Let be an open subset of the complex plane and let be analytic. Denote the -th iterate of by , i.e. and . Then the Julia set of is the subset of characterized by the following property: if then the restriction of to any neighborhood of is not a normal family.
It can also be shown that the Julia set of is the closure of the set of repelling periodic points of . (Repelling periodic point means that, for some , we have and .)
A simple example is afforded by the map ; in this case, the Julia set is the unit circle. In general, however, things are much more complicated and the Julia set is a fractal.