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 iteration of ƒ.[1] Alexandre Eremenko proved that the escaping set always contains at least one point, and that the boundary of the escaping set is exactly the Julia set.[1]
For the complex quadratic polynomial ƒ(z) = z2, the escaping set is the set of z for which | z | > 1. The Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the non-rational points). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.