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A note on the topology of escaping endpoints

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

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
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Abstract

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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
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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