Rippon, P. J. and Stallard, G. M.
(2015).
Annular itineraries for entire functions.
Transactions of the American Mathematical Society, 367(1) pp. 377–399.
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_{0}s_{1} . . . defined by
f^{n}(z) ∈ A_{s}_{n}(R), for n ≥ 0,
where A_{0}(R) = {z : |z| < R} and
A_{n}(R) = {z : M^{n−1}(R) ≤ |z|<M^{n}(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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