Definition:Julia Set
Definition
Let $f$ be an iterated rational function defined on the extended complex plane $\overline \C$.
Definition $1$
The Julia set of $f$ is the closed subset of $\overline \C$ which is invariant under $f$.
Definition $2$
The Julia set of $f$ is the boundary of the set of those points in $\overline \C$ whose orbits under $f$ are bounded.
Filled Julia Set
The filled Julia set of $f$ is the set of those points in $\overline \C$ whose orbits under $f$ are bounded.
Examples
Julia Set for $-0.13 + 0.75 i$
The below is a graphical representation of the Julia set for the rational function $z \mapsto z^2 + c$ for the point $c = -0.13 + 0.75i$:
Filled Julia Set for $-0.75$
The below is a graphical representation of the filled Julia set for the rational function $z \mapsto z^2 + c$ for the point $c = -0.75$:
Also see
- Results about Julia sets can be found here.
Source of Name
This entry was named for Gaston Maurice Julia.
Historical Note
The concept of the Julia set was introduced by Gaston Maurice Julia in $1918$.
Julia Set Generator
Julia sets presented on this page were generated using the Javascript Julia Set Generator from Mark McClure's "marksmath" site.