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On the Hausdorff dimension of the escaping set of certain meromorphic functions


Authors: Walter Bergweiler and Janina Kotus
Journal: Trans. Amer. Math. Soc. 364 (2012), 5369-5394
MSC (2010): Primary 37F10; Secondary 30D05, 30D15
DOI: https://doi.org/10.1090/S0002-9947-2012-05514-0
Published electronically: April 30, 2012
MathSciNet review: 2931332
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Abstract: Let $ f$ be a transcendental meromorphic function of finite order $ \rho $ for which the set of finite singularities of $ f^{-1}$ is bounded. Suppose that $ \infty $ is not an asymptotic value and that there exists $ M \in \mathbb{N}$ such that the multiplicity of all poles, except possibly finitely many, is at most $ M$. For $ R>0$ let $ I_R(f)$ be the set of all $ z\in \mathbb{C}$ for which $ \liminf _{n\to \infty }\vert f^n(z)\vert\geq R$ as $ n\to \infty $. Here $ f^n$ denotes the $ n$-th iterate of $ f$. Let $ I(f)$ be the set of all $ z\in \mathbb{C}$ such that $ \vert f^n(z)\vert\to \infty $ as $ n\to \infty $; that is, $ I(f)=\bigcap _{R>0} I_R(f)$. Denote the Hausdorff dimension of a set $ A\subset \mathbb{C}$ by $ \mathrm {HD}(A)$. It is shown that $ \lim _{R \to \infty } \mathrm {HD}(I_R(f))\leq 2 M \rho /(2+ M\rho )$. In particular, $ \mathrm {HD}(I(f))\leq 2 M \rho /(2+ M\rho )$. These estimates are best possible: for given $ \rho $ and $ M$ we construct a function $ f$ such that $ \mathrm {HD}(I(f))= 2 M \rho /(2+ M\rho )$ and $ \mathrm {HD}(I_R(f))> 2 M \rho /(2+ M\rho )$ for all $ R>0$.

If $ f$ is as above but of infinite order, then the area of $ I_R(f)$ is zero. This result does not hold without a restriction on the multiplicity of the poles.


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Additional Information

Walter Bergweiler
Affiliation: Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D–24098 Kiel, Germany
Email: bergweiler@math.uni-kiel.de

Janina Kotus
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland
Email: J.Kotus@impan.pl

DOI: https://doi.org/10.1090/S0002-9947-2012-05514-0
Received by editor(s): January 21, 2009
Received by editor(s) in revised form: May 13, 2010, and November 22, 2010
Published electronically: April 30, 2012
Additional Notes: The authors were supported by the EU Research Training Network CODY. The first author was also supported by the G.I.F., the German–Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003, the ESF Research Networking Programme HCAA and the Deutsche Forschungsgemeinschaft, Be 1508/7-1. The second author was also supported by PW Grant 504G 1120 0011 000 and Polish MNiSW Grant “Chaos, fraktale i dynamika konforemna II”
Article copyright: © Copyright 2012 American Mathematical Society

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