The filled-in Julia set K(f) of a polynomial f is a Julia set and its interior, non-escaping set.

Formal definition

The filled-in Julia set K(f) of a polynomial f is defined as the set of all points z of the dynamical plane that have bounded orbit with respect to f where:

Relation to the Fatou set

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:

Relation between Julia, filled-in Julia set and attractive basin of infinity

The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity where: denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for f 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 f are pre-periodic. Such critical points are often called Misiurewicz points.

Spine

The most studied polynomials are probably those of the form, which are often denoted by f_c, where c is any complex number. In this case, the spine S_c of the filled Julia set K is defined as arc between \beta-fixed point and -\beta, with such properties: Algorithms for constructing the spine: Curve R: divides dynamical plane into two components.

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