Preperiodic and Misiurewicz points







Note : this page hase some bugs (preperiod) !!!! I should change it.

Misiurewicz points are named Misiurewicz points after Michal Misiurewicz

Definition ( from Evgeny Demidov ):
A point of parameter plane is Misiurewicz
if its critical orbit ( on dynamical plane ) becomes periodic with period n after k (a finite number) steps.

c = Mk,n
where
k = preperiod, k > 0
n= period

How one can compute Misiurewicz points of  Mandelbrot function ?



"... 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




They are are roots of equation:
Fc (k)(0) = Fc (k+n)(0)
Fc (k+n)(0) - Fc (k)(0) = 0

Misiurewicz points "are solutions (in c) to equations of the form
(1) p^j(0,c) = p^k(0,c),
where p(z,c) = z^2+c and p^k means iterate p k times.
Suppose k>j in (1), and let m = k-j.
Then (1) has 2^k solutions.
2^m of these are also solutions to :
(2) p^m(0,c) = 0.
The 2^m solutions to (2) are periodic: these are locations of atom centers,
and are not Misiurewicz points.
The remaining 2^k-2^m (not necessarily distinct) solutions to (1) are Misiurewicz points."

Scott Huddleston
Definition of function for finding Misiurewicz points :
# define polynomial in Maxima
P(n):=if n=0 then 0 else P(n-1)^2+c;
(%o1) P(n):=if n=0 then 0 else P(n-1)^2+c
# first function
(%i3) S(preperiod,period):=solve([P(preperiod)=P(preperiod+period)], [c]);
(%o3) S(preperiod,period):=solve([P(preperiod)=P(preperiod+period)],[c])
# second function
(%i5) M(preperiod,period):=float(rectform(solve([P(preperiod)=P(preperiod+period)],[c])));
(%o5) M(preperiod,period):=float(rectform(solve([P(preperiod)=P(preperiod+period)],[c])))
# Example 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]

M2,1
Fc (2)(0) = Fc (3)(0)
c2 +c = (c2 +c)2 +c
solutions:
c=-2 it is the tip of the main antenna and it is the only M2,1 and orbit = {0, -2, 2, 2,...}
c=0 this is periodic point so we discard this solution

M3,1
Fc (3)(0) = Fc (4)(0)
(c2 +c)2 + c = ((c2 +c)2 +c)2 + c
solutions:
c=-1.115142508039935*%i-0.22815549365396 this is antenna tip that reach farthest in negative imaginary direction
c= 1.115142508039935*%i-0.22815549365396 this is antenna tip that reach farthest in positive imaginary direction
c=-1.543689012692072 this is "band merging point"
we discard this solutions
c=-2 because it is M2,1
c=0 because this is periodic point

M4,1
Fc (4)(0) = Fc (5)(0)
((c2 +c)2 + c)2 + c = (((c2 +c)2 + c)2 + c)2 + c
solutions:
c=-2.0
c=-1.115142508039935*%i-0.22815549365396
c=1.115142508039935*%i-0.22815549365396
c=-1.543689012692072
c=0.0

M5,1
Fc (5)(0) = Fc (6)(0)
c=-2.0
c=-1.115142508039935*%i-0.22815549365396
c=1.115142508039935*%i-0.22815549365396
c=-1.543689012692072
c=0.0

{{-1.892910987907820}, {-1.296355138173036 - 0.4418516057351964 I}, {-1.296355138173036 + 0.4418516057351964 I,}, {-0.1010963638456222 - 0.9562865108091413 I,}, {-0.1010963638456222 + 0.9562865108091413 I,}, {0.,}, {0.3439069959725680 - 0.7006200202350049 I,}, {0.3439069959725680 + 0.7006200202350049 I, }} M6,1
Fc (6)(0) = Fc (7)(0)

c=-2.0,
c=-1.115142508039938*%i-0.22815549365396,
c=1.115142508039938*%i-0.22815549365396,
c=-1.543689012692077,
c=0.0,
0.0=c^15+8.0*c^14+28.0*c^13+60.0*c^12+94.0*c^11+116.0*c^10+114.0*c^9+94.0*c^8+70.0*c^7+48.0*c^6+ 32.0*c^5+20.0*c^4+10.0*c^3+4.0*c^2+2.0*c+2.0,0.0=c^31+16.0*c^30+120.0*c^29+568.0*c^28+1932.0*c^27+5096.0*c^26+10948.0*c^25+ 19788.0*c^24+30782.0*c^23+41944.0*c^22+50788.0*c^21+55308.0*c^20+54746.0*c^19+49700.0*c^18+41658.0*c^17+32398.0*c^16+23462.0* c^15+15872.0*c^14+10096.0*c^13+6096.0*c^12+3528.0*c^11+1976.0*c^10+1072.0*c^9+564.0*c^8+290.0*c^7+144.0*c^6+68.0*c^5+28.0*c^4+ 10.0*c^3+4.0*c^2+2.0*c+2.0
0.0=c^7+4.0*c^6+6.0*c^5+6.0*c^4+6.0*c^3+4.0*c^2+2.0*c+2.0

M2,2
Fc (2)(0) = Fc (4)(0)
c^2+c = ((c^2+c)^2+c)^2+c
Solutions:
c =-i
c = i and its orbit is {0, I, I-1, I, I-1, ...}
Ignore this solutions :
c =-2 = M2,1
c =-1
c = 0  this is periodic point so we discard this solution

M(4,3)
c=-1.115142508039938*%i-0.22815549365396,
c=1.115142508039938*%i-0.22815549365396,
c=-1.543689012692077,
c=-1.754877666246686,
c=0.74486176661974*%i-0.12256116687666,
c=-0.74486176661974*%i-0.12256116687666,
Ignore this solutions :
c=-2.0 = M2,1
c=0.0 this is periodic point
0.0=c^6+2.0*c^5+2.0*c^4+2.0*c^3+c^2+1.0,0.0=c^21+10.0*c^20+45.0*c^19+125.0*c^18+249.0*c^17+ 384.0*c^16+472.0*c^15+473.0*c^14+400.0*c^13+295.0*c^12+201.0*c^11+135.0*c^10+84.0*c^9+44.0*c^8+19.0*c^7+8.0*c^6+6.0*c^5+5.0*c^4+ 2.0*c^3-1.0*c^2-1.0*c+1.0,0.0=c^12+6.0*c^11+14.0*c^10+18.0*c^9+18.0*c^8+16.0*c^7+10.0*c^6+6.0*c^5+5.0*c^4+2.0*c^3+1.0, 0.0=c^7+4.0*c^6+6.0*c^5+6.0*c^4+6.0*c^3+4.0*c^2+2.0*c+2.0

Misiurewicz points are boundary points = All Misiurewicz points belong to the boundary of the Mandelbrot set.

External arguments of Misiurewicz points are rational numbers with even denominator

Misiurewicz points can be :


c:=-2
Iteration of z=0 for c:=-2
orbit(z=0 ; c=-2) = { 0, -2, (2) } so preperiod =2 and period = 1
arg(orbit(z,c)) = { 0 , 1/2 , (0/2) }

If one wants use external angles of c-point (on parameter plane) instead of c-points
then :
  uses DoublingMap instead of Mandelbot map
  must add 0 angle ( = arg(z=0) ) at the beginning.

external angle=1/2 under doubling map = 1/2 , (0/2)
so we add 0 angle:
arg(orbit(z=0,c=-2)) = { 0 , 1/2 , (0/2) }
as it should be (:-))

c:=i
orbit(z=0 ; c=i) = { 0, i, (i-1, -i) } so preperiod =2 and period = 2



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Autor: Adam Majewski adammaj1-at-o2-dot--pl

Feel free to e-mail me. (:-))
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