Annular itineraries for entire functions

Rippon, P.; Stallard, G.

doi:10.1090/S0002-9947-2014-06354-X

Annular itineraries for entire functions
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by P. J. Rippon and G. M. Stallard PDF

Trans. Amer. Math. Soc. 367 (2015), 377-399 Request permission

Abstract:

In order to analyse the way in which the size of the iterates of a transcendental entire function $f$ can behave, we introduce the concept of the annular itinerary of a point $z$. This is the sequence of non-negative integers $s_0s_1\ldots$ defined by \[ f^n(z)\in A_{s_n}(R),\;\;\text {for }n\ge 0, \] where $A_0(R)=\{z:|z|<R\}$ and \[ A_n(R)=\{z:M^{n-1}(R)\le |z|<M^n(R)\},\;\;n\ge 1. \] Here $M(r)$ is the maximum modulus of $f$ on $\{z:|z|=r\}$ and $R>0$ 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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