ANNULAR ITINERARIES FOR ENTIRE FUNCTIONS

作者：Rippon P J^*; Stallard G M

来源：Transactions of the American Mathematical Society, 2015, 367(1): 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 s(0)s(1) .... defined by
f(n)(z) is an element of A(sn) (R), for n >= 0,
where A(0)(R) = {z:vertical bar z vertical bar < R} and
An(R) = {z : Mn-1 (R) <= vertical bar z vertical bar < M-n(R)}, n >= 1.
Here M(r) is the maximum modulus of f on {z : vertical bar z vertical bar = 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.

• 出版日期2015-1