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
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
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 are pre-periodic. Such critical points are often called Misiurewicz points.
The most studied polynomials are probably those of the form , which are often denoted by , where is any complex number. In this case, the spine of the filled Julia set is defined as arc between -fixed point and ,
Algorithms for constructing the spine: