Filled Julia set

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The filled-in Julia set \ K(f) of a polynomial \ f  is :

Formal definition[edit]

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) \  \overset{\underset{\mathrm{def}}{}}{=} \  \{ z \in  \mathbb{C}  : f^{(k)} (z)  \not\to  \infty\  as\  k \to \infty \}
where :

\mathbb{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[edit]

The filled-in Julia set is the (absolute) complement of the attractive basin of infinity.
K(f) = \mathbb{C} \setminus A_{f}(\infty)

The attractive basin of infinity is one of the components of the Fatou set.
A_{f}(\infty) = F_\infty

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

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

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

where :
A_{f}(\infty) denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for f

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 are pre-periodic. Such critical points are often called Misiurewicz points.

Spine[edit]

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 \beta\,-fixed point and -\beta\,,

S_c = \left [ - \beta , \beta  \right ]\,

with such properties:

  • spine lies inside \ K \,.[1] This makes sense when K\, is connected and full [2]
  • spine is invariant under 180 degree rotation,
  • spine is a finite topological tree,
  • Critical point  z_{cr} = 0  \, always belongs to the spine.[3]
  • \beta\,-fixed point is a landing point of external ray of angle zero \mathcal{R}^K  _0,
  • -\beta\, is landing point of external ray \mathcal{R}^K  _{1/2}.

Algorithms for constructing the spine:

  • Simplified version of algorithm:
    • connect - \beta\, and  \beta\, within K\, by an arc,
    • when K\, has empty interior then arc is unique,
    • otherwise take the shortest way that contains 0.[5]

Curve R\, :

R\   \overset{\underset{\mathrm{def}}{}}{=} \  R_{1/2}\ \cup\  S_c\  \cup \ R_0 \,

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.