Slow escape in tracts

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.