one variable complex functions which are especially associated to the Julia sets, the Fatou sets, the Sigel disks, the Herman rings, fixed points as well as the Mandelbrot sets. For the sake of interest, we consider mainly research works which have been done from 2015 to 2019. To achieve our purpose, some interesting and selected themes are considered to show some recent advances in the dynamics of one variable complex functions.

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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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A criterion to generate carpet Julia sets

Cited by 6 publications

References 20 publications

A brief introduction to one-dimensional complex dynamics

A brief introduction to one-dimensional complex dynamics

On Some Advances in Dynamics of One Variable Complex Functions

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

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