Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-19T05:44:26.834Z Has data issue: false hasContentIssue false
  • English
  • Français

Entire functions for which the escaping set is a spider's web

Published online by Cambridge University Press:  05 September 2011

D. J. SIXSMITH
D. J. SIXSMITH*
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA. e-mail: sixsmiths@hotmail.co.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct several new classes of transcendental entire functions, f, such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider's web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f) and A(f) are spiders' webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders' webs to give new results concerning functions with no unbounded Fatou components.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 3 , November 2011 , pp. 551 - 571
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abramowitz, M. and Stegun, I. A. Eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, 1972.Google Scholar
[2]Anderson, J. M. and Hinkkanen, A.Unbounded domains of normality. Proc. Amer. Math. Soc. 126, 11 (1998), 3243–3252.CrossRefGoogle Scholar
[3]Baker, I. N.Zusammensetzungen ganzer Funktionen. Math. Z. 69 (1958), 121–163.CrossRefGoogle Scholar
[4]Baker, I. N.The iteration of polynomials and transcendental entire functions. J. Austral. Math. Soc. Ser. A 30, 4 (1980/81), 483–495.CrossRefGoogle Scholar
[5]Bell, E. T.Exponential numbers. Amer. Math. Monthly 41, 7 (1934), 411–419.CrossRefGoogle Scholar
[6]Bergweiler, W.Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S.) 29, 2 (1993), 151–188.CrossRefGoogle Scholar
[7]Bergweiler, W. and Hinkkanen, A.On semiconjugation of entire functions. Math. Proc. Camb. Phil. Soc. 126, 3 (1999), 565–574.CrossRefGoogle Scholar
[8]Bouroubi, S.Bell numbers and Engel's conjecture. Rostock. Math. Kolloq. 62 (2007), 61–70.Google Scholar
[9]Cao, C. and Wang, Y.Boundedness of Fatou components of holomorphic maps. J. Dynam. Differential Equations 16, 2 (2004), 377–384.CrossRefGoogle Scholar
[10]Eremenko, A. E.On the iteration of entire functions. Dynamical systems and ergodic theory (Warsaw 1986) 23 (1989), 339–345.Google Scholar
[11]Fuchs, W. H. J.Proof of a conjecture of G. Pólya concerning gap series. Illinois J. Math. 7 (1963), 661–667.CrossRefGoogle Scholar
[12]Hayman, W. K.Angular value distribution of power series with gaps. Proc. London Math. Soc. 24 (3) (1972), 590–624.CrossRefGoogle Scholar
[13]Hayman, W. K.The local growth of power series: a survey of the Wiman–Valiron method. Canad. Math. Bull. 17, 3 (1974), 317–358.CrossRefGoogle Scholar
[14]Hinkkanen, A. Entire functions with bounded Fatou components. In Transcendental dynamics and complex analysis, vol. 348 of London Math. Soc. Lecture Note Ser. (Cambridge University Press, 2008), pp. 187–216.CrossRefGoogle Scholar
[15]Juneja, O. P.On the coefficients of an entire series. J. Anal. Math. 24 (1971), 395–401.CrossRefGoogle Scholar
[16]Mihaljević–Brandt, H. and Peter, J. Poincaré functions with spiders' webs. Preprint, arXiv:1010.0299v1.Google Scholar
[17]Osborne, J. W. The structure of spider's web fast escaping sets. Preprint, arXiv:1011.0359v1.Google Scholar
[18]Rippon, P. J. and Stallard, G. M. Regularity and fast escaping points of entire functions. In preparation.Google Scholar
[19]Rippon, P. J. and Stallard, G. M.Functions of small growth with no unbounded Fatou components. J. Anal. Math. 108 (2009), 61–86.CrossRefGoogle Scholar
[20]Rippon, P. J. and Stallard, G. M. Fast escaping points of entire functions. Preprint, arXiv:1009.5081v1 (2010).Google Scholar
[21]Singh, A. P.Composite entire functions with no unbounded Fatou components. J. Math. Anal. Appl. 335, 2 (2007), 907–914.CrossRefGoogle Scholar
[22]Wang, Y.On the Fatou set of an entire function with gaps. Tohoku Math. J. (2) 53, 1 (2001), 163–170.CrossRefGoogle Scholar