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Dimensions of slowly escaping sets and annular itineraries for exponential functions

Published online by Cambridge University Press:  13 April 2015

D. J. SIXSMITH Open the ORCID record for D. J. SIXSMITH [Opens in a new window]
D. J. SIXSMITH*
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK email david.sixsmith@open.ac.uk
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Abstract

We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity ‘slowly’, and which have Hausdorff dimension equal to $1$. We prove these results by using the idea of an annular itinerary. In the case of a general transcendental entire function we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2273 - 2292
Copyright
© Cambridge University Press, 2015 

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