Dimensions of slowly escaping sets and annular itineraries for exponential functions

TL;DR

This paper investigates the structure of slowly escaping points in exponential functions, constructing sets with Hausdorff dimension one and analyzing their dynamical properties using annular itineraries.

Contribution

It introduces new sets of slowly escaping points with Hausdorff dimension one and characterizes their properties for exponential and general transcendental entire functions.

Findings

Constructed sets of slowly escaping points with Hausdorff dimension one.

Proved the existence of uniformly slowly escaping sets with strong dynamical properties.

Provided necessary and sufficient conditions for the non-emptiness of these sets.

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.