Slow escape in tracts

TL;DR

This paper extends previous results on slow escaping points in transcendental entire functions, demonstrating their existence within specific types of tracts and providing illustrative examples.

Contribution

It introduces new covering techniques to prove slow escape points exist in logarithmic and boundary-specific tracts, expanding prior work.

Findings

Existence of slow escaping points in logarithmic tracts

Existence of slow escaping points in tracts with certain boundary properties

Examples illustrating the theoretical results

Abstract

Let $f$ be a transcendental entire function. By a result of Rippon and Stallard, there exist points whose orbit escapes arbitrarily slowly. By using a range of techniques to prove new covering results, we extend their theorem to prove the existence of points which escape arbitrarily slowly within logarithmic tracts and tracts with certain boundary properties. We then give examples to illustrate our results in a variety of tracts.