Fractals/Iterations in the complex plane/misiurewicz

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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.



Misiurewicz points c

  • with period 1 are of the type:[4]
    • 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:[5]

  • slow
  • fast
  • 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:[6]

  • branch tips = terminal points of the branches[7] or tips of the midgets[8]
  • centers of spirals = fast spiral
  • branch point = points where branches meet[9] = centers of slow spirals with more then 1 arm
  • band-merging points of chaotic bands (the separator of the chaotic bands and )[10] = 2 arm spiral

angles of external rays[edit]

  • endpoint = 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

named types[edit]


The principal Misiurewicz point of the limb :[11]

  • 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 :[12]

  • 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


Misiurewicz Points, part of the Mandelbrot set:

  • Centre 0.4244 + 0.200759i; Max. Iterations 100; View radius 0.00479616 [13]





"... 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 [14]

Computing Misiurewicz points of complex quadratic mapping[edit]

roots of polynomial[edit]

Misiurewicz points [15] 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 polynomials[edit]

" 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 domains[edit]

Newton method[edit]

Finding external angles of rays that land on the Misiurewicz point[edit]



  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. W Jung : Homeomorphisms on Edges of the Mandelbrot Set Ph.D. thesis of 2002
  5. Book : Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set, page 461, by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
  6. Fractal Geometry from Yale University by Michael Frame, Benoit Mandelbrot (1924-2010), and Nial NegerFebruary 2, 2013
  7. Terminal Point by  Robert P. Munafo, 2008 Mar 9.
  8. mathoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
  9. Branch Point by Robert P. Munafo, 1997 Nov 19.
  10. Symbolic sequences of one-dimensional quadratic map points by G Pastor, Miguel Romera, Fausto Montoya Vitini
  11. Families of Homeomorphic Subsets of the Mandelbrot Set by Wolf Jung page 7
  12. 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
  13. example
  15. MIsiurewicz point in wikipedia