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Maximally and non-maximally fast escaping points of transcendental entire functions

Published online by Cambridge University Press:  28 January 2015

D. J. SIXSMITH
D. J. SIXSMITH*
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA. e-mail: david.sixsmith@open.ac.uk
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Abstract

We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest.

It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 2 , March 2015 , pp. 365 - 383
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1]Baker, I. N.Multiply connected domains of normality in iteration theory. Math. Z. 81 (1963), 206–214.Google Scholar
[2]Baker, I. N.Wandering domains in the iteration of entire functions. Proc. London Math. Soc. (3) 49 3 (1984), 563–576.Google Scholar
[3]Bergweiler, W.Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S.) 29 2 (1993), 151–188.CrossRefGoogle Scholar
[4]Bergweiler, W.An entire function with simply and multiply connected wandering domains. Pure Appl. Math. Q. 7 2 (2011), 107–120.Google Scholar
[5]Bergweiler, W. and Hinkkanen, A.On semiconjugation of entire functions. Math. Proc. Camb. Phil. Soc. 126 3 (1999), 565–574.Google Scholar
[6]Carleson, L. and Gamelin, T. W.Complex dynamics. Universitext: Tracts in Mathematics. (Springer-Verlag, New York, 1993).Google Scholar
[7]Eremenko, A. E.On the iteration of entire functions. Dynamical Systems and Ergodic Theory (Warsaw 1986) 23 (1989), 339–345.Google Scholar
[8]Hardy, G. H.The maximum modulus of an integral function. Quarterly J. Math. 41 (1909), 1–9.Google Scholar
[9]Hayman, W. K.Subharmonic functions. Vol. 2, vol. 20 of London Math. Soc. Monogr.. (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989).Google Scholar
[10]Herring, M. E.Mapping properties of Fatou components. Ann. Acad. Sci. Fenn. Math. 23 2 (1998), 263–274.Google Scholar
[11]Hurewicz, W. and Wallman, H.Dimension Theory. (Princeton University Press, Princeton, N. J., 1948).Google Scholar
[12]Osborne, J.The structure of spider's web fast escaping sets. Bulletin of the London Mathematical Society 44 3 (2012), 503–519.CrossRefGoogle Scholar
[13]Osborne, J.Connectedness properties of the set where the iterates of an entire function are bounded. Math. Proc. Camb. Phil. Soc. 155 3 (2013), 391–410.Google Scholar
[14]Rempe, L.The escaping set of the exponential. Ergodic Theory Dynam. Systems 30 2 (2010), 595–599.CrossRefGoogle Scholar
[15]Rippon, P. and Stallard, G. M. Annular itineraries for entire functions. Trans. Amer. Math. Soc. electronically published on June 26, 2014, DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06354-X (to appear in print) (2014).Google Scholar
[16]Rippon, P. J. and Stallard, G. M.Escaping points of meromorphic functions with a finite number of poles. J. Anal. Math. 96 (2005), 225–245.Google Scholar
[17]Rippon, P. J. and Stallard, G. M.On questions of Fatou and Eremenko. Proc. Amer. Math. Soc. 133 4 (2005), 1119–1126.Google Scholar
[18]Rippon, P. J. and Stallard, G. M.Escaping points of entire functions of small growth. Math. Z. 261 3 (2009), 557–570.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.Google Scholar
[20]Rippon, P. J. and Stallard, G. M.Slow escaping points of meromorphic functions. Trans. Amer. Math. Soc. 363 8 (2011), 4171–4201.Google Scholar
[21]Rippon, P. J. and Stallard, G. M.Fast escaping points of entire functions. Proc. London Math. Soc. (3) 105 4 (2012), 787–820.Google Scholar
[22]Rippon, P. J. and Stallard, G. M.Baker's conjecture and Eremenko's conjecture for functions with negative zeros. J. Anal. Math. 120 (2013), 291–309.Google Scholar
[23]Sixsmith, D. J.Entire functions for which the escaping set is a spider's web. Math. Proc. Camb. Phil. Soc. 151 3 (2011), 551–571.Google Scholar
[24]Sixsmith, D. J.Simply connected fast escaping Fatou components. Pure Appl. Math. Q. 8 4 (2012), 1029–1046.Google Scholar
[25]Sixsmith, D. J.On fundamental loops and the fast escaping set. J. Lond. Math. Soc. (2) 88 3 (2013), 716–736.Google Scholar
[26]Tyler, T. F.Maximum curves and isolated points of entire functions. Proc. Amer. Math. Soc. 128 9 (2000), 2561–2568.Google Scholar
[27]Valiron, G.Lectures on the General Theory of Integral Functions. (Chelsea, 1949).Google Scholar