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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the Hausdorff dimension of the escaping set of certain meromorphic functions
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by Walter Bergweiler and Janina Kotus PDF
Trans. Amer. Math. Soc. 364 (2012), 5369-5394 Request permission

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 $\liminf _{n\to \infty }|f^n(z)|\geq R$ 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 $|f^n(z)|\to \infty$ 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.

References
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Additional Information
  • Walter Bergweiler
  • Affiliation: Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D–24098 Kiel, Germany
  • MR Author ID: 35350
  • 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
  • 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”
  • © Copyright 2012 American Mathematical Society
  • 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
  • MathSciNet review: 2931332