Re: M-set: from internal angles to external

It appears Linus Vespas idea of what Farey numbers are and what they
are usually accepted to be are somewhat different.
My friend Gary Adamson, after reading one of the seminal works
in the area by Dr. Jay Kappraff, introduced them to me in the early 90's.
Since then I have been an avid follower of the area.
Linus Vespas seems to be advancing an approach based on the special linear group of 2by2 matrices of the modular type SL(2,Q) ( he says SL(2,Z)) : that is rational number matrices which have determinant of one. The actual definition that is accepted for Farey numbers is functionally: In true Basic form
350 DEF bis(x)
351 REM Mandelbrot Multifractal cartoon function of two functions defined
352 REM 0<x1<1
353 REM 0<y1<1
354 REM unit square domain
355 REM lines continuous
356 REM page 33 of "Multifractal and 1/f Noise"
360 LET x1=1/2
370 LET y1=1
400 LET x= mod(x,1)
410 IF x>=0 and x <= x1 then LET y=x/(1-x)
420
430 IF x>x1 and x<=1 then LET y=(1-x)/x
440 LET bis=y
450 END DEF
In Mathematica terms the function is:
f[x_] := (x/(1 - x)) /; 0 <= x <= 1/2
f[x_] := ((1 - x)/x) /; 1/2 < x <= 1
ff[x_] = f[Mod[Abs[x], 1]]
That is if the Rational number p/q is in the domain [0,1]
and it is less than 1/2 in value it has the form"
Farey[0<=p/q<=1/2)=(p/q)/(1-p/q)
Farey[1>=p/q>=1/2)=(1-p/q)/(p/q)
The resulting angles will be then as complex sine /cosine harmonics:
f[p,q]=Exp[I*Farey[p/q]*2*Pi]
As I explained, the actual cycle numbers of the Mandelbrot set bulbs/ tongues are found to be solutions to polynomials that are different than these (irrational numbers complex roots) where the Farey numbers only give "bad approximations" in the KAM ( Kolmogorov, Arnold and Moser, Russians) theory of tori.
The whole idea of the Farey limits is from Per Bak
and involves the idea of what is called phase locking
( devil's staircase steps).
What it states in rough terms is that iterative chaotic functions tend toward some rational "quantum" limit based on integers.
This effect rational is actually observed in the asteroid belts
in astronomy where it is given a name due the discoverer.
What Douady was talking about was that if you had some Farey angle,
then you could expect some "feature" in the Mandelbrot set to mirror it as a phase locking effect and not that they appeared "exactly" on
those numbers as Linus Vespas seems to expect.

So two points:
1) wrong Farey numbers used
2) only "bad approximations" of rational numbers are actually expected

I'm not saying that Linus Vespas is totally wrong,
just that his results need to be better defined in terms of accepted Farey theory and the theory of Mandelbrot cycles.
In fact I think his works show great promise
and that he is a gifted mathematician who should be encouraged.

adam majewski wrote: 
Roger Bagula napisał(a):

The Linus.org isn't coming up on this end of the Internet.
I'm also having trouble with Australia today...

The natural cyclotomic rotations are:
r(n,m)=Exp(I*m*2*Pi/n)
They are solutions to the
x^n -1=0
cyclotomic polynomials.
Linus Vepstas seems to be using some 2 based scale
that I don't actually understand: you should ask him.

The usual way to get the "cycles" in a Mandelbrot set is to do
a constant iterative procedure: ( from page 129 of Complex Dynamics by Carleson and Gamelin, Spring, 1991)
P[1,c]=c
P[2,c]=c+c^2
p[3,c]=c+c^2+2*c^3+c^4
p[4,c]=c+c^2+2*c^3+5*c^4+6*c^5+6*c^6+4*c^7+c^8
croot[m_,n_]=c/. Table[Solve[P[n,c]==0,c],{n,1,4}][[m,n]]
is solved for the roots and the Julia of the roots gives
Douady type angular cycles: one of the roots give Douady's rabbit.
As you can see these polynomial are not only different than the
cyclotomic polynomials, but have very different solutions.
If Linus Vespas has a better way, I'd like to see it.
That gives angles on the circle of 2*Pi radians of :
rational[m,n]=Arg[croot[m,n]]/(2*Pi)

Maybe you mean some other angles?
adam majewski wrote:

Roger Bagula napisał(a):

Several people have done work on this type of problem if I understand your question right. In "The Beauty of Fractals" Peitgen et al,Springer-verlag 1986 A. Duuady : Julia Sets and the Mandelbrot set pape 161 decusses such angles.
http://math.bu.edu/people/bob/
Bob Devaney has some papers related at:
http://math.bu.edu/people/bob/papers.html
Geometry of the Antennas in the Mandelbrot Set
With Monica Moreno Rocha

This paper proves that one can determine (using harmonic measure) the p/q bulb in the Mandelbrot set by looking at the geometry of its antenna. Fractals. 10 (2002), 39-46.

    * Postscript Version
    * pdf Version: http://math.bu.edu/people/bob/papers/monica.pdf
adam majewski wrote:

Hi
How compute externla angles of 2 landing rays on the root point of secondary components on the main carioid?

Adam Majewski

republika.pl/fraktal




Thx for that informations.
In Devaney method is conversion from binary number ( intinerary) to real number ( external angle).
Do you know different methods?
Linas Vepstas shows method that doesn't use conversion:
http://linas.org/art-gallery/escape/phase/arcs.html

rotation number=p/q
for p:=1 external_angle_1=1/(2^q - 1)
externa;_angle_2:=2/( 2^q -1)
is it possible to compute external angles for any rotation number using this method?

Adam Majewski
republika.pl/fraktal


http://web.archive.org/web/20030406171912/linas.org/art-gallery/escape/phase/arcs.html 

here is copy of linas page

adam Majewski 
.

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