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The escaping set of transcendental self-maps of the punctured plane

Published online by Cambridge University Press:  13 October 2016

DAVID MARTÍ-PETE
DAVID MARTÍ-PETE*
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK email [email protected]
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Abstract

We study the different rates of escape of points under iteration by holomorphic self-maps of $\mathbb{C}^{\ast }=\mathbb{C}\setminus \{0\}$ for which both zero and infinity are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either zero, infinity or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points, each of which has $J(f)$ as its boundary.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018 , pp. 739 - 760
Copyright
© Cambridge University Press, 2016 

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