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 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.

References in corpus (4)

• Fast escaping points of entire functions
• Hausdorff dimensions of escaping sets of transcendental entire functions
• On the Hausdorff dimension of the Julia set of a regularly growing entire function
• Escape rate and Hausdorff measure for entire functions

Cited by in corpus (2)

• The escaping set in transcendental dynamics
• On the dimension of points which escape to infinity at given rate under exponential iteration