Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-04T18:04:41.440Z Has data issue: false hasContentIssue false

A note on the topology of escaping endpoints

Published online by Cambridge University Press:  13 January 2020

DAVID S. LIPHAM*
Affiliation:
Department of Mathematics, Auburn University at Montgomery, Montgomery, AL 36117, USA email [email protected], [email protected]

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.

Type
Original Article
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), 897918.10.1090/S0002-9947-1993-1182980-3CrossRefGoogle Scholar
Alhabib, N. and Rempe-Gillen, L.. Escaping endpoints explode. Comput. Methods Funct. Theory 17(1) (2017), 65100.CrossRefGoogle Scholar
Baker, I. N. and Domínguez, P.. Residual Julia sets. J. Anal. 8 (2000), 121137.Google Scholar
Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29(2) (1993), 151188.10.1090/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), 529534.CrossRefGoogle Scholar
Charatonik, W. J.. The Lelek fan is unique. Houston J. Math. 15 (1989), 2734.Google Scholar
Devaney, R. L. and Krych, M.. Dynamics of exp(z). Ergod. Th. & Dynam. Sys. 4 (1984), 3552.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), 162.Google Scholar
Eremenko, A. E.. On the iteration of entire functions. Banach Center Publ. 23(1) (1989), 339345.CrossRefGoogle Scholar
Eremenko, A. E. and Lyubich, M. Y.. Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42(4) (1992), 9891020.CrossRefGoogle Scholar
Kawamura, K., Oversteegen, L. G. and Tymchatyn, E. D.. On homogeneous totally disconnected 1-dimensional spaces. Fund. Math. 150 (1996), 97112.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), 177183.10.1017/S0143385700005460CrossRefGoogle Scholar
Oversteegen, L. G. and Tymchatyn, E. D.. On the dimension of certain totally disconnected spaces. Proc. Amer. Math. Soc. 122(3) (1994), 885891.CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M.. Fast escaping points of entire functions. Proc. Lond. Math. Soc. 105 (2012), 787820.CrossRefGoogle Scholar