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.



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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