Dimensions of slowly escaping sets and annular itineraries for exponential functions

Ergodic Theory and Dynamical Systems FirstView (10 2015), 1–20. Also available on the arXiv.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 an annular itinerary. In the case of a general transcendental entire function we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.