Escaping set

In mathematics, and particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ.

belongs to the escaping set if and only if the sequence defined by

, the origin belongs to the escaping set, since the sequence tends to infinity.

The iteration of transcendental entire functions was first studied by Pierre Fatou in 1926[2] The escaping set occurs implicitly in his study of the explicit entire functions

The first study of the escaping set for a general transcendental entire function is due to Alexandre Eremenko who used Wiman-Valiron theory.

[3] He conjectured that every connected component of the escaping set of a transcendental entire function is unbounded.

[1][4] There are many partial results on this problem but as of 2013 the conjecture is still open.

In 2021 a paper by Martí-Pete, Rempe & Waterman constructed a counterexample to Eremenko's conjecture[5] Eremenko also asked whether every escaping point can be connected to infinity by a curve in the escaping set; it was later shown that this is not the case.

Indeed, there exist entire functions whose escaping sets do not contain any curves at all.

[4] The following properties are known to hold for the escaping set of any non-constant and non-linear entire function.

Note that the final statement does not imply Eremenko's Conjecture.

A polynomial of degree 2 extends to an analytic self-map of the Riemann sphere, having a super-attracting fixed point at infinity.

is an open and connected subset of the complex plane, and the Julia set is the boundary of this basin.

For instance the escaping set of the complex quadratic polynomial

consists precisely of the complement of the closed unit disc: For transcendental entire functions, the escaping set is much more complicated than for polynomials: in the simplest cases like the one illustrated in the picture it consists of uncountably many curves, called hairs or rays.

In other examples the structure of the escaping set can be very different (a spider's web).

[7] As mentioned above, there are examples of transcendental entire functions whose escaping set contains no curves.