Fractals/Iterations in the complex plane/julia

Various types of dynamics needs various algorithms

Types edit

Classification of Julia sets according to :

  • topology
    • of filled Julia set's interior
    • shape[1]
  • local dynamics near periodic points
  • connectedness
  • position of parameter c on the parameter plane [2]

Filled Julia set can have :

  • an non-empty interior ( Julia set is connected )
    • parabolic: filled Julia set have parabolic cycle ( c is on boundary of hyperbolic component )
    • Siegel : filled Julia sets containing Siegel disc. Julia set can be locally connected or not. That depends on the rotation number. ( c is on boundary of hyperbolic component )
    • attracting : filled Julia set have attracting cycle ( c is inside hyperbolic component )
    • superattracting : filled Julia set have superattracting cycle( c is in the center of hyperbolic component ). Examples : Airplane Julia set, Douady's Rabbit, Basillica.
  • empty interior
    • disconnected ( c is outside of Mandelbrot set )[3]
    • connected
      • Cremer Julia sets ( c is on boundary of hyperbolic component , Julia set is connected but not locally connected)
      • dendrits ( Julia set is connected and locally connected )
        • Misiurewicz Julia sets (c is a Misiurewicz point )
        • Feigenbaum Julia sets ( )
        • others which have no [4]

topology edit

Julia set of root point is topologically the same as the Julia set of the child period center, but

  • the center ( nucleus) Julia set is very easy to draw ( superattracting basin = very fast dynamics because critical point is also periodic point)
  • while the root Julia set ( parabolic) is hard to draw ( parabolic basin and lasy dynamics)


  • t = 1/2
    • Julia set of the root point = fat Basilica Julia set: c = -3/4 = - 0.75
    • Julia set of period 2 center = (slim) Basicica Julia set: c = -1
  • t = 1/3
    • Julia set of the root point = fat Douady Rabbit: c = -0.125000000000000 +0.649519052838329i
    • Julia set of period 3 center = (slim) Doudy Rabbit Julia set: c = -0.122561166876654 +0.744861766619744i period = 3

Models of Julia set edit

Lamination of rabbit Julia set

components of Julia set's interior edit

Douady’s pinched disk model edit

Dynamic rays and their landing properties are a key tool to understanding (the topology of) Julia sets of polynomials. In particular, the structure of the Julia set is determined by rays that land at a common point: at least in good cases (under the assumption of local connectivity), the knowledge of which rays land together gives a homeomorphic model for the Julia set that is known as Douady’s pinched disk model [11]

Algorithms edit

"... a single algorithm for computing all quadratic Julia sets does not exist."[12]


Types edit

equicontinouty edit

"The Julia set of f then is the set of all points of G, at which this sequence of iterated functions is not equicontinous. The Fatou set is its complement. Laxly said the action of the iterated functions on near points is examined. Places, where points, which are near enough, remain near during iterations, belong to the Fatou set. Places, where points, as near they may be, are teared apart, belong to the Julia set. In the following I only consider functions, which map the Riemann sphere, i.e. the complex plane with an ideal point "infinity" added, to itself. The Julia sets are white, the Fatou sets black." Michael Becker

accuracy edit

points edit

"We know the periodic points are dense in the Julia set, but in the case of weird ones (like the ones with Cremer points, or even some with Siegel disks where the disk itself is very 'deep' within the Julia set, as measured by the external rays), the periodic points tend to avoid certain parts of the Julia set as long as possible. This is what causes the 'inverse method' of rendering images of Julia sets to be so bad for those cases." ( answered Oct 26 '14 at 14:52 by Jacques Carette )[17]

Distribution of points of inverse orbit of repelling fixed point of complex quadratic polynomial.
Periodic points of f(z) = z*z-0.75 for period =6 as intersections of 2 implicit curves

compare with:

Special cases edit

  • "The Julia set is nonempty when d ≥ 2, but the Fatou set may be empty (as demonstrated by Latt`es examples)."[18]

Names edit

  • cauliflower = Julia set for c = 1/4 (c is the cusp of main cardioid), parabolic Julia set
    • an imploded cauliflower is a Julia set for   with  [19], disconnected Julia set
  • airplane Julia set. C is the center of period 3 component on the real axis: c = -1.75487766624669276
  • helicopter z → z^3 − 0.2634 − 1.2594i
  • (Douady) rabbit. C is a the center period 3 hyperbolic component of Mandelbrot set for complex quadratic map
    • Cubic Rabbit z → z^3 + 0.545 + 0.539i
  • dendrite. C is a tip
  • the Kokopelli Julia set   [20] The angle 3/15 = p0011 = 0.(0011) has preperiod = 0 and period = 4. The conjugate angle on the parameter plane is 4/15 or p0100. The kneading sequence is AAB* and the internal address is 1-3-4. The corresponding parameter rays are landing at the root of a primitive component of period 4.
  • Basilica = = Julia set for c = -1 (c is the center of period 2 component)

See also edit

References edit

  1. blog scientific american : fractal-kitties-illustrate-the-endless-possibilities-for-julia-sets By Evelyn Lamb on September 26, 2012
  2. math.stackexchange question : classification-of-points-in-the-mandelbrot-set
  3. images of disconnected Julia sets
  4. Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle by Mikhail Lyubich
  5. Simple topological models of Julia sets by L. Oversteegen
  6. Combinatorial Julia Sets (1) By Jim Belk
  7. Jacek Skryzalin: On Quadratic Mappings With and Attracting Cycle
  8. Eugene1806's Blog
  9. Adrien Douady, Descriptions of compact sets in C. , in: Topological Methods in Modern Mathematics, Publish or Perish (1993), 429– 465.
  10. Hausdorff dimension of biaccessible angles of quadratic Julia sets and of the Mandelbrot set by Henk Bruin and Dierk Schleicher
  12. Computability of Julia sets by Mark Braverman, Michael Yampolsky
  13. Some Julia sets by Michael Becker, 6/2003. Last modification: 2/2004.
  14. orbit traps example video
  15. orbit traps making
  16. Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees by Robin Stoll, Dierk Schleicher
  17. Clustering of periodic points for a polynomial iteration of C
  18. Lucjan Emil Böttcher and his mathematical legacy Stanisław Domoradzki, Małgorzata Stawiska
  19. [6] Douady A., Buff X., Devaney RL. and Sentenac P. Baby Mandelbrot sets are born in cauliflowers. In: Lei T. editor. The Mandelbrot set, theme and variations. Cambridge: University Press, 2000, p. 19-36.
  20. The Thurston Algorithm for quadratic matings by Wolf Jung