Fractals/Iterations in the complex plane/misiurewicz
Misiurewicz point is the parameter c ( point oc parameter plane) where the critical orbit is pre-periodic.
Properities[edit | edit source]
- "Around Misiurewicz points (which are only countable but dense), the Mandelbrot set looks locally like the corresponding Julia set, in particular it looks locally very different for each Misiurewicz point." Pablo Shmerkin
- The Mandelbrot is asymptotically self-similar about pre-periodic Misiurewicz points.
notation[edit | edit source]
Misiurewicz polynomial ( map) can be marked by:
- 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  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[edit | edit source]
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 "
types[edit | edit source]
period[edit | edit source]
Misiurewicz points c
Topological[edit | edit source]
all Misiurewicz points are centers of the spirals, which are turning:
- if the Misiurewicz point is a real number, it does not turn at all
Spirals can also be classified by the number of arms.
- branch tips = terminal points of the branches or tips of the midgets
- centers of spirals = fast spiral
- branch point = points where branches meet = centers of slow spirals with more then 1 arm
- band-merging points of chaotic bands (the separator of the chaotic bands and ) = 2 arm spiral
number of external rays[edit | edit source]
- endpoint = tip = 1 angle
- primitive type = 2 angles of primitive cycle
- satellite type = 2 or more angles from satellite cycle
preperiod and period[edit | edit source]
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 types[edit | edit source]
principal[edit | edit source]
- hase m external angles, that are preimages (under doubling) of the external angles of
characteristic[edit | edit source]
Characteristic Misiurewicz point of the chaotic band of the Mandelbrot set is :
- 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
Examples[edit | edit source]
Misiurewicz Points, part of the Mandelbrot set:
- Centre 0.4244 + 0.200759i; Max. Iterations 100; View radius 0.00479616 
videos[edit | edit source]
- Kalles Fraktaler - Dive into Misiurewicz
- Embedded Julia set similar to Misiurewicz Julia set by Wolf Jung
demos[edit | edit source]
- Mandel demo 6 page 1
Images[edit | edit source]
Computing[edit | edit source]
"... 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 
Computing Misiurewicz points of complex quadratic mapping[edit | edit source]
roots of polynomial[edit | edit source]
Misiurewicz points  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 | edit source]
" 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 | edit source]
- misiurewicz_domains by Claude Heiland-Allen
- Misiurewicz domain coordinates and size estimates by Claude Heiland-Allen
Newton method[edit | edit source]
- Newton's method for Misiurewicz points by Claude Heiland-Allen
- Preperiodic Mandelbrot set Newton basins by Claude Heiland-Allen
Finding external angles of rays that land on the Misiurewicz point[edit | edit source]
- Devaney algorithm for principle Misiurewicz point
- Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen
- External angles of Misiurewicz points by Claude Heiland-Allen
- principal Misiurewicz point of wake p/q
Questions[edit | edit source]
- Questions from math.SE related with Misiurewicz point
- Questions from MO related with Misiurewicz point
References[edit | edit source]
- mathoverflow question: is-there-some-known-way-to-create-the-mandelbrot-set-the-boundary-with-an-ite
- Finite Subdivision Rules from Matings of Quadratic Functions: Existence and Constructions by Mary E. Wilkerson
- The Thurston Algorithm for quadratic matings by Wolf Jung
- 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. https://doi.org/10.1155/2007/45920
- W Jung : Homeomorphisms on Edges of the Mandelbrot Set Ph.D. thesis of 2002
- Book : Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set, page 461, by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
- Fractal Geometry from Yale University by Michael Frame, Benoit Mandelbrot (1924-2010), and Nial NegerFebruary 2, 2013
- Terminal Point by Robert P. Munafo, 2008 Mar 9.
- mathoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
- Branch Point by Robert P. Munafo, 1997 Nov 19.
- Symbolic sequences of one-dimensional quadratic map points by G Pastor, Miguel Romera, Fausto Montoya Vitini
- Families of Homeomorphic Subsets of the Mandelbrot Set by Wolf Jung page 7
- 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
- VISUALIZATION OF THE BIFURCATION LOCUS OF CUBICPOLYNOMIAL FAMILY by HIROYUKI INOU
- MIsiurewicz point in wikipedia