Hausdorff dimension of escaping sets of meromorphic functions II

Abstract

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 $4$ .

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

“…As a consequence of our main result, we also have the following theorem which completes the study begun in [2] and continued in [3].…”

“…Our results can be seen as a contribution to recent efforts to understand the differences and similarities between the classes B and S. We refer to [2,3,10,15,16,17,21] and the survey [37] for results in this direction. As an example we mention the recent result of Albrecht and Bishop [1] who showed that given δ > 0 there exists an entire function f ∈ S such that 1 < dim J(f ) < 1 + δ.…”

“…A major difference to [2,3] is that we do not use quasiconformal surgery. Our main tools are a recent result of Mayer and Urbański [33] on the dimension of escaping sets, a method of MacLane [32] and Vinberg [42] to construct real entire functions with preassigned critical values, and a result of Warschawski [43] on the angular derivative.…”

A brief introduction to one-dimensional complex dynamics

“…若 R(f ) 分别为空、有限或无限集, 则称 f 分别为不可重整、(至多) 有限可重整或无穷可重整. 对于可重整的单临界多项式, 根据重整的 组合可将其划分成 primitive 和 satellite 等类型 17) (参见文献 [499, 第 7.3 小节]).…”

“…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].…”