On the Hausdorff dimension of the escaping set of certain meromorphic functions
HTML articles powered by AMS MathViewer
- 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
- Steven B. Bank and Robert P. Kaufman, On meromorphic solutions of first-order differential equations, Comment. Math. Helv. 51 (1976), no. 3, 289–299. MR 430370, DOI 10.1007/BF02568158
- Krzysztof Barański, Hausdorff dimension of hairs and ends for entire maps of finite order, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 3, 719–737. MR 2464786, DOI 10.1017/S0305004108001515
- Krzysztof Barański, Bogusława Karpińska, and Anna Zdunik, Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts, Int. Math. Res. Not. IMRN 4 (2009), 615–624. MR 2480096, DOI 10.1093/imrn/rnn141
- Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. MR 1216719, DOI 10.1090/S0273-0979-1993-00432-4
- Walter Bergweiler, Philip J. Rippon, and Gwyneth M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, 368–400. MR 2439666, DOI 10.1112/plms/pdn007
- P. Domínguez, Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 225–250. MR 1601879
- A. È. Erëmenko, On the iteration of entire functions, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 339–345. MR 1102727
- A. Eremenko, Transcendental meromorphic functions with three singular values, Illinois J. Math. 48 (2004), no. 2, 701–709. MR 2085435
- A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102
- Anatoly A. Goldberg and Iossif V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. Translated from the 1970 Russian original by Mikhail Ostrovskii; With an appendix by Alexandre Eremenko and James K. Langley. MR 2435270, DOI 10.1090/mmono/236
- W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
- Janina Kotus, On the Hausdorff dimension of Julia sets of meromorphic functions. II, Bull. Soc. Math. France 123 (1995), no. 1, 33–46 (English, with English and French summaries). MR 1330786
- Janina Kotus and Mariusz Urbański, Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions, Bull. London Math. Soc. 35 (2003), no. 2, 269–275. MR 1952406, DOI 10.1112/S0024609302001686
- Janina Kotus and Mariusz Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions, Transcendental dynamics and complex analysis, London Math. Soc. Lecture Note Ser., vol. 348, Cambridge Univ. Press, Cambridge, 2008, pp. 251–316. MR 2458807, DOI 10.1017/CBO9780511735233.013
- James K. Langley, Critical values of slowly growing meromorphic functions, Comput. Methods Funct. Theory 2 (2002), no. 2, [On table of contents: 2004], 539–547. MR 2038137, DOI 10.1007/BF03321864
- R. Daniel Mauldin and Mariusz Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105–154. MR 1387085, DOI 10.1112/plms/s3-73.1.105
- Volker Mayer, The sizes of the Julia set of meromorphic functions, Math. Nachr. 282 (2009), no. 8, 1189–1194. MR 2547715, DOI 10.1002/mana.200710794
- Curt McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), no. 1, 329–342. MR 871679, DOI 10.1090/S0002-9947-1987-0871679-3
- Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330
- Lasse Rempe and Sebastian Van Strien, Absence of line fields and Mañé’s theorem for nonrecurrent transcendental functions, Trans. Amer. Math. Soc. 363 (2011), no. 1, 203–228. MR 2719679, DOI 10.1090/S0002-9947-2010-05125-6
- P. J. Rippon and G. M. Stallard, Iteration of a class of hyperbolic meromorphic functions, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3251–3258. MR 1610785, DOI 10.1090/S0002-9939-99-04942-4
- H. Schubert, Über die Hausdorff-Dimension der Juliamenge von Funktionen endlicher Ordnung. Dissertation, University of Kiel, 2007.
- Gwyneth M. Stallard, Entire functions with Julia sets of zero measure, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 3, 551–557. MR 1068456, DOI 10.1017/S0305004100069437
- Gwyneth M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. Soc. (2) 49 (1994), no. 2, 281–295. MR 1260113, DOI 10.1112/S0024610799008029
- Gwyneth M. Stallard, Dimensions of Julia sets of transcendental meromorphic functions, Transcendental dynamics and complex analysis, London Math. Soc. Lecture Note Ser., vol. 348, Cambridge Univ. Press, Cambridge, 2008, pp. 425–446. MR 2458811, DOI 10.1017/CBO9780511735233.017
- O. Teichmüller, Eine Umkehrung des zweiten Hauptsatzes der Wertverteilungstheorie. Deutsche Math. 2 (1937), 96–107; Gesammelte Abhandlungen, Springer, Berlin, Heidelberg, New York, 1982, pp. 158–169.
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