Parameter rays of Misiurewicz points






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


Preperiod is used in 2 meanings : Note that :
k = K -1

Period p is the same for critical value and citical point

"... the usual convention is to use the preperiod of the critical value.
This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane." ( Wolf Jung )


Use cpp program : symbolic dynamics of quadratic polynomials with help of functions by Wolf Jung to check it

n numerator numerator of angle n/d
d denominator of angle n/d
k preperiod of angle n/d under doubling map
p period of angle n/d under doubling map



External angle of ray landing on Misirewicz point

Principal value of angle is measured in turns modulo 1.

Forms of angles :
The explicit normalized form of formula for denominator of angle :
d = ( 2^P -1) * 2^k

k = preperiod of the critical value

"P is either p or 2p, depending on the cases" (see paper Pastor03)
p = period




Angles of the form : n/ 2 ;   preperiod = 2 , period = 1
1/2 
landing point of external ray R(1/2) is c = M2,1 = - 2
it is a tip of the main antenna
orbit(z=0 ; c=-2) = { 0, -2, (2) } so preperiod =2 and period = 1
arg(orbit(z=0,c=-2)) = { 0 , 1/2 , (0/2) }
orbit of external angle 1/2 under doubling map = { 1/2 , ( 0/2) }

external argument :


Angles of the form : n/ 4 ;   preperiod =3 , period = 1
1/4  orbit = {0, 1/4 , 2/4 , ( 0/4) }
3/4  orbit = {0, 3/4 , 2/4 , ( 0/4) }

One should exclude angle:
2/4 = 1/2 because external ray for this angle lands on M2,1

Angles of the form : n/ 6 ;   preperiod = 2 , period = 2
1/6   orbit = {0, 1/6 , ( 2/6 , 4/6 ) }    land on the point c= i
5/6   orbit = {0, 5/6 , ( 4/6 , 2/6 ) }    land on the point c= - i

One should exclude angles :
2/6 = 1/3 because external ray for this angle is periodic and lands on the root point
3/6 = 1/2 because external ray for this angle lands on M2,1
4/6 = 2/3 because external ray for this angle is periodic and lands on the root point

Angles of the form : n/ 8 ;   preperiod =4 , period = 1
External rays landing on the end points of the filaments with arguments:
1/8   orbit = {0, 1/8 , 2/8 , 4/8 , ( 0/8) }
3/8   orbit = {0, 3/8 , 6/8 , 4/8 , ( 0/8) }
5/8  orbit = {0, 5/8 , 2/8 , 4/8 , ( 0/8) }
7/8  orbit = {0, 7/8 , 6/8 , 4/8 , ( 0/8) }

One should exclude angles :
2/8 = 1/4 because external ray for this angle lands on M3,1
4/8 = 1/2 because external ray for this angle lands on M2,1
6/8 = 3/4 because external ray for this angle lands on M3,1


Angles of the form : n/ 10 ;   preperiod 2 , period = 4
1/10  orbit = {0, 1/10 , ( 2/10 , 4/10 , 8/10 , 6/10 ) }  
3/10  orbit = {0, 3/10 , ( 6/10 , 2/10 , 4/10 , 8/10 ) }
7/10  orbit = {0, 7/10 , ( 4/10 , 8/10 , 6/10 , 2/10 ) }
9/10  orbit = {0, 9/10 , ( 8/10 , 6/10 , 2/10 , 4/10 ) }

2/10 = 1/5
4/10 = 2/5
5/10=1/2
6/10 = 3/5
8/10=4/5

Angles of the form : n/ 12 ;   preperiod 3 , period = 2
1/12  orbit = {0, 1/12 , 2/12 ,( 4/12 , 8/12 ) }  
5/12  orbit = {0, 5/12 , 10/12 ,( 8/12 , 4/12 ) }  
7/12  orbit = {0, 7/12 , 2/12 ,( 4/12 , 8/12 ) }  
11/12  orbit = {0, 11/12 , 10/12 ,( 8/12 , 4/12 ) }  



Here are :
- one point with 2 external rays : (5/12 , 7/12) near c = -1,54368901269109
- 2 points with 1 external rays : 1/12 and 11/12

orbit of 1/12 ( only numerators ) : [1,2,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8,4,8]
1/12 goes to period 2 cycle (4/12,8/12)= (1/3,2/3)

One should exclude angles :
2/12  = 1/6
3/12  = 1/4
4/12  = 1/3
6/12  = 1/2
8/12  = 2/3
9/12  = 3/4
10/12  = 5/6

Angles of the form : n/ 56 ;   preperiod = 4 , period = 3
1/56   orbit = {0, 1/56 , 2/56 , 4/56 , ( 8/56 , 16/56 , 32/56 ) }

3/56   orbit = {0, 3/56 , 6/56 , 12/56 , ( 24/56 , 48/56 , 40 /56 ) }
5/56  orbit = {0, 5/56 , 10/56 , 20/56 , ( 40/56 , 24/56 , 48 /56 ) }
6/56  orbit = {0, 6/56 , 12/56 , 24/56 , ( 40/56 , 24/56 , 48 /56 ) }

Principial Misiurewicz point of the 1/3 limb has 3 external rays :
 9/56   orbit = {0, 9/56 , 18/56 , 36/56 , ( 16/56 , 32/56 , 8/56 ) }
11/56   orbit = {0, 11/56 , 22/56 , 44/56 , ( 32/56 , 8/56 , 16/56 ) }
15/56   orbit = {0, 15/56 , 30/56 , 4/56 , ( 8/56 , 16/56 , 32/56 ) }


Principial Misiurewicz point of the 2/3 limb M has 3 external rays :
41/56  numerators of orbit = {0,41,26,52,(48,40,24)}
45/56  numerators of orbit = {0,45,34,12,(24,48,40)}
47/56  numerators of orbit = (0,47,38,20,(40,24,48)}

point c= -0.101096 + i * 0.956287 is landing point for rays : 9/56, 11/56 and 15/56


2/56 = 1/28
4/56 = 1 /14
6/56 = (2*3)/ (2*28)= 3/28
7/56 = 1/8

Angles of the form : n/112 ;   preperiod = , period =
Misiurewicz point of the limb M1/3 has 3 external rays :
23/112
25/112
29/112

Angles of the form : n/224 ;   preperiod = , period =
Misiurewicz point of the limb M2/3 has 3 external rays :
181/224
185/224
187/224

Angles of the form : n/992 ;   preperiod = , period =
Misiurewicz point of the limb M3/5 has 5 external rays :
?/992
39/992
47/992
?/992

5/16
3/16
19/240


images made with Mandel - program for DOS by Wolf Jung, version 3.5
Probably mathemathically most advanced program for drawing mandel / Fatou / Julia sets

1< enter > ; q = 2 ; e = 0/1 < enter > ; F9 ; < end >
changed to grayscale, resized and converted to jpg using IrfanView




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

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