Fractals/Iterations in the complex plane/Mandelbrot set/MFpoint12

What is the Myrberg-Feigenbaum point of $F_{1/2}$ family ?

name

• MF = the Myrberg-Feigenbaum point
• the Feigenbaum Point
• Accumulation point of period-doubling cascade

images

• 
  Zoom toward Feigenbaum point
• 
  The Feigenbaum point (red arrow) is limit of the bifurcation point
• 
  Feigenbaum point Julia set

properities

The Myrberg-Feigenbaum point is

• a point c of parameter plane
• a Misiurewicz point
• a biaccesible point. It means that it is a landing point of 2 external rays with irrational angles. The rays are not spiralling at all (no turn), because if the Misiurewicz point is a real number, it does not turn at all
• boundary point between chaotic (-2 < c < MF) and periodic region (MF< c < 1/4)
• the accumulation point is the limit of the disk centers
• it is a limit of a series of bifurcation parameters ( root points $c_{n}$ ) of period-2ⁿ component. In other words period-doubling cascade finishes at the Myrberg-Feigenbaum point.
• it is a limit of a series of band-merging points $m_{n}$. In other words a period-doubling cascade of chaotic bands also finishes, from the opposite side, at the MF point.

$\lim _{n\to \infty }c_{n}=c_{\infty }=c_{F}$

$\lim _{n\to \infty }m_{n}=m_{\infty }=c_{F}$

What is the address of Feigenbaum point ?

Angled internal address:

$\mathbf {MF} _{1/2}{\xleftarrow {1/2}}....{\xleftarrow {1/2}}\ 16\quad {\xleftarrow {1/2}}\ 8\quad {\xleftarrow {1/2}}\ 4\quad {\xleftarrow {1/2}}\ 2\quad {\xleftarrow {1/2}}\ 1$

What is the value of Feigenbaum point ?

the Myrberg-Feigenbaum point

$\mathbf {MF} _{1/2}=c=-1.401155\ldots$

What external rays land on the Myrberg-Feigenbaum point ?

Decimal values of external angles t of rays that lands on the Myrberg-Feigenbaum point are (0.412454... , 0.58755...)

How to compute angles of external rays ?

To compute angles one can use 2 methods:

• find a limit of a series of bifurcation parameters ( root points $c_{n}$ ) of period-2ⁿ component.
• find a limit of a series of band-merging points $m_{n}$:

How to compute limit of angles landing on bifurcation parameters ?

The candidate upper external angle is obtained by using the substitution (string replacing): 0 -> 01 and 1 -> 10 repeatedly:

• 0
• 01
• 0110
• 01101001
• 0110100110010110
• ...

But it is not known whether the rays actually lands; maybe M is not locally connected at the Feigenbaum point and some long decorations are shielding it from external rays.

One can compute it using Maxima CAS program :

kill(all);
remvalue(all);

f(x):=if (x=0) then [0,1] else [1,0];
compile(all);

a:[];
a:endcons([0],a);

for n:2 thru 10 step 1 do (
   a:endcons([],a),
   for x in a[n-1] do (
      a[n]:endcons(first(f(x)),a[n]),
      a[n]:endcons(second(f(x)),a[n])),
      print(n,a[n])
);

How to compute points of perid n tupling bifurcations ?

• UNIVERSALIiTY FOR PERIOD n-TUPLINGS IN COMPLEX MAPPINGS by Predrag CVITANOVIC and Jan MYRHEIM. Physics Letters A Volume 94, Issue 8, 28 March 1983, Pages 329-333
• period tripling:
  • point of Golberg - Sinai - Khanin (GSK) lGSK = 0.0236411685 + 0.7836606508i
  • Corresponded critical point (GSK point) is situated at λc = 0.0236411685377 + 0.7836606508052i and characterized by following critical indexes [2], namely, by critical multiplier µc, scale factor α, and parameter scaling constant δ :
    • µc = −0.47653179 − 1.05480867i
    • α = −2.0969 + 2.3583i
    • δ = 4.6002 − 8.9812i
  • Near the critical point GSK a structure of "leaves" of the Mandelbrot set possesses a property of scale invariance in respect to rescaling of g-gGSK with the complex factor d=4.60022558 -8.98122473i.

zoom

• "sequence of illustrations, each view is centered at the Feigenbaum point and the magnification increases by 4.6692 (the Feigenbaum Constant) each time. The filaments become steadily denser until they fill the view."

References

• Decimal expansion of the accumulation point of the logistic map.