Filled Julia set

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The filled-in Julia set K ( f ) {\displaystyle K(f)} {\displaystyle K(f)} of a polynomial f {\displaystyle f} {\displaystyle f} is the union of a Julia set and its interior, non-escaping set.

Formal definition

The filled-in Julia set K ( f ) {\displaystyle K(f)} {\displaystyle K(f)} of a polynomial f {\displaystyle f} {\displaystyle f} is defined as the set of all points z {\displaystyle z} {\displaystyle z} of the dynamical plane that have bounded orbit with respect to f {\displaystyle 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\}} {\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}} where:

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 )} {\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 }} {\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}.} {\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 )} {\displaystyle J(f)=\partial K(f)=\partial A_{f}(\infty )} where: A f ( ∞ ) {\displaystyle A_{f}(\infty )} {\displaystyle A_{f}(\infty )} denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for f {\displaystyle 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 \}.} {\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} {\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} {\displaystyle f(z)=z^{2}+c}, which are often denoted by f c {\displaystyle f_{c}} {\displaystyle f_{c}}, where c {\displaystyle c} {\displaystyle c} is any complex number. In this case, the spine S c {\displaystyle S_{c}} {\displaystyle S_{c}} of the filled Julia set K {\displaystyle K} {\displaystyle K} is defined as arc between β {\displaystyle \beta } {\displaystyle \beta }-fixed point and − β {\displaystyle -\beta } {\displaystyle -\beta }, S c = [ − β , β ] {\displaystyle S_{c}=\left[-\beta ,\beta \right]} {\displaystyle S_{c}=\left[-\beta ,\beta \right]} with such properties:

  • spine lies inside K {\displaystyle K} {\displaystyle K}.[1] This makes sense when K {\displaystyle K} {\displaystyle K} is connected and full[2]
  • spine is invariant under 180 degree rotation,
  • spine is a finite topological tree,
  • Critical point z c r = 0 {\displaystyle z_{cr}=0} {\displaystyle z_{cr}=0} always belongs to the spine.[3]
  • β {\displaystyle \beta } {\displaystyle \beta }-fixed point is a landing point of external ray of angle zero R 0 K {\displaystyle {\mathcal {R}}_{0}^{K}} {\displaystyle {\mathcal {R}}_{0}^{K}},
  • − β {\displaystyle -\beta } {\displaystyle -\beta } is landing point of external ray R 1 / 2 K {\displaystyle {\mathcal {R}}_{1/2}^{K}} {\displaystyle {\mathcal {R}}_{1/2}^{K}}.

Algorithms for constructing the spine:

  • detailed version is described by A. Douady[4]
  • Simplified version of algorithm:
    • connect − β {\displaystyle -\beta } {\displaystyle -\beta } and β {\displaystyle \beta } {\displaystyle \beta } within K {\displaystyle K} {\displaystyle K} by an arc,
    • when K {\displaystyle K} {\displaystyle K} has empty interior then arc is unique,
    • otherwise take the shortest way that contains 0 {\displaystyle 0} {\displaystyle 0}.[5]

Curve R {\displaystyle 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}} {\displaystyle R{\overset {\mathrm {def} }{{}={}}}R_{1/2}\cup S_{c}\cup R_{0}} divides dynamical plane into two components.

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References

  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.