• Source: Filled Julia set

      • The filled-in Julia set



      K
      (
      f
      )


      {\displaystyle K(f)}

      of a polynomial



      f


      {\displaystyle f}

      is a Julia set and its interior, non-escaping set.


      Formal definition


      The filled-in Julia set



      K
      (
      f
      )


      {\displaystyle K(f)}

      of a polynomial



      f


      {\displaystyle f}

      is defined as the set of all points



      z


      {\displaystyle z}

      of the dynamical plane that have bounded orbit with respect to



      f


      {\displaystyle f}





      K
      (
      f
      )






      =





      d
      e
      f




      {

      z
      ∈

      C

      :

      f

      (
      k
      )


      (
      z
      )
      ↛
      ∞


      as


      k
      →
      ∞

      }



      {\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}}


      where:





      C



      {\displaystyle \mathbb {C} }

      is the set of complex numbers





      f

      (
      k
      )


      (
      z
      )


      {\displaystyle f^{(k)}(z)}

      is the



      k


      {\displaystyle k}

      -fold composition of



      f


      {\displaystyle f}

      with itself = iteration of function



      f


      {\displaystyle 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


      (
      ∞
      )


      {\displaystyle K(f)=\mathbb {C} \setminus A_{f}(\infty )}


      The attractive basin of infinity is one of the components of the Fatou set.





      A

      f


      (
      ∞
      )
      =

      F

      ∞




      {\displaystyle A_{f}(\infty )=F_{\infty }}


      In other words, the filled-in Julia set is the complement of the unbounded Fatou component:




      K
      (
      f
      )
      =

      F

      ∞


      C


      .


      {\displaystyle K(f)=F_{\infty }^{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


      (
      ∞
      )


      {\displaystyle J(f)=\partial K(f)=\partial A_{f}(\infty )}


      where:




      A

      f


      (
      ∞
      )


      {\displaystyle A_{f}(\infty )}

      denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for



      f


      {\displaystyle f}






      A

      f


      (
      ∞
      )



      =



      d
      e
      f





      {
      z
      ∈

      C

      :

      f

      (
      k
      )


      (
      z
      )
      →
      ∞

      a
      s

      k
      →
      ∞
      }
      .


      {\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



      f


      {\displaystyle 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


      {\displaystyle f(z)=z^{2}+c}

      , which are often denoted by




      f

      c




      {\displaystyle f_{c}}

      , where



      c


      {\displaystyle c}

      is any complex number. In this case, the spine




      S

      c




      {\displaystyle S_{c}}

      of the filled Julia set



      K


      {\displaystyle K}

      is defined as arc between



      β


      {\displaystyle \beta }

      -fixed point and



      −
      β


      {\displaystyle -\beta }

      ,





      S

      c


      =

      [

      −
      β
      ,
      β

      ]



      {\displaystyle S_{c}=\left[-\beta ,\beta \right]}


      with such properties:

      spine lies inside



      K


      {\displaystyle K}

      . This makes sense when



      K


      {\displaystyle K}

      is connected and full
      spine is invariant under 180 degree rotation,
      spine is a finite topological tree,
      Critical point




      z

      c
      r


      =
      0


      {\displaystyle z_{cr}=0}

      always belongs to the spine.




      β


      {\displaystyle \beta }

      -fixed point is a landing point of external ray of angle zero






      R



      0


      K




      {\displaystyle {\mathcal {R}}_{0}^{K}}

      ,




      −
      β


      {\displaystyle -\beta }

      is landing point of external ray






      R



      1

      /

      2


      K




      {\displaystyle {\mathcal {R}}_{1/2}^{K}}

      .
      Algorithms for constructing the spine:

      detailed version is described by A. Douady
      Simplified version of algorithm:
      connect



      −
      β


      {\displaystyle -\beta }

      and



      β


      {\displaystyle \beta }

      within



      K


      {\displaystyle K}

      by an arc,
      when



      K


      {\displaystyle K}

      has empty interior then arc is unique,
      otherwise take the shortest way that contains



      0


      {\displaystyle 0}

      .
      Curve



      R


      {\displaystyle R}

      :




      R






      =





      d
      e
      f




      R

      1

      /

      2


      ∪

      S

      c


      ∪

      R

      0




      {\displaystyle R{\overset {\mathrm {def} }{{}={}}}R_{1/2}\cup S_{c}\cup R_{0}}


      divides dynamical plane into two components.


      Images

































      Names


      airplane
      Douady rabbit
      dragon
      basilica or San Marco fractal or San Marco dragon
      cauliflower
      dendrite
      Siegel disc


      Notes




      References


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

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