Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Annular itineraries for entire functions


Authors: P. J. Rippon and G. M. Stallard
Journal: Trans. Amer. Math. Soc. 367 (2015), 377-399
MSC (2010): Primary 37F10; Secondary 30D05
DOI: https://doi.org/10.1090/S0002-9947-2014-06354-X
Published electronically: June 26, 2014
MathSciNet review: 3271265
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In order to analyse the way in which the size of the iterates of a transcendental entire function $ f$ can behave, we introduce the concept of the annular itinerary of a point $ z$. This is the sequence of non-negative integers $ s_0s_1\ldots $ defined by

$\displaystyle f^n(z)\in A_{s_n}(R),\;\;$$\displaystyle \text {for }n\ge 0, $

where $ A_0(R)=\{z:\vert z\vert<R\}$ and

$\displaystyle A_n(R)=\{z:M^{n-1}(R)\le \vert z\vert<M^n(R)\},\;\;n\ge 1. $

Here $ M(r)$ is the maximum modulus of $ f$ on $ \{z:\vert z\vert=r\}$ and $ R>0$ is so large that $ M(r)>r$, for $ r\ge R$.

We consider the different types of annular itineraries that can occur for any transcendental entire function $ f$ and show that it is always possible to find points with various types of prescribed annular itineraries. The proofs use two new annuli covering results that are of wider interest.


References [Enhancements On Off] (What's this?)

  • [1] I. N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc. (3) 49 (1984), no. 3, 563-576. MR 759304 (86d:58066), https://doi.org/10.1112/plms/s3-49.3.563
  • [2] Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151-188. MR 1216719 (94c:30033), https://doi.org/10.1090/S0273-0979-1993-00432-4
  • [3] Walter Bergweiler and A. Hinkkanen, On semiconjugation of entire functions, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 565-574. MR 1684251 (2000c:37057), https://doi.org/10.1017/S0305004198003387
  • [4] W. Bergweiler, P. J. Rippon, and G. M. Stallard, Multiply connected wandering domains of entire functions, Proc. London Math. Soc. (3) 107 (2013), no. 6, 1261-1301. MR 3149847
  • [5] Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383 (94h:30033)
  • [6] 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 (88e:58057), https://doi.org/10.1017/S0143385700003655
  • [7] 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 (92c:30027)
  • [8] 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 (93k:30034)
  • [9] W. K. Hayman, Subharmonic functions. Vol. 2, London Mathematical Society Monographs, vol. 20, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989. MR 1049148 (91f:31001)
  • [10] M. E. Herring, Mapping properties of Fatou components, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 2, 263-274. MR 1642181 (2000b:30032)
  • [11] Masashi Kisaka, On the connectivity of Julia sets of transcendental entire functions, Ergodic Theory Dynam. Systems 18 (1998), no. 1, 189-205. MR 1609471 (99a:30033), https://doi.org/10.1017/S0143385798097570
  • [12] J. W. Osborne, The structure of spider's web fast escaping sets, Bull. Lond. Math. Soc. 44 (2012), no. 3, 503-519. MR 2966997, https://doi.org/10.1112/blms/bdr112
  • [13] 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 (2011b:30070), https://doi.org/10.1007/s11854-009-0018-z
  • [14] P. J. Rippon and G. M. Stallard, Slow escaping points of meromorphic functions, Trans. Amer. Math. Soc. 363 (2011), no. 8, 4171-4201. MR 2792984 (2012d:37108), https://doi.org/10.1090/S0002-9947-2011-05158-5
  • [15] P. J. Rippon and G. M. Stallard, Fast escaping points of entire functions, Proc. Lond. Math. Soc. (3) 105 (2012), no. 4, 787-820. MR 2989804, https://doi.org/10.1112/plms/pds001
  • [16] P. J. Rippon and G. M. Stallard, Regularity and fast escaping points of entire functions, Int. Math. Res. Not., DOI 10.1093/imrn/rnt111(2013).
  • [17] Günter Rottenfusser, Johannes Rückert, Lasse Rempe, and Dierk Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. (2) 173 (2011), no. 1, 77-125. MR 2753600 (2012b:37121), https://doi.org/10.4007/annals.2011.173.1.3
  • [18] 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 (2003b:37069), https://doi.org/10.1017/S0305004101005813

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37F10, 30D05

Retrieve articles in all journals with MSC (2010): 37F10, 30D05


Additional Information

P. J. Rippon
Affiliation: Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
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
Email: g.m.stallard@open.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2014-06354-X
Received by editor(s): January 24, 2013
Published electronically: June 26, 2014
Additional Notes: Both authors were supported by the EPSRC grant EP/H006591/1
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society