Fractals/Iterations in the complex plane/Parameter plane


real sliceEdit

" each period-doubling cascade superstable orbit ... generate the corresponding chaotic band   and the Misiurewicz point   that separates the chaotic bands   and   "[1]



  • B is a chaotic region from -2 to MF
  •   = Feigenbaum point = MF = the Myrberg-Feigenbaum
  • A is a periodic region : from MF to 1/4

chaotic regionEdit



  •   is chaotic band  

See also

periodic regionEdit

period doubling cascade

  • 2^n = the powers of 2 in decreasing order, ending with 2^0 = 1."[2]


Mini Mandelbrot setsEdit


  • midget
  • mini mandelbrot set
  • baby Mandelbrots
  • island = island mu-molecule = island mu-unit [3]
  • primitive component

Parameter rays of mini Mandelbrot sets [4]

" a saddle-node (parabolic) periodic point in a complex dynamical system often admits homoclinic points, and in the case that these homoclinic points are nondegenerate, this is accompanied by the existence of infinitely many baby Mandelbrot sets converging to the saddle-node parameter value in the corresponding parameter plane." Devaney [5]

" Warped midgets in the Mandelbrot set have been measured, using an algorithm that allows the positions of the head, and cardioid atoms (north and south) of any midget to be found, once one has placed the cursor on the computer terminal somewhere inside any midget. We describe two distortions of midgets: linear distortions and angular distortions. When the north and south angles are plotted in the north/south angle plane, families of points are formed. The angle and distance measures of warped midgets from the Sea Horse Valley of the Mandelbrot set and from other sea horse valleys of midgets, whether on the Spike or on tendrils above atoms, all fall closely together in one part of the north/south plane. Measures of warped midgets from tendrils above the major atoms on the surface of the Cardioid fall closely together in another part of the north/south plane. This different way of looking at the Mandelbrot set offers an interesting way of studying the distortions of midgets."  A. G. Davis Philip, Michael Frame, Adam Robucci: Warped midgets in the Mandelbrot set. Computers & Graphics 18(2): 239-248 (1994)


Julia island = Embedded Julia Set = virtual Julia SetsEdit

  • "A structure comprised of filaments, resembling a Julia set in appearance, which has a higher delta Hausdorff dimension than filaments in the immediately surrounding region. They are sometimes also called Julia islands or virtual Julia Sets." Robert Munafo[6]
  • "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot." [7]

order of hyperbolic componenetsEdit

Parts of parameter planeEdit

Parts of Mandelbrot set according to M Romera et al.:[10]

    • main cardioid
    • q/p family (= q/p limb)
      • period doubling cascade of hyperbolic components which ends at the Myrberg-Feigenbaum point
      • shrub

Not that her q/p not p/q notation is used


  • wake / limb
  • migdet / island / mu-molecule[11]
  • hyperbolic component of Mandelbrot set



  • mu-ency [14]
  • Mandlebrot

How to choose a point from parameter plane ?Edit

How to move on parameter plane ?Edit


The dynamics of the polynomials of moving along curves is : [20]

  • for external ray: ”stretch” the dynamic on the basin of infinity. The argument of φPa,b (2a) stay unchanged ( fixed)
  • for equipotential: twist the dynamics in the annulus between the Green level curves of the escaping critical point and of the critical value. The modulus of φPa,b (2a is fixed


Examples :

 // glsl code by iq from
 float ltime = 0.5-0.5*cos(time*0.12);
 vec2 c = vec2( -0.745, 0.186 ) - 0.045*zoom*(1.0-ltime);
 // glsl code by xylifyx  from
 vec2 c = vec2( 0.37+cos(iTime*1.23462673423)*0.04, sin(iTime*1.43472384234)*0.10+0.50);
 // by Marco Gilardi
 vec2 c = vec2(-0.754, 0.05*(abs(cos(0.1*iTime))+0.8));

Plane typesEdit

The phase space of a quadratic map is called its parameter plane. Here:

  •   is constant
  •   is variable

There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.

The parameter plane consists of :

  • The Mandelbrot set
    • The bifurcation locus = boundary of Mandelbrot set
    • Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set [21]

There are many different types of the parameter plane[22][23]

  • plain ( c-plane ):  
  • inverted c-plane = 1/c plane:  
  • lambda plane   where  
  • exponential plane ( map) [24][25][26]
  • unrolled plain (flatten' the cardiod = unroll ) [27][28] = "A region along the cardioid is continuously blown up and stretched out, so that the respective segment of the cardioid becomes a line segment. .." ( Figure 4.22 on pages 204-205 of The Science Of Fractal Images)[29]
  • transformations [30]
  • log : "To illustrate the complexity of the boundary of the Mandelbrot set, Figure 8 renders the image of dM under the transformation log(z - c) for a certain c e dM ? Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to

zooming in towards the point c. (Namely, c = -0.39054087... - 0.58678790i... the point on the boundary of the main cardioid corresponding to the golden mean Siegel disk.). Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to zooming in towards the point c. "[31]

    • "Legendary side scrolling fractal zoom. 1 Month + (Interpolator+Video Editor) = Log(z). This means logarithmic projection for this location, that gives this interesting side-scrolling plane ^^)"[32]
    • " There are no program that can render this fractal on log(Z) plane. But you can make it in Ultra Fractal or in similar software with programmable distributive. Formula is:C = exp(D), for D - is your zoomable coordinates" SeryZone X

Description by NIKOLA UBAVIĆ:

  • Inverted c-plane : "Geometrically, the ... relationship between the parameters represents the composition of the inversion with respect to the unit circle centered at zero, and the conjugation (axial symmetry with respect to the real line). Due to this connection, the cardioid from the boundary of the Mandelbrot set in the "standard" parameterization corresponds to the tear-shaped curve in the alpha parameterization."[33]
  • "if translation is performed for before the inversion 1/4, then the cardioid is imaged in a parabola"  
  • "By inverting a complex plane around a unit circle with center at zero, one of these circles remains invariant, while the other image is inside it."  
  • "If ... translation is performed before the inversion1, then the two circles are mapped into two parallel lines. In this way the second of the next two figures was obtained."  


point c descriptionEdit

  • c value
    • Cartesion description
      • real part
      • imaginary part
    • polar description:
      • (external or internal ) angle
      • ( external or internal) radius

Point TypesEdit

point =pixel of parameter plane = c parameter


Criteria for classification of parameter plane points :

  • arithmetic properties of internal angle (rotational number) or external angle
    • in case of exterior point:
      • type of angle : rational, irrational, ....
      • preperiod and period of angle under doubling map
    • in case of boundary point :
      • preperiod and period of external angle under doubling map
      • preperiod and period of internal angle under doubling map
  • set properties ( relation with the Mandelbrot set and wakes)
    • interior
    • boundary
    • exterior
      • inside wake, subwake
      • outside all the wakes, belonging to a parameter ray landing at a Siegel or Cremer parameter,
  • geometric properities
    • number of external rays that land on the boundary point : tips ( 1 ray), biaccesible, triaccesible, ....
    • position of critical point with relation to the Julia set
  • Renormalization


There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.

Simple classificationEdit

  • exterior of Mandelbrot set
  • Mandelbrot set
    • boundary of Mandelbrot set
    • interior of Mandelbrot set
      • centers,
      • other internal points ( points of internal rays )

partial classification of boundary pointsEdit

Classification :[34]

  • Boundaries of primitive and satellite hyperbolic components:
    • Parabolic (including 1/4 and primitive roots which are landing points for 2 parameter rays with rational external angles = biaccesible ).
    • Siegel ( a unique parameter ray landing with irrational external angle)
    • Cremer ( a unique parameter ray landing with irrational external angle)
  • Boundary of M without boundaries of hyperbolic components:
    • non-renormalizable (Misiurewicz with rational external angle and other).
    • renormalizable
      • finitely renormalizable (Misiurewicz and other).
      • infinitely renormalizble (Feigenbaum and other). Angle in turns of external rays landing on the Feigenbaum point are irrational numbers
  • non hyperbolic components ( we believe they do not exist but we cannot prove it ) Boundaries of non-hyperbolic components would be infinitely renormalizable as well.

Here "other" has not a complete description. The polynomial may have a locally connected Julia set or not, the critical point may be rcurrent or not, the number of branches at branch points may be bounded or not ...




Structure of the Mandelbrot set

See alsoEdit


  1. [Pastor97a] : Harmonic structure of one-dimensional quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini
  2. The On-Line Encyclopedia of Integer Sequences : A005408 = The odd numbers: a(n) = 2n+1
  3. the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018
  4. Parameter rays of mini mandelbrot sets
  5. Devaney In Global Analysis of Dynamical Systems, ed.: H. Broer, B. Krauskopf, G. Vegter. IOP Publishing (2001), 329-338 or Homoclinic Points in Complex Dynamical Systems
  6. embedded julia set from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017.
  9. M. Romera et al, Int. J. Bifurcation Chaos 13, 2279 (2003). Shrubs in the Mandelbrot Set Ordering
  10. SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya
  11. mumolecule From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020.
  12. : how-distorted-can-a-minibrot-be
  13. distribution From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020
  14. Mu-Ency - The Encyclopedia of the Mandelbrot Set by R Munafo
  15. fractalforums : parameter-adjustment-art-or-luck ?
  16. interesting c points by Owen Maresh
  17. Visual Guide To Patterns In The Mandelbrot Set by Miqel
  18. fractalforums : deep-zooming-to-interesting-areas
  19. math.stackexchange question: parameter-plane-dynamics-of-fixed-points-and-their-preimages-for-standard-quadra
  20. Reading escaping trees from Hubbard trees in Sn by Matthieu Arfeux
  21. Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
  22. Alternate Parameter Planes by David E. Joyce
  23. exponentialmap by Robert Munafo
  24. mu-ency : exponential map by R Munafo
  25. Exponential mapping and OpenMP by Claude Heiland-Allen
  26. exponential_mapping_with_kalles_fraktaler by Claude Heiland-Allen
  27. Linas Vepstas : Self Similar?
  28. the flattened cardioid of a Mandelbrot by Tom Rathborne
  29. Stretching cusps by Claude Heiland-Allen
  30. Twisted Mandelbrot Sets by Eric C. Hill
  32. youtube video : Mandelbrot deep zoom to 2^142 or 5.5*10^42. Log(z) by SeryZone X
  33. Julie's and Mandelbrot's set by NIKOLA UBAVIĆ
  34. stackexchange : classification-of-points-in-the-mandelbrot-set
  35. fractalforums : tricky-mandelbrot-problem