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arXiv · 1301.1328

Annular itineraries for entire functions

Philip J. Rippon·Gwyneth M. Stallard
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Abstract

In order to analyse the way in which the size of the iterates $(f^n(z))$ of a transcendental entire function $f$ can behave, we introduce the concept of the {\it annular itinerary} of a point $z$. This is the sequence of non-negative integers $s_0s_1...$ defined by \[ f^n(z)\in A_{s_n}(R),\;\;\text{for}n\ge 0, \] where $A_0(R)=\{z:|z| 0$ is so large that $M(r)>r$, for $r\ge R$. We consider the different types of annular itineraries that can occur for any transcendental entire function $f$ and show that it is always possible to find points with various types of prescribed annular itineraries. The proofs use two new annuli covering results that are of wider interest.

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Philip J. Rippon, Gwyneth M. Stallard. 2013-01-07. Annular itineraries for entire functions. https://arxiv.org/abs/1301.1328

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