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

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

# name

• MF = the Myrberg-Feigenbaum point
• the Feigenbaum Point[1][2]
• Accumulation point of period-doubling cascade[3]

# 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)[4]
• the accumulation point is the limit of the disk centers
• it is a limit of a series of bifurcation parameters ( root points ${\displaystyle c_{n}}$  ) 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 ${\displaystyle m_{n}}$ . In other words a period-doubling cascade of chaotic bands also finishes, from the opposite side, at the MF point.[5]

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

# What is the address of Feigenbaum point ?

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


# What is the value of Feigenbaum point ?

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


# 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 ${\displaystyle c_{n}}$  ) of period-2n component.
• find a limit of a series of band-merging points ${\displaystyle m_{n}}$ :

### 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])
);


# 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."