The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

Bergweiler, Walter; Cui, Weiwei

doi:10.1007/s00209-022-03142-0

Cited by 3 publications

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“…再结合 J(tan z) = R 和 Misiurewicz 的结果 J(e z ) = C [525] 可知, 有界型超越亚纯函数 Julia 集的 Hausdorff 维数可以取到区间 (0, 2] 中的任何值. Bergweiler 和崔巍巍 [80] 证明了对于任意 s ∈ (0, 2], 存在一个有限型的超越亚纯函数 f (事实上仅有 3 个奇异值), 使得 dim H J(f ) = s. 对于有限型超越亚纯函数逃逸集的 Hausdorff 维数研究可参见文献 [16,17].…”

Section: 超越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: 超越unclassified

A brief introduction to one-dimensional complex dynamics

Fei

2024

Sci. Sin.-Math.

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Hausdorff dimension of escaping sets of meromorphic functions II

Aspenberg¹,

Cui²

2022

Ergod. Th. Dynam. Sys.

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A function which is transcendental and meromorphic in the plane has at least two singular values. On the one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either $2$ or $1/2$ . On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in $[0,2]$ (cf. [M. Aspenberg and W. Cui. Hausdorff dimension of escaping sets of meromorphic functions. Trans. Amer. Math. Soc.374(9) (2021), 6145–6178]). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than $4$ .

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The Hausdorff Dimension of Escaping Sets of Meromorphic Functions in the Speiser Class

Bergweiler

Cui

2023

International Mathematics Research Notices

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Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko–Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show that these bounds are also sharp in the Speiser class. We apply this method also to construct meromorphic functions in the Speiser class with preassigned dimensions of the Julia set and the escaping set.

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The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

Abstract: We show that for each $$d\in (0,2]$$ d ∈ ( 0 , 2 ] there exists a meromorphic function f such that the inverse function of f has three singularities and the Julia set of f has Hausdorff dimension d.

Cited by 3 publications

References 52 publications

A brief introduction to one-dimensional complex dynamics

A brief introduction to one-dimensional complex dynamics

Hausdorff dimension of escaping sets of meromorphic functions II

The Hausdorff Dimension of Escaping Sets of Meromorphic Functions in the Speiser Class

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