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Hausdorff dimension of escaping sets of meromorphic functions II

Aspenberg, Magnus LU and Cui, Weiwei LU (2023) In Ergodic Theory and Dynamical Systems 43(5). p.1471-1491
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 <![CDATA[ $2$ ]]> or <![CDATA[ $1/2$ ]]>. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in <![CDATA[ $[0,2]$ ]]> (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 <![CDATA[ $4$... (More)

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 <![CDATA[ $2$ ]]> or <![CDATA[ $1/2$ ]]>. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in <![CDATA[ $[0,2]$ ]]> (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 <![CDATA[ $4$ ]]>.

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author
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
escaping sets, meromorphic functions, quasiconformal surgery, singular values, Speiser functions
in
Ergodic Theory and Dynamical Systems
volume
43
issue
5
pages
1471 - 1491
publisher
Cambridge University Press
external identifiers
  • scopus:85124914323
ISSN
0143-3857
DOI
10.1017/etds.2022.5
language
English
LU publication?
yes
additional info
Publisher Copyright: © The Author(s), 2022. Published by Cambridge University Press.
id
4b38ee14-b77c-46b0-a927-dd536a74745e
date added to LUP
2022-04-12 16:16:46
date last changed
2023-10-26 14:58:51
@article{4b38ee14-b77c-46b0-a927-dd536a74745e,
  abstract     = {{<p>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 &lt;![CDATA[ $2$ ]]&gt; or &lt;![CDATA[ $1/2$ ]]&gt;. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in &lt;![CDATA[ $[0,2]$ ]]&gt; (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 &lt;![CDATA[ $4$ ]]&gt;. </p>}},
  author       = {{Aspenberg, Magnus and Cui, Weiwei}},
  issn         = {{0143-3857}},
  keywords     = {{escaping sets; meromorphic functions; quasiconformal surgery; singular values; Speiser functions}},
  language     = {{eng}},
  number       = {{5}},
  pages        = {{1471--1491}},
  publisher    = {{Cambridge University Press}},
  series       = {{Ergodic Theory and Dynamical Systems}},
  title        = {{Hausdorff dimension of escaping sets of meromorphic functions II}},
  url          = {{http://dx.doi.org/10.1017/etds.2022.5}},
  doi          = {{10.1017/etds.2022.5}},
  volume       = {{43}},
  year         = {{2023}},
}