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Dynamical sets whose union with infinity is connected

Published online by Cambridge University Press:  10 August 2018

DAVID J. SIXSMITH Open the ORCID record for DAVID J. SIXSMITH [Opens in a new window]
DAVID J. SIXSMITH*
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK email [email protected]
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Abstract

Suppose that $f$ is a transcendental entire function. In 2011, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected and an example of a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class.

It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider’s web. We use our results to give a large class of functions in the Eremenko–Lyubich class for which the escaping set is not a spider’s web. Finally, we give a novel topological criterion for certain sets to be a spider’s web.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 3 , March 2020 , pp. 789 - 798
Copyright
© Cambridge University Press, 2018 

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