The filled-in Julia set
of a polynomial
is a Julia set and its interior, non-escaping set.
The filled-in Julia set
of a polynomial
is defined as the set of all points
of the dynamical plane that have bounded orbit with respect to
{\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}}
where: The filled-in Julia set is the (absolute) complement of the attractive basin of infinity.
The attractive basin of infinity is one of the components of the Fatou set.
In other words, the filled-in Julia set is the complement of the unbounded Fatou component:
The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity
denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for
{\displaystyle A_{f}(\infty )\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to \infty \ as\ k\to \infty \}.}
If the filled-in Julia set has no interior then the Julia set coincides with the filled-in Julia set.
This happens when all the critical points of
Such critical points are often called Misiurewicz points.
The most studied polynomials are probably those of the form
is any complex number.
In this case, the spine
of the filled Julia set
is defined as arc between
-fixed point and
with such properties: Algorithms for constructing the spine: Curve
divides dynamical plane into two components.