A Review on the Structure and Properties of the Escaping Set of Transcendental Entire Functions

Abstract

For a transcendental entire function f, we study the structure and properties of the escaping set I(f) which consists of points whose iterates under f escape to infinity. We concentrate on Eremenko’s conjecture and we review some attempts of its proofs. A significant amount of progress in Eremenko’s conjecture has been made possible via fast escaping set A(f) which consists points that escape to infinity as fast as possible. This set can be written as union of closed sets, called levels of A(f). We review classes of functions for which A(f) and each of its levels has the structure of infinite spider’s web. In general, we study classes of entire functions for which the escaping set I(f) is a spider’s web. Spider’s web is a recently investigated structure of I(f) that gives new results in the direction of Eremenko’s conjecture.