In mathematics, and particularly complex dynamics

Complex dynamics

Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions.-Techniques:*General** Montel's theorem...

, the escaping set of an entire function

Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

 ƒ consists of all points that tend to infinity under the repeated iteration of ƒ.
Alexandre Eremenko proved that the escaping set always contains at least one point, and that the boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

 of the escaping set is exactly the Julia set

Julia set

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...

.

Example

For the complex quadratic polynomial

Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.-Forms:When the quadratic polynomial has only one variable , one can distinguish its 4 main forms:...

 , the escaping set is the set of z for which The Julia set is the unit circle and on this the iteration is given by doubling of angles

Dyadic transformation

The dyadic transformation is the mapping produced by the rule...

 (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.