parameter rays of root points
of components of family F(1/2)






internal angle of main cardioid = 1/2
Periods of the form: 2 n where n is natural number:
infinity ... , 32, 16, 8, 4, 2 , 1

Pair of external angles = (1/3, 2/3) landing on the point c1= - 3/4 = - 0.75, which is root point of period 2 component
with center c=-1 and radius = 1/4 ... and ...



Pair of external angles = (6/15 = 2/5, 9/15) landing on the point c2= -5/4 = -1.25 , which is root point of period 4 component
with center c=-1.3107 and radius = ... and ...


Pair of external angles = (7/17=105/255,150/255) landing on the point c3 = -1.3680989394 ..., which is root point of period 8 component


cn = the point of attachment of the 2n-cycle and 2n-1-cycle components in the period-doubling cascade = root point of 2n period component:
period 2 : c1 = -0.75 with pair of external rays (1/3 ; 2/3) where 0.(01) = p01 = 1/3

period 4 : c2 = -1.25 with pair of external rays 6/15 ; 9/15 where 0.(0110) =2/5 = 6/15

period 8 : c3 = -1.3680989394 ... with pair of external rays 0.(01101001) = 7/17=105/255 = 0.(4117647058823529)

period 16 : c4 = -1.3940461566 ... with pair of external rays 0.(0110100110010110) = 106/257 = 0.(412451361867704280155642023346303501945525291828793774319066147 8599221789883268482490272373540856031128404669260700389105058365758754863813229571984435797665369649805447470817120622568093385214007782101167315175097276264591439688715953307392996108949416342)

period 32 : c5 = -1.3996312389 ... with pair of external rays where first is 0.(01101001100101101001011001101001) = 27031/65537 ( decimal period length = 65537 )
period 64 : c6 = -1.4008287424 ... with pair of external rays where first is 0.(0110100110010110100101100110100110010110011010010110100110010110); There are more than 100000 fractional digits in the decimal expansion

period 128 : c7 = -1.4010852713 ... with pair of external rays where first is 0.(011010011001011010010110011010011001011001101001011010011001011010010110011010010110100110010110011010 01100101101001011001101001)

period 256 : c8 = -1.401140214699 ... with pair of external rays where first is 0.(01101001100101101001011001101001100101100110100101 1010011001011010010110011010010110100110010110011010 011001011010010110011010011001011001101001011010011 0010110011010011001011010010110011010010110100110010 110100101100110100110010110011010010110100110010110)

period 512 : c9 = -1.401151982029 ... with pair of external rays where first is 0.(01101001100101101001011001101001100101100110100101 1010011001011010010110011010010110100110010110011010 011001011010010110011010011001011001101001011010011 0010110011010011001011010010110011010010110100110010 110100101100110100110010110011010010110100110010110 1001011001101001011010011001011001101001100101101001 011001101001011010011001011010010110011010011001011 0011010010110100110010110011010011001011010010110011 010011001011001101001011010011001011010010110011010 01011010011001011001101001100101101001011001101001)

period 1024 : c10 = -1.401154502237 ... with pair of external rays where first is
...
Feigenbaum-Myrberg point = -1.4011551890 ...
> What external rays land on the Myrberg-Feigenbaum point ?
The candidate angle form above is obtained by using the substitution
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.( Wolf Jung )

upper external angles of roots are:
angle(c1)=0.(01)...2=1/3
angle(c2)=0.(0110)...2=6/15
angle(c3)=0.(01101001)...2=7/17
angle(c4)=0.(0110100110010110)...2=106/257
...
One can compute it using Maxima CAS program :
Morse-Thue Number


Finch, Steven R. :Mathematical constants / Steven R. Finch. Published/Created: New York : Cambridge University Press, 2003.

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

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