Filled Julia 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

  K ( f )   = d e f   { z ∈ C : f ( k ) ( z ) ↛ ∞   a s   k → ∞ }
where :

C is the set of complex numbers

  f ( k ) ( z ) is the   k -fold composition of f with itself = iteration of function f

Relation to the Fatou set

The filled-in Julia set is the (absolute) complement of the attractive basin of infinity.
K ( f ) = C ∖ A f ( ∞ )

The attractive basin of infinity is one of the components of the Fatou set.
A f ( ∞ ) = F ∞

In other words, the filled-in Julia set is the complement of the unbounded Fatou component:
K ( f ) = F ∞ C .

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

J ( f ) = ∂ K ( f ) = ∂ A f ( ∞ )

where :
A f ( ∞ ) denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for f

A f ( ∞ )   = d e f   { z ∈ C : f ( k ) ( z ) → ∞   a s   k → ∞ } .

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 f ( z ) = z 2 + c , 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 β -fixed point and − β ,

S c = [ − β , β ]

with such properties:

Algorithms for constructing the spine:

Curve R  :

R   = d e f   R 1 / 2   ∪   S c   ∪   R 0

divides dynamical plane into two components.