Annular itineraries for entire functions

Rippon, P. J. and Stallard, G. M. (2015). Annular itineraries for entire functions. Transactions of the American Mathematical Society, 367(1) pp. 377–399.

DOI: https://doi.org/10.1090/S0002-9947-2014-06354-X

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 s0s1 . . . defined by

fn(z)Asn(R), for n ≥ 0,

where A0(R) = {z : |z| < R} and

An(R) = {z : Mn−1(R) ≤ |z|<Mn(R)}, n≥ 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 ≥ 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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