Английская Википедия:Filled Julia set

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The filled-in Julia set <math>K(f) </math> of a polynomial <math>f </math> is a Julia set and its interior, non-escaping set.

Formal definition

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

Relation to the Fatou set

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

The attractive basin of infinity is one of the components of the Fatou set. <math display="block">A_{f}(\infty) = F_\infty </math>

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

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

Шаблон:Wikibooks

The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity <math display="block">J(f) = \partial K(f) = \partial A_{f}(\infty)</math> where: <math>A_{f}(\infty)</math> denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for <math>f</math>

<math display="block">A_{f}(\infty) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C} : f^{(k)} (z) \to \infty\ as\ k \to \infty \}. </math>

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 <math>f</math> are pre-periodic. Such critical points are often called Misiurewicz points.

Spine

The most studied polynomials are probably those of the form <math>f(z) = z^2 + c</math>, which are often denoted by <math>f_c</math>, where <math>c</math> is any complex number. In this case, the spine <math>S_c</math> of the filled Julia set <math>K </math> is defined as arc between <math>\beta</math>-fixed point and <math>-\beta</math>, <math display="block">S_c = \left [ - \beta , \beta \right ]</math> with such properties:

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

Algorithms for constructing the spine:

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

Curve <math>R</math>: <math display="block">R \overset{\mathrm{def}}{{}={}} R_{1/2} \cup S_c \cup R_0 </math> divides dynamical plane into two components.

Images

Names

Notes

Шаблон:Reflist

References

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

Шаблон:Fractals

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester Шаблон:Webarchive
  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, Georgia, USA, 1986.
  5. ↑ K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257
  6. ↑ The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher
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