Filled Julia set

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The filled-in Julia set of a polynomial is :

Formal definition[edit]

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


where :

is the set of complex numbers

is the -fold composition of with itself = iteration of function

Relation to the Fatou set[edit]

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[edit]

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

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.

Spine[edit]

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 ,

with such properties:

  • spine lies inside .[1] This makes sense when is connected and full [2]
  • spine is invariant under 180 degree rotation,
  • spine is a finite topological tree,
  • Critical point always belongs to the spine.[3]
  • -fixed point is a landing point of external ray of angle zero ,
  • is landing point of external ray .

Algorithms for constructing the spine:

  • detailed version is described by A. Douady[4]
  • Simplified version of algorithm:
    • connect and within by an arc,
    • when has empty interior then arc is unique,
    • otherwise take the shortest way that contains .[5]

Curve  :

divides dynamical plane into two components.

Images[edit]

Notes[edit]

References[edit]

  1. Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0-387-15851-8.
  2. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, MAT-Report no. 1996-42.