Fractals/Iterations in the complex plane/misiurewicz

Misiurewicz point is the parameter c ( point oc parameter plane) where the critical orbit is pre-periodic.



Misiurewicz polynomial ( map) can be marked by:[2]

  • the parameter coordinate c ∈ M
  • the external angle   of the ray that lands:
    • at z = c in J(f) on the dynamic plane
    • at c in M on the parameter plane





  • the Kokopelli Julia set   [3] 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.


Preperiod is used in 2 meanings :

  • K =preperiod of critical point
  • k = preperiod of critical value

Note that :

 k = K -1

Period p is the same for critical value and citical point

Wolf Jung uses preperiod of critical value : "... the usual convention is to use the preperiod of the critical value. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane."

Pastor uses preperiod of critical point : "all the Misiurewicz points are given with one unit more in their preperiods, therefore this M2,1 is given as M3,1 "[4]



Misiurewicz points c

  • with period 1 are of the type:[5]
    • alpha, i.e.  
    • beta, i.e  
  • with period > 1

where alfa and beta are fixed points of complex quadratic polynomial


all Misiurewicz points are centers of the spirals, which are turning:[6]

  • fast
  • slow
  • no turn : if the Misiurewicz point is a real number, it does not turn at all

Spirals can also be classified by the number of arms.

Visual types:[7]

  • branch tips = terminal points of the branches[8] or tips of the midgets[9]
  • branch point = points where branches meet[10]
    • centers of slow spirals with more then 1 arm
    • centers of spirals = fast spiral
  • band-merging points of chaotic bands (the separator of the chaotic bands   and   )[11] = 2 arm spiral

number of external raysEdit

  • endpoint = tip = 1 angle
  • primitive type = 2 angles of primitive cycle
  • satellite type = 2 or more angles from satellite cycle

where primitive and satelite are the types of hyperbolic components

preperiod and periodEdit

In general a preperiodic critical value has a preperiod k, a period p, a ray period rp, and v angles. There are three cases,

  • tip: r = 1 and v = 1
  • primitive: r = 1 and v = 2
  • satellite: r > 1 and v = r

named typesEdit


The principal Misiurewicz point   of the limb  :[12]

  • hase m external angles, that are preimages (under doubling) of the external angles of  


Characteristic Misiurewicz point of the chaotic band of the Mandelbrot set is :[13]

  • the most prominent and visible Misiurewicz point of a chaotic band
  • have the same period as the band
  • have the same period as the gene of the band

primary separatorEdit
  • band-merging point = point merging 2 chaotic bands   and  

Examples: [14]

  •   = tip of main antenna , external angle = 1/2[15]
  •  , external angles 5 and 7/12
  •  , external angles 33 and 47/80
  •  , external angles 1795 and 2557/4352
  • ...
  •   = Feigenbaum point = MF = the Myrberg-Feigenbaum

In the text form:

double m[12] = {


const complex double cf = -1.401155189093314712; //the Feigenbaum point -1.401155 = m[infinity]
secondary separatorEdit
  • tree separators ( tree is a subset of band )


  • have not the same period as the band


Misiurewicz Points, part of the Mandelbrot set:

  • Centre 0.4244 + 0.200759i; Max. Iterations 100; View radius 0.00479616 [16]





"... we do not know how to compute (...) Misiurewicz parameters (with high (pre)periods) for the family of quadratic rational maps. One might need to and a non-rigorous method to and Misiurewicz parameter in a reasonable time like Biham-Wenzel's method." HIROYUKI INOU [17]

Computing Misiurewicz points of complex quadratic mappingEdit

roots of polynomialEdit

Misiurewicz points [18] are special boundary points.

Define polynomial in Maxima CAS :

P(n):=if n=0 then 0 else P(n-1)^2+c;

Define a Maxima CAS function whose roots are Misiurewicz points, and find them.


Examples of use :

(%i6) M(2,1);
(%o6) [c=-2.0,c=0.0]
(%i7) M(2,2);
(%o7) [c=-1.0*%i,c=%i,c=-2.0,c=-1.0,c=0.0]

factorizing the polynomialsEdit

" factorizing the polynomials that determine Misiurewicz points. I believe that you should start with

  ( f^(p+k-1) (c) + f^(k-1) (c) ) / c

This should already have exact preperiod k , but the period is any divisor of p . So it should be factorized further for the periods.

Example: For preperiod k = 1 and period p = 2 we have

  c^3 + 2c^2 + c + 2 .

This is factorized as

(c + 2)*(c^2 + 1)  

for periods 1 and 2 . I guess that these factors appear exactly once and that there are no other factors, but I do not know."Wolf Jung

Misiurewicz domainsEdit

Newton methodEdit

Finding external angles of rays that land on the Misiurewicz pointEdit

Relation between preperiod/period of external angles and Misurewicz point:



  1. mathoverflow question: is-there-some-known-way-to-create-the-mandelbrot-set-the-boundary-with-an-ite
  2. Finite Subdivision Rules from Matings of Quadratic Functions: Existence and Constructions by Mary E. Wilkerson
  3. The Thurston Algorithm for quadratic matings by Wolf Jung
  4. G. Pastor, M. Romera, G. Alvarez, J. Nunez, D. Arroyo, F. Montoya, "Operating with External Arguments of Douady and Hubbard", Discrete Dynamics in Nature and Society, vol. 2007, Article ID 045920, 17 pages, 2007.
  5. W Jung : Homeomorphisms on Edges of the Mandelbrot Set Ph.D. thesis of 2002
  6. Book : Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set, page 461, by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
  7. Fractal Geometry from Yale University by Michael Frame, Benoit Mandelbrot (1924-2010), and Nial NegerFebruary 2, 2013
  8. Terminal Point by  Robert P. Munafo, 2008 Mar 9.
  9. mathoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
  10. Branch Point by Robert P. Munafo, 1997 Nov 19.
  11. Symbolic sequences of one-dimensional quadratic map points by G Pastor, Miguel Romera, Fausto Montoya Vitini
  12. Families of Homeomorphic Subsets of the Mandelbrot Set by Wolf Jung page 7
  13. G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya, "On periodic and chaotic regions in the Mandelbrot set", Chaos, Solitons & Fractals, 32 (2007) 15-25
  14. A scaling constant equal to unity in 1D- quadratic maps by M. ROMERA, G. PASTOR and F. MONTOYA
  15. G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71
  16. example
  18. MIsiurewicz point in wikipedia