Fractals/Iterations in the complex plane/misiurewicz

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
• The Mandelbrot is asymptotically self-similar about pre-periodic Misiurewicz points.

notation

Misiurewicz polynomial ( map) can be marked by:

• the parameter coordinate c ∈ M
• the external angle $\theta$ 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 $c=\gamma _{M}(3/15)=0.156520166833755+1.032247108922832i$ 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:
  • 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:

• 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:

• 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 $B_{i-1}$ and $B_{i}$ ) = 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 $M_{k/m}$:

• $f^{m}(b)=\alpha _{b}$
• 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 :

• 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

videos

• Kalles Fraktaler - Dive into Misiurewicz
• Embedded Julia set similar to Misiurewicz Julia set by Wolf Jung

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

Computing Misiurewicz points of complex quadratic mapping

roots of polynomial

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.

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

• misiurewicz_domains by Claude Heiland-Allen
• Misiurewicz domain coordinates and size estimates by Claude Heiland-Allen

Newton method

• 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

• 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
  • How to compute external angles of principal Misiurewicz point of wake p/q using Devaney's algorithm ?
  • program Mandel by Wolf Jung

Questions

• Questions from math.SE related with Misiurewicz point
  • math.stackexchange question: is there a Golden spiral in the Mandelbrot set?
• Questions from MO related with Misiurewicz point

References

• 7. Periodic and preperiodic points in the Mandelbrot set by : Professor Douglas C. Ravenel
• M & J-sets similarity for preperiodic points. Lei's theorem by Douglas C. Ravenel
• Misiurewicz points and the M-set self-similarity by Douglas C. Ravenel
• Misiurewicz Point by Robert P. Munafo, 2013 May 28