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

description at wikipedia

Properities

"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^[1]
The Mandelbrot is asymptotically self-similar about pre-periodic Misiurewicz points.

notation

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

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

$c=\gamma _{M}(p/q)$

so

$z^{2}+c=z^{2}+\gamma _{M}(p/q)$

Examples:

the Kokopelli Julia set $c=\gamma _{M}(3/15)=0.156520166833755+1.032247108922832i$ ^[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.

types

period

Misiurewicz points c

with period 1 are of the type:^[4]
- alpha, i.e. $f_{c}^{k}(c)=\alpha _{c}$
- beta, i.e $f_{c}^{k}(c)=\beta _{c}$
with period > 1

where alfa and beta are fixed points of complex quadratic polynomial

Topological

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 $B_{i-1}$ and $B_{i}$ )^[10] = 2 arm spiral

angles of external rays

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

principal

The principal Misiurewicz point $c=b$ of the limb $M_{k/m}$ :^[11]

$f^{m}(b)=\alpha _{b}$
hase m external angles, that are preimages (under doubling) of the external angles of $\alpha _{b}$

characteristic

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

Examples

Misiurewicz Points, part of the Mandelbrot set:

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

videos

demos

Mandel demo 6 page 1

Images

commons : category:Misiurewicz point

Computing

"... 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

roots of polynomial

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.

M(preperiod,period):=allroots(%i*P(preperiod+period)-%i*P(preperiod));

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

" 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

Newton method

Finding external angles of rays that land on the Misiurewicz point

Questions

References

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[1] thoverflow 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] thoverflow 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] xample

[14] VISUALIZATION OF THE BIFURCATION LOCUS OF CUBICPOLYNOMIAL FAMILY by HIROYUKI INOU

[15] MIsiurewicz point in wikipedia