Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T20:03:29.018Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the topology of escaping endpoints

Part of: Entire and meromorphic functions, and related topics Complex dynamical systems Spaces with richer structures

Published online by Cambridge University Press:  13 January 2020

DAVID S. LIPHAM Open the ORCID record for DAVID S. LIPHAM [Opens in a new window]
DAVID S. LIPHAM*
Affiliation:
Department of Mathematics, Auburn University at Montgomery, Montgomery, AL 36117, USA email dsl0003@auburn.edu, dlipham@aum.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp (z)+a$ when $a\in (-\infty ,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function $f$, the escaping Julia set $I(f)\cap J(f)$ is first category.

Keywords

Julia setescaping endpointsalmost zero-dimensional

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 54E52: Baire category, Baire spaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 1156 - 1159
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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

Aarts, J. M. and Oversteegen, L. G.. The geometry of Julia sets. Trans. Amer. Math. Soc. 338(2) (1993), 897–918.10.1090/S0002-9947-1993-1182980-3CrossRefGoogle Scholar
Alhabib, N. and Rempe-Gillen, L.. Escaping endpoints explode. Comput. Methods Funct. Theory 17(1) (2017), 65–100.CrossRefGoogle Scholar
Baker, I. N. and Domínguez, P.. Residual Julia sets. J. Anal. 8 (2000), 121–137.Google Scholar
Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29(2) (1993), 151–188.10.1090/S0273-0979-1993-00432-4CrossRefGoogle Scholar
Bula, W. D. and Oversteegen, L. G.. A characterization of smooth Cantor bouquets. Proc. Amer. Math. Soc. 108 (1990), 529–534.CrossRefGoogle Scholar
Charatonik, W. J.. The Lelek fan is unique. Houston J. Math. 15 (1989), 27–34.Google Scholar
Devaney, R. L. and Krych, M.. Dynamics of exp(z). Ergod. Th. & Dynam. Sys. 4 (1984), 35–52.CrossRefGoogle Scholar
Dijkstra, J. J. and Lipham, D. S.. On cohesive almost zero-dimensional spaces. Preprint.Google Scholar
Dijkstra, J. J. and van Mill, J.. Erdős space and homeomorphism groups of manifolds. Mem. Amer. Math. Soc. 208(979) (2010), 1–62.Google Scholar
Eremenko, A. E.. On the iteration of entire functions. Banach Center Publ. 23(1) (1989), 339–345.CrossRefGoogle Scholar
Eremenko, A. E. and Lyubich, M. Y.. Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42(4) (1992), 989–1020.CrossRefGoogle Scholar
Kawamura, K., Oversteegen, L. G. and Tymchatyn, E. D.. On homogeneous totally disconnected 1-dimensional spaces. Fund. Math. 150 (1996), 97–112.Google Scholar
Mayer, J. C.. An explosion point for the set of endpoints of the Julia set of 𝜆exp(z). Ergod. Th. & Dynam. Sys. 10(1) (1990), 177–183.10.1017/S0143385700005460CrossRefGoogle Scholar
Oversteegen, L. G. and Tymchatyn, E. D.. On the dimension of certain totally disconnected spaces. Proc. Amer. Math. Soc. 122(3) (1994), 885–891.CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M.. Fast escaping points of entire functions. Proc. Lond. Math. Soc. 105 (2012), 787–820.CrossRefGoogle Scholar