Filled Julia set

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The filled-in Julia set \ K(f_c) of a polynomial \ f _c is defined as the set of all points z\, of dynamical plane that have bounded orbit with respect to \ f _c

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

\mathbb{C} is set of complex numbers

z\, is complex variable of function \ f _c (z)
c\, is complex parameter of function \ f _c (z)

f_c:\mathbb C\to\mathbb C

\ f_c may be various functions. In typical case \ f_c is complex quadratic polynomial.

\ f^{(k)} _c (z) is the \ k -fold compositions of f _c\, with itself = iteration of function f _c\,

Contents

[edit] Relation to the Fatou set

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

Attractive basin of infinity is one of components of the Fatou set.
A_{f_c}(\infty) = F_\infty

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

[edit] Relation between Julia, filled-in Julia set and attractive basin of infinity

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Julia set is common boundary of filled-in Julia set and attractive basin of infinity

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

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

A_{f_c}(\infty) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C}  : f^{(k)} _c (z) \to \infty\ as\ k \to \infty \}.

If filled-in Julia set has no interior then Julia set coincides with filled-in Julia set. It happens when c\, is Misiurewicz point.

[edit] Spine

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 properities:

  • spine lays 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 constructiong 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 shorest 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 2 components.

[edit] Images

Filled Julia set for fc, c=φ−2=-0.4 where φ means Golden_ratio

Filled Julia with no interior = Julia set. It is for c=i.

Filled Julia set for c=-1+0.1*i. Here Julia set is the boundary of filled-in Julia set.

Douady rabbit

[edit] Notes

  1. ^ Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester
  2. ^ John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)
  3. ^ Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case
  4. ^ A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Ga, USA, 1986.
  5. ^ K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257

[edit] References

  1. Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0387158518.
  2. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathemathics Technical University of Denmark , MAT-Report no. 1996-42.
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