Slow escaping points of meromorphic functions
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Abstract:
We show that for any transcendental meromorphic function $f$ there is a point $z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is, tend to $\infty$, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which $f^n(z)$ tends to $\infty$ at a bounded rate, and establish the connections between these sets and the Julia set of $f$. To do this, we show that the iterates of $f$ satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a quasi-nested wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.References
- I. N. Baker, An entire function which has wandering domains, J. Austral. Math. Soc. Ser. A 22 (1976), no. 2, 173–176. MR 419759, DOI 10.1017/s1446788700015287
- I. N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc. (3) 49 (1984), no. 3, 563–576. MR 759304, DOI 10.1112/plms/s3-49.3.563
- I. N. Baker and P. Domínguez, Boundaries of unbounded Fatou components of entire functions, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 2, 437–464. MR 1724391
- Irvine N. Baker and L. S. O. Liverpool, Picard sets for entire functions, Math. Z. 126 (1972), 230–238. MR 344473, DOI 10.1007/BF01110727
- Krzysztof Barański, Trees and hairs for some hyperbolic entire maps of finite order, Math. Z. 257 (2007), no. 1, 33–59. MR 2318569, DOI 10.1007/s00209-007-0114-7
- Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
- 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, Invariant domains and singularities, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 3, 525–532. MR 1317494, DOI 10.1017/S0305004100073345
- Walter Bergweiler, On the Julia set of analytic self-maps of the punctured plane, Analysis 15 (1995), no. 3, 251–256. MR 1357963, DOI 10.1524/anly.1995.15.3.251
- W. Bergweiler, An entire function with simply and multiply connected wandering domains, Pure Appl. Math. Quarterly, 7 (2011), 107–120.
- Walter Bergweiler and A. Hinkkanen, On semiconjugation of entire functions, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 565–574. MR 1684251, DOI 10.1017/S0305004198003387
- 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
- Robert L. Devaney and Folkert Tangerman, Dynamics of entire functions near the essential singularity, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 489–503. MR 873428, DOI 10.1017/S0143385700003655
- 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
- Núria Fagella, Dynamics of the complex standard family, J. Math. Anal. Appl. 229 (1999), no. 1, 1–31. MR 1664296, DOI 10.1006/jmaa.1998.6134
- P. Fatou, Sur l’itération des fonctions transcendantes Entières, Acta Math. 47 (1926), no. 4, 337–370 (French). MR 1555220, DOI 10.1007/BF02559517
- Dieter Gaier, Lectures on complex approximation, Birkhäuser Boston, Inc., Boston, MA, 1987. Translated from the German by Renate McLaughlin. MR 894920, DOI 10.1007/978-1-4612-4814-9
- W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
- W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672
- A. Hinkkanen, Julia sets of polynomials are uniformly perfect, Bull. London Math. Soc. 26 (1994), no. 2, 153–159. MR 1272301, DOI 10.1112/blms/26.2.153
- Boguslawa Karpińska, Hausdorff dimension of the hairs without endpoints for $\lambda \exp z$, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 11, 1039–1044 (English, with English and French summaries). MR 1696203, DOI 10.1016/S0764-4442(99)80321-8
- Masashi Kisaka, On the connectivity of Julia sets of transcendental entire functions, Ergodic Theory Dynam. Systems 18 (1998), no. 1, 189–205. MR 1609471, DOI 10.1017/S0143385798097570
- John C. Mayer, An explosion point for the set of endpoints of the Julia set of $\lambda \exp (z)$, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 177–183. MR 1053806, DOI 10.1017/S0143385700005460
- Lasse Rempe, Topological dynamics of exponential maps on their escaping sets, Ergodic Theory Dynam. Systems 26 (2006), no. 6, 1939–1975. MR 2279273, DOI 10.1017/S0143385706000435
- Lasse Rempe, Philip J. Rippon, and Gwyneth M. Stallard, Are Devaney hairs fast escaping?, J. Difference Equ. Appl. 16 (2010), no. 5-6, 739–762. MR 2675603, DOI 10.1080/10236190903282824
- P. J. Rippon, Asymptotic values of continuous functions in Euclidean space, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 2, 309–318. MR 1142750, DOI 10.1017/S030500410007540X
- P. J. Rippon, Baker domains of meromorphic functions, Ergodic Theory Dynam. Systems 26 (2006), no. 4, 1225–1233. MR 2247639, DOI 10.1017/S0143385706000162
- P. J. Rippon and G. M. Stallard, On sets where iterates of a meromorphic function zip towards infinity, Bull. London Math. Soc. 32 (2000), no. 5, 528–536. MR 1767705, DOI 10.1112/S002460930000730X
- P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1119–1126. MR 2117213, DOI 10.1090/S0002-9939-04-07805-0
- P. J. Rippon and G. M. Stallard, On multiply connected wandering domains of meromorphic functions, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 405–423. MR 2400399, DOI 10.1112/jlms/jdm118
- P. J. Rippon and G. M. Stallard, Escaping points of entire functions of small growth, Math. Z. 261 (2009), no. 3, 557–570. MR 2471088, DOI 10.1007/s00209-008-0339-0
- P. J. Rippon and G. M. Stallard, Functions of small growth with no unbounded Fatou components, J. Anal. Math. 108 (2009), 61–86. MR 2544754, DOI 10.1007/s11854-009-0018-z
- P.J. Rippon and G.M. Stallard, Fast escaping points of entire functions, arXiv:1009.5081.
- Dierk Schleicher and Johannes Zimmer, Escaping points of exponential maps, J. London Math. Soc. (2) 67 (2003), no. 2, 380–400. MR 1956142, DOI 10.1112/S0024610702003897
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
- Jian-Hua Zheng, On uniformly perfect boundary of stable domains in iteration of meromorphic functions. II, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 3, 531–544. MR 1891688, DOI 10.1017/S0305004101005813
- Jian-Hua Zheng, On multiply-connected Fatou components in iteration of meromorphic functions, J. Math. Anal. Appl. 313 (2006), no. 1, 24–37. MR 2178719, DOI 10.1016/j.jmaa.2005.05.038
Additional Information
- P. J. Rippon
- Affiliation: Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
- MR Author ID: 190595
- Email: p.j.rippon@open.ac.uk
- G. M. Stallard
- Affiliation: Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
- MR Author ID: 292621
- Email: g.m.stallard@open.ac.uk
- Received by editor(s): September 5, 2008
- Received by editor(s) in revised form: June 23, 2009
- Published electronically: March 15, 2011
- © Copyright 2011 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 363 (2011), 4171-4201
- MSC (2010): Primary 37F10; Secondary 30D05
- DOI: https://doi.org/10.1090/S0002-9947-2011-05158-5
- MathSciNet review: 2792984