Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-14T06:18:27.016Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

Part of: Complex dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  20 November 2020

MAREIKE WOLFF
MAREIKE WOLFF*
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24118 Kiel, Germany
*
e-mail: wolff@math.uni-kiel.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge \exp (|z|^\alpha )$ for some $\alpha>0$ and all z outside a set of finite Lebesgue measure.

Keywords

exponential polynomialfast escaping setFatou setJulia setLebesgue measure

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3821 - 3840
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Aspenberg, M. and Bergweiler, W.. Entire functions with Julia sets of positive measure. Math. Ann. 352(1) (2012), 27–54.10.1007/s00208-010-0625-0CrossRefGoogle Scholar
Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N. S.) 29(2) (1993), 151–188.CrossRefGoogle Scholar
Bergweiler, W.. Lebesgue measure of Julia sets and escaping sets of certain entire functions. Fund. Math. 242(3) (2018), 281–301.10.4064/fm477-11-2017CrossRefGoogle Scholar
Bergweiler, W. and Chyzhykov, I.. Lebesgue measure of escaping sets of entire functions of completely regular growth. J. Lond. Math. Soc. (2) 94(2) (2016), 639–661.CrossRefGoogle Scholar
Bergweiler, W. and Hinkkanen, A.. On semiconjugation of entire functions. Math. Proc. Cambridge Philos. Soc. 126(3) (1999), 565–574.CrossRefGoogle Scholar
Bock, H.. On the dynamics of entire functions on the Julia set. Results Math. 30(1–2) (1996), 16–20.CrossRefGoogle Scholar
Eremenko, A. E.. On the iteration of entire functions. Dynamical Systems and Ergodic Theory (Warsaw 1986) ( Banach Center Publications , 23). PWN, Warsaw, 1989, pp. 339–345.Google Scholar
Hemke, J.-M.. Recurrence of entire transcendental functions with simple post-singular sets. Fund. Math. 187(3) (2005), 255–289.10.4064/fm187-3-4CrossRefGoogle Scholar
McMullen, C.. Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300(1) (1987), 329–342.CrossRefGoogle Scholar
Pommerenke, C.. Univalent Functions ( Studia Mathematica/Mathematische Lehrbücher , XXV). Vandenhoeck & Ruprecht, Göttingen, 1975.Google Scholar
Pommerenke, C.. Boundary Behaviour of Conformal Maps ( Grundlehren der Mathematischen Wissenschaften , 299). Springer, Berlin, 1992.CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M.. Fast escaping points of entire functions. Proc. Lond. Math. Soc. (3) 105(4) (2012), 787–820.CrossRefGoogle Scholar
Schubert, H.. Area of Fatou sets of trigonometric functions. Proc. Amer. Math. Soc. 136(4) (2008), 1251–1259.10.1090/S0002-9939-07-09015-6CrossRefGoogle Scholar
Sixsmith, D. J.. Julia and escaping set spiders’ webs of positive area. Int. Math. Res. Not. 2015(19) (2015), 9751–9774.10.1093/imrn/rnu245CrossRefGoogle Scholar
Zhang, S. and Yang, F.. Area of the complement of the fast escaping set of a family of entire functions. Kodai Math. J. 41(3) (2018), 531–553.CrossRefGoogle Scholar
Zheng, J.-H.. On multiply-connected Fatou components in iteration of meromorphic functions. J. Math. Anal. Appl. 313(1) (2006), 24–37.CrossRefGoogle Scholar