On the Partition of Fast Escaping Sets of a Transcendental Entire Function

For a transcendental entire function f, the set of form I ( f ) = {z ∈  : (z) → ∞  as n → ∞} is called an escaping set. The major open question in transcendental dynamics is the conjecture of Eremenko, which states that for any transcendental entire function f, the escaping set I ( f ) has no bounded component. This conjecture in a special case has been proved by defining the fast escaping set A ( f ), which consists of points that move to infinity as fast as possible. Very recent studies in the field of transcendental dynamics have concentrated on the partition of fast escaping sets into maximally and non-maximally fast escaping sets. It is well known that a fast escaping set has no bounded component, but in contrast, there are entire transcendental functions for which each maximally and non-maximally fast escaping set has uncountably many singleton components.