Dimensions of slowly escaping sets and annular itineraries for exponential functions

Abstract

We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity ‘slowly’, and which have Hausdorff dimension equal to$1$. We prove these results by using the idea of anannular itinerary. In the case of a general transcendental entire function we show that one of these sets, theuniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.