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Distinguishing endpoint sets from Erdős space

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

Published online by Cambridge University Press:  15 February 2022

DAVID S. LIPHAM
DAVID S. LIPHAM*
Affiliation:
Department of Mathematics, Auburn University at Montgomery, Montgomery AL 36117, U.S.A. e-mails: dlipham@aum.edu, dsl0003@auburn.edu
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Abstract

We prove that the set of all endpoints of the Julia set of $f(z)=\exp\!(z)-1$ which escape to infinity under iteration of f is not homeomorphic to the rational Hilbert space $\mathfrak E$ . As a corollary, we show that the set of all points $z\in \mathbb C$ whose orbits either escape to $\infty$ or attract to 0 is path-connected. We extend these results to many other functions in the exponential family.

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 54F50: Spaces of dimension $leq 1$; curves, dendrites
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 3 , November 2022 , pp. 635 - 646
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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