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. That is, a complex numberbelongs to the escaping set if and only if the sequence defined by converges to infinity as gets large. The escaping set of is denoted by. For example, for, the origin belongs to the escaping set, since the sequence tends to infinity.
History
The iteration of transcendental entire functions was first studied by Pierre Fatou in 1926 The escaping set occurs implicitly in his study of the explicit entire functions and. The first study of the escaping set for a general transcendental entire function is due to Alexandre Eremenko who used Wiman-Valiron theory. He conjectured that every connected component of the escaping set of a transcendental entire function is unbounded. This has become known as Eremenko's conjecture. There are many partial results on this problem but as of 2013 the conjecture is still open. 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.
Properties
The following properties are known to hold for the escaping set of any non-constant and non-linear entire function.
The escaping set contains at least one point.
For a transcendental entire function, the escaping set always intersects the Julia set. In particular, the escaping set is open if and only if is a polynomial.
Every connected component of the closure of the escaping set is unbounded.
The escaping set always has at least one unbounded connected component.
For a transcendental entire function, the escaping set is connected or it has infinitely many components.
The set is connected.
Note that the final statement does not imply Eremenko's Conjecture.
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 on uncountably many curves, called hairs or rays. In other examples the structure of the escaping set can be very different. As mentioned above, there are examples of entire functions whose escaping set contains no curves.
