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The escaping set of the exponential

Published online by Cambridge University Press:  29 June 2009

LASSE REMPE
LASSE REMPE*
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK (email: [email protected])
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Abstract

We show that the set I(f) of points that converge to infinity under iteration of the exponential map f(z)=exp (z) is a connected subset of the complex plane.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 595 - 599
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Devaney, R. L.. Knaster-like continua and complex dynamics. Ergod. Th. & Dynam. Sys. 13(4) (1993), 627–634.CrossRefGoogle Scholar
[2]Devaney, R. L. and Jarque, X.. Indecomposable continua in exponential dynamics. Conform. Geom. Dyn. 6 (2002), 1–12.CrossRefGoogle Scholar
[3]Devaney, R. L. and Krych, M.. Dynamics of exp(z). Ergod. Th. & Dynam. Sys. 4(1) (1984), 35–52.CrossRefGoogle Scholar
[4]Eremenko, A. È.. On the iteration of entire functions. Dynamical systems and ergodic theory (Warsaw, 1986), Vol. 23. Banach Center Publ., PWN, Warsaw, 1989, pp. 339–345.Google Scholar
[5]Eremenko, A. È. and Lyubich, M. Y.. Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42(4) (1992), 989–1020.CrossRefGoogle Scholar
[6]Förster, M., Rempe, L. and Schleicher, D.. Classification of escaping exponential maps. Proc. Amer. Math. Soc. 136(2) (2008), 651–663.CrossRefGoogle Scholar
[7]Mihaljević-Brandt, H.. Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds. Manuscript, 2008.Google Scholar
[8]Rempe, L.. On a question of Eremenko concerning escaping sets of entire functions. Bull. London Math. Soc. 39(4) (2007), 661–666, arXiv:math.DS/0610453.CrossRefGoogle Scholar
[9]Rempe, L.. On nonlanding dynamic rays of exponential maps. Ann. Acad. Sci. Fenn. Math. 32 (2007), 353–369, arXiv:math.DS/0511588.Google Scholar
[10]Rippon, P. J. and Stallard, G. M.. On questions of Fatou and Eremenko. Proc. Amer. Math. Soc. 133(4) (2005), 1119–1126.CrossRefGoogle Scholar
[11]Rippon, P. J. and Stallard, G. M.. Escaping points of entire functions of small growth. Math. Z. 261(3) (2008), 557–570.CrossRefGoogle Scholar
[12]Rottenfußer, G., Rückert, J., Rempe, L. and Schleicher, D.. Dynamic rays of entire functions. Ann. of Math. (2) to appear, arXiv:0704.3213Google Scholar
[13]Schleicher, D. and Zimmer, J.. Escaping points of exponential maps. J. London Math. Soc. (2) 67(2) (2003), 380–400.CrossRefGoogle Scholar