Julia set

Let $U$ be an open subset of the complex plane and let $f:U\to U$ be analytic. Denote the $n$-th iterate of $f$ by $f^n$, i.e. $f^1=f$ and $f^{n+1}=f\circ f^n$. Then the Julia set of $f$ is the subset $J$ of $U$ characterized by the following property: if $z\in J$ then the restriction of $\{f^n\mid n\in \mathbb{N}\}$ to any neighborhood of $z$ is not a normal family.

It can also be shown that the Julia set of $f$ is the closure of the set of repelling periodic points of $f$. (Repelling periodic point means that, for some $n$, we have $f^n(z)=z$ and $|f^{\prime }(z)|>1$.)

A simple example is afforded by the map $f(z)=z^2$; 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.

From the definition, it follows that the Julia set is closed under $f$ and its inverse — $f(J)=J$ and $f^{-1}(J)=J$. Topologically, Julia sets are perfect and have empty interior.

Title	Julia set
Canonical name	JuliaSet
Date of creation	2013-03-22 17:15:26
Last modified on	2013-03-22 17:15:26
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	7
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 28A80