Fractals/Iterations in the complex plane/p misiurewicz

How to compute external angles of principal Misiurewicz point of wake p/q using Devaney's algorithm ? =names=
 * a principal Misiurewicz points of the wake ( or the limb or the shrub )
 * the main node of the shrub
 * the hub = center part of shrub ( Pastor notation), the point where spokes join
 * a junction point of q spokes which is attached directly to the p/q bulb ( Devaney notation )
 * "the first dominating α-Misiurewicz point in M p/q, i.e., the one of lowest pre-period"
 * Eye of elephant resting on internal angle 1/4 of main cardioid ( Curtis McMullen)

=notes= Principal misiurewicz point of p/q-wake is $$c = M_{q,1}$$
 * it has q arms ( spokes, branches) numbered from 0 to q-1 in a clockwise direction
 * it is a landing point for q external angles
 * critical point has preperiod q and period p = 1 under complex quadratic map for $$c = M_{q,1}$$

External angles of q rays landing on $$c = M_{q,p}$$
 * in the binary expansion length of preperiodic and periodic part is q
 * period and preperiod of angle under doubling map is q

Important differences:
 * Romero-Pastor notation uses q/p not p/q
 * Preperiod: the usual convention is to use the preperiod of the critical value $$z_{cv} = c$$, not preperiod of critical point $$z_{cr} = 0$$. 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

=introduction=

How to work with the shift map ?

If length of string s is q then

$$ \sigma^qs = \sigma^0s = s $$

shifting q digits in blocks of b digits
Note that

$$ b < q $$

q=5 b=1
$$\sigma^0( s ) = \sigma^0(00001) =   {\color{Red}00001}   $$ $$\sigma^1( s ) = \sigma^1(00001) =   00010 $$ $$\sigma^2( s ) = \sigma^2(00001) =   00100    $$ $$\sigma^3( s ) = \sigma^3(00001) =   01000  $$ $$\sigma^4( s ) = \sigma^4(00001) =   10000  $$ $$\sigma^5( s ) = \sigma^5(00001) =   {\color{Red}00001} $$

q=5 b=2
$$\sigma^0( s ) = \sigma^0(00001) =   {\color{Red}00001}   $$ $$\sigma^2( s ) = \sigma^2(00001) =   00100    $$ $$\sigma^4( s ) = \sigma^4(00001) =   10000  $$

q=5 b=3
$$\sigma^0( s ) = \sigma^0(00001) =   {\color{Red}00001}   $$ $$\sigma^3( s ) = \sigma^3(00001) =   01000  $$

q=5 b=4
$$\sigma^0( s ) = \sigma^0(00001) =   {\color{Red}00001}   $$ $$\sigma^4( s ) = \sigma^4(00001) =   10000  $$

=Algorithm= Algorithm is based on the Theorem 5.3 in: Geometry of the Antennas in the Mandelbrot Set by R L Devaney and M Moreno-Rocha, April 11, 2000

External Angles of Hub ( see section 3.9 of the Book by Claude) or spoke

The $$p/q$$ bulb ( = hyperbolic component) has 2 external angles landing on it's root point (bond) :

$$\theta_{\color{blue}-}(p/q) = 0.({\color{blue}s_-})$$ $$\theta_{\color{red}+}(p/q) = 0.({\color{red}s_+})$$

such that :

$$ \theta_{\color{blue}-} < \theta_{\color{red}+}$$

These angles have :
 * repeating binary expansion denoted by round brackets or overline
 * length of repeating ( periodic ) part is $$q$$

Other names of these angles are angles of the wake.

The junction point of its hub ( principal Misiurewicz point) $$M$$ has external angles in increasing order $$0.s_-(s_+)$$ $$0.s_-(\sigma^b s_+)$$ $$ \vdots$$ $$ .s_-(\sigma^{(q-p-1)b} s_+)$$ $$ .s_+(\sigma^{(q-p)b} s_+)$$ $$ \vdots$$ $$0.s_+(s_-)$$

where
 * s is a finite string of q binary digits = s consist of q binary digits = length(s)= q
 * $$\sigma$$ is the shift map
 * $$\frac{p}{q}$$ fraction has Farey parents a/b and r/s
 * b is a denominator of lower Farey parent

$$ \begin{cases} \frac{a}{b} < \frac{p}{q} < \frac{r}{s}\\ \frac {a} {b} \oplus \frac {r} {s} =  \frac {p} {q}\\ \end{cases} $$

Implementation:

input and output

 * input : 2 external angles of the wake $$p/q $$
 * output : $$q$$ external angles of principal Misiurewicz point ( hub)

steps

 * input = $$p/q$$
 * check input
 * both p and q are:
 * integers
 * > 0
 * proper fraction : p < q
 * irreducible fraction = in lowest terms ( An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 A fraction is in lowest terms when the greatest common factor (GCF) of the numerator and denominator is 1 )
 * if input is good then there are $$ q$$ angles to compute
 * compute 2 angles of the wake : $$s_-(p/q)$$ and $$s_+(p/q)$$
 * compute first 2 of q angles : $$0.s_-(s_+)$$ and  $$0.s_+(s_-)$$
 * compute last $$q -2$$ angles
 * compute Farey parents of $$p/q $$
 * compute $$ q - p$$
 * ( to do )

=Examples by wake = wake           angles of the wake      angle of principal Misiurewicz point    angles that land on z=0 on the dynamical plane                  period(c)               c --- k/r = 1/2	wake 1 ; 2/3		Mis 5/12				zcr 5 ; 17/24	                    				period_landing = 1	c -0.2281554936539618 ; 1.115142508039937 k/r = 1/3	wake 1 ; 2/7		Mis 9/56				zcr 9 ; 65/112 						period_landing = 1	c -0.1010963638456222 ; 0.9562865108091415 k/r = 1/4	wake 1 ; 2/15		Mis 17/240				zcr 17 ; 257/480						period_landing = 1	c -0.01718797733835019 ; 1.037652343793215 k/r = 1/5	wake 1 ; 2/31		Mis 33/992				zcr 33 ; 1025/1984						period_landing = 1	c -0.01660571692147523 ; 1.006001828834065 k/r = 1/6	wake 1 ; 2/63		Mis 65/4032				zcr 65 ; 4097/8064						period_landing = 1	c 0.002241106093233115 ; 1.006987004324957 k/r = 1/7	wake 1 ; 2/127		Mis 129/16256				zcr 129 ; 16385/32512						period_landing = 1	c -0.001369133815686842 ; 1.002755660363466 k/r = 1/8	wake 1 ; 2/255		Mis 257/65280				zcr 257 ; 65537/130560						period_landing = 1	c 0.001159450074256577 ; 1.000609019839529 k/r = 1/9	wake 1 ; 2/511		Mis 513/261632				zcr 513 ; 262145/523264 					period_landing = 1	c 0.0001701882004481036 ; 1.000517331884371 k/r = 1/10	wake 1 ; 2/1023		Mis 1025/1047552			zcr 1025 ; 1048577/2095104      				period_landing = 1	c 0.0002217350415235168 ; 0.9999309294242422 k/r = 1/11	wake 1 ; 2/2047		Mis 2049/4192256			zcr 2049 ; 4194305/8384512      				period_landing = 1	c 8.600871635354104e-05 ; 1.000043520609493 k/r = 1/12	wake 1 ; 2/4095		Mis 4097/16773120			zcr 4097 ; 16777217	/33546240				period_landing = 1	c 1.907198794976112e-05 ; 0.9999636227152136 k/r = 1/13	wake 1 ; 2/8191		Mis 8193/67100672			zcr 8193 ; 67108865	/134201344				period_landing = 1	c 1.619607246569189e-05 ; 0.9999946863543573 k/r = 1/14	wake 1 ; 2/16383	Mis 16385/268419072			zcr 16385 ; 268435457	/536838144				period_landing = 1	c -2.164159763572468e-06 ; 0.9999930692712914 k/r = 1/15	wake 1 ; 2/32767	Mis 32769/1073709056			zcr 32769 ; 1073741825	/2147418112				period_landing = 1	c 1.36020585022823e-06 ; 0.9999973111035358 k/r = 1/16	wake 1 ; 2/65535	Mis 65537/4294901760			zcr 65537 ; 4294967297	/8589803520				period_landing = 1	c -1.136844998313359e-06 ; 0.9999994042152635 k/r = 1/17	wake 1 ; 2/131071	Mis 131073/17179738112			zcr 131073 ; 17179869185	/34359476224			period_landing = 1	c -1.660928890362016e-07 ; 0.9999994938657326 k/r = 1/18	wake 1 ; 2/262143	Mis 262145/68719214592			zcr 262145 ; 68719476737	/137438429184			period_landing = 1	c -2.165774171377629e-07 ; 1.000000067631949 k/r = 1/19	wake 1 ; 2/524287	Mis 524289/274877382656			zcr 524289 ; 274877906945	/549754765312			period_landing = 1	c -8.402826966472988e-08 ; 0.9999999574950604 k/r = 1/20	wake 1 ; 2/1048575	Mis 1048577/1099510579200		zcr 1048577 ; 1099511627777	/2199021158400			period_landing = 1	c -1.861820421561348e-08 ; 1.000000035526125 k/r = 1/21	wake 1 ; 2/2097151	Mis 2097153/4398044413952		zcr 2097153 ; 4398046511105	/8796088827904			period_landing = 1	c -1.581664298449309e-08 ; 1.000000005190412 k/r = 1/22	wake 1 ; 2/4194303	Mis 4194305/17592181850112		zcr 4194305 ; 17592186044417	/35184363700224			period_landing = 1	c 2.11348855536603e-09 ; 1.000000006768042 k/r = 1/23	wake 1 ; 2/8388607	Mis 8388609/70368735789056		zcr 8388609 ; 70368744177665	/140737471578112		period_landing = 1	c -1.32827905765734e-09 ; 1.000000002625882 k/r = 1/24	wake 1 ; 2/16777215	Mis 16777217/281474959933440		zcr 16777217 ; 281474976710657	/562949919866880		period_landing = 1	c 1.110191297822782e-09 ; 1.000000000581819 k/r = 1/25	wake 1 ; 2/33554431	Mis 33554433/1125899873288192		zcr 33554433 ; 1125899906842625	/2251799746576384		period_landing = 1	c 1.62200284270896e-10 ; 1.00000000049427 k/r = 1/26	wake 1 ; 2/67108863	Mis 67108865/4503599560261632		zcr 67108865 ; 4503599627370497	/9007199120523264		period_landing = 1	c 2.115013311798569e-10 ; 0.9999999999339535 k/r = 1/27	wake 1 ; 2/134217727	Mis 134217729/18014398375264256		zcr 134217729 ; 18014398509481985	/36028796750528512	period_landing = 1	c 8.205882795347896e-11 ; 1.000000000041509 k/r = 1/28	wake 1 ; 2/268435455	Mis 268435457/72057593769492480		zcr 268435457 ; 72057594037927937	/144115187538984960	period_landing = 1	c 1.818186256603596e-11 ; 0.9999999999653065 k/r = 1/29	wake 1 ; 2/536870911	Mis 536870913/288230375614840832	zcr 536870913 ; 288230376151711745	/576460751229681664	period_landing = 1	c 1.544590637441404e-11 ; 0.9999999999949313 k/r = 1/30	wake 1 ; 2/1073741823	Mis 1073741825/1152921503533105152	zcr 1073741825 ; 1152921504606846977	/2305843007066210304	period_landing = 1	c -2.063955458366402e-12 ; 0.9999999999933906 k/r = 1/31	wake 1 ; 2/2147483647	Mis 2147483649/4611686016279904256	zcr 2147483649 ; 4611686018427387905	/9223372032559808512	period_landing = 1	c 1.29718610843552e-12 ; 0.9999999999974356 k/r = 1/32	wake 1 ; 2/4294967295	Mis 4294967297/18446744069414584320	zcr 4294967297 ; 1	/18446744065119617024			period_landing = 1	c -1.084197223871117e-12 ; 0.9999999999994318 k/r = 1/33	wake 1 ; 2/8589934591	pow error

1/2
So here are 4 angles (q+2) in increasing order :
 * 2 rays landing on the root point ( s+ and s- )
 * q=2 rays landing on the Misiurewicz point

Farey parents of 1/2 are 0/1  and  1/1

0/1 < 1/2 < 1/1 	 0.0000000000000000 < 0.5000000000000000 < 1.0000000000000000

The denominator of smaller parent :

$$ b = 1 $$

$$\begin{align} 0.(s_-) = 0.(01) = \frac{1}{3 } = 0.33333333333333333333\\ 0.s_-(s_+) = 01(10) = \frac{5}{12} = 0.41666666666666666666\\ 0.s_+(s_-) = 10(01) = \frac{7}{12} = 0.5833333333333333333 \\ 0.(s_+) = 0.(10) = \frac{2}{3 } = 0.66666666666666666666\\ \end{align} $$

The angle 5/12  or  01p10 has  preperiod = 2  and  period = 2. The corresponding parameter ray is landing at a Misiurewicz point of preperiod 2 and period dividing 2.

$$ M_{2,1} = c = -1.543689012692076 +0.000000000000000 i   $$

Compare with
 * $$\mathbf{MF}_{1/2} $$ is the Myrberg-Feigenbaum point c = −1.401155 with external angles = (0.412454..., 0,58755...)

$$\begin{cases} 0.(s_-) = 0.(01) = \frac{1}{3} = \frac{4}{12} = 0.(3) = wake    \\ 0.s_-(s_+) = 0.01(10) = \frac{5}{12} = 0.41(6) = PrincipalMis = M_{2,2}\\ 0.0(1) = \frac{1}{2} = \frac{6}{12} = 0.5 = tip = M_{1,1} = c = -2\\ 0.s_+(s_-) = 0.10(01) = \frac{7}{12} = 0.58(3) = PrincipalMis = M_{2,2}\\ 0.(s_+) = 0.(10) = \frac{2}{3} = \frac{8}{12} = 0.(6)= wake     \\ \end{cases} $$

1/3


The $\frac{p}{q} = \frac{1}{3}$ bulb ( = period 3 hyperbolic component) has 2 external angles landing on it's root point (bond) :

$$\theta_-(1/3) = 0.({\color{Blue}s_-}) = 0.({\color{Blue}001}) $$ $$\theta_+(1/3) = 0.({\color{Red}s_+}) = 0.({\color{Red}010}) $$

such that :

$$ \theta_- < \theta_+$$

Principal Misiurewicz point $$M_{3,1}$$ of $$\frac{1}{3}$$ wake is a landing point for $$ q = 3$$ external angles. It is denoted by

$$c = M_{3,1} = -0.101096363845622 +0.956286510809142 i $$

where :
 * first number denotes preperiod
 * second number denotes period

Two of them one can easly compute from angles the wake :

$$ \theta_-(M) = 0.{\color{Blue}s_-}({\color{Red}s_+}) = 0.{\color{Blue}001}({\color{Red}010}) $$ $$ \theta_+(M) = 0.{\color{Red}s_+}({\color{Blue}s_-}) = 0.{\color{Red}010}({\color{Blue}001}) $$

such that :

$$ {\color{Blue}s_-} < {\color{Blue}s_-}({\color{Red}s_+}) < {\color{Red}s_+}({\color{Blue}s_-}) < {\color{Red}s_+}$$

So the problem is to compute only 1 ray.

First find Farey parents of $$\frac{1}{3}$$

$$ \frac{0}{1} \oplus \frac{1}{2} =  \frac {0 + 1}{1 + 2}   = \frac {1}{3} $$

such that :

$$ \frac{0}{1} < \frac {1}{3} < \frac{1}{2}$$

Take denominator of smaller parent :

$$ b = 1 $$

and compute last fraction.

First find periodic part :
 * remember that shift map works on the infinite sequence
 * take only first q digits from result of shift map

$$\sigma^b({\color{Red}s_+}) = \sigma( \color{Red}010\ 010\ 010 \color{Black}...) = \color{Green}100$$

then last angle is :

$$0.{\color{Blue}s_-}({\color{Green}\sigma^b s_+}) = 0.{\color{Blue}001}({\color{Green}100}) $$

So here are 5 angles (q+2) in increasing order :

$$\begin{cases} &\theta_-(1/3) &=&\ 0.({\color{Blue}s_-})    &=& 0.({\color{Blue}001}) && \text{ lower angle of the wake} \\ &\theta_-(M)  &=&\ 0.\ {\color{Blue}s_-}({\color{Red}s_+})  &=& 0.\ {\color{Blue}001}\ ({\color{Red}010})  = \frac{9}{56} && = M_{3,1}\\ &\theta_m(M)  &=&\ 0.\ {\color{Blue}s_-}({\color{Green}\sigma s_+}) &=& 0.\ {\color{Blue}001}\ ({\color{Green}100})  = \frac{11}{56} && = M_{3,1}\\ &\theta_+(M)  &=&\ 0.\ {\color{Red}s_+}({\color{Blue}s_-}) &=& 0.\ {\color{Red}010}\ ({\color{Blue}001})  = \frac{15}{56} && = M_{3,1}\\ &\theta_+(1/3) &=&\ 0.({\color{Red}s_+}) &=& 0.({\color{Red}010}) && \text{upper angle of the wake}\\ \end{cases} $$

One can check it with Mandel:

The angle 9/56  or  001p010 has preperiod = 3  and  period = 3. The corresponding parameter ray is landing at a Misiurewicz point of preperiod 3 and period dividing 3. Do you want to draw the ray and to shift c to the landing point?

1/4
The $$\frac{p}{q} = \frac{1}{4}$$ bulb ( = period 4 hyperbolic component) has 2 external angles landing on it's root point (bond) :

$$\theta_-(1/4) = 0.(s_-) = 0.({\color{Blue}0001}) $$ $$\theta_+(1/4) = 0.(s_+) = 0.({\color{Red} 0010}) $$

Principal Misiurewicz point $$M$$ of $$\frac{1}{4}$$ wake is a landing point for $$ q = 4$$ external angles.

$$ M_{4,1} = c = 0.366362983422764  +0.591533773261445 i$$

Two of them one can easly compute from angles the wake :

$$ \theta_-(M) = 0.s_-(s_+) = 0.{\color{Blue}0001}({\color{Red}0010}) $$ $$ \theta_+(M) = 0.s_+(s_-) = 0.{\color{Red}0010}({\color{Blue}0001}) $$

So the problem is to compute only $$ q-2 = 2$$ rays.

First find Farey parents of $$\frac{1}{4}$$



$$ \frac{0}{1} \oplus \frac{1}{3} =  \frac {0 + 1}{1 + 3}   = \frac {1}{4} $$

Take denominator of lower parent :

$$ b = 1 $$

and compute last fractions.

First find periodic parts for n :

$$\sigma^1( s_+) = \sigma^1( {\color{Red} 0010\ 0010\ 0010} ...)  =  \color{Green}  0100   $$ $$\sigma^2( s_+) = \sigma^2( {\color{Red} 0010\ 0010\ 0010} ...)  =  \color{Magenta}1000 $$

then 2 last angles are :

$$0.s_-(\sigma^1 s_+) = 0.{\color{Blue}0001}({\color{Green}0100}) $$ $$0.s_-(\sigma^2 s_+) = 0.{\color{Red}0010}({\color{Magenta}1000}) $$

So here are $$ q+2 = 6$$ angles in increasing order :

$$\begin{align} &\theta_-(1/4) &=& 0.(s_-) &=& 0. ({\color{Blue}0001}) && \text{ lower angle of the wake} \\ &\theta_-(M) &=& 0.s_-\ (\quad  s_+) &=& 0.\ {\color{Blue}0001}\ ({\color{Red}0010})   && \text{ lower angle of M}\\ &\theta_{m-}(M)  &=&\ 0.s_-(\sigma^1 s_+) &=& 0.\ {\color{Blue}0001}\ ({\color{Green}0100})  && \text{ middle angle of M}\\ &\theta_{m+}(M)  &=&\ 0.s_-(\sigma^2 s_+) &=& 0.\ {\color{Blue}0001}\ ({\color{Magenta}1000})  && \text{ middle angle of M}\\ &\theta_+(M) &=& 0.s_+\ (\quad  s_-) &=& 0.\ {\color{Red}0010}\ ({\color{Blue}0001})  && \text{ upper angle of M}\\ &\theta_+(1/4) &=& 0.(s_+) &=& 0. ({\color{Red} 0010}) && \text{upper angle of the wake}\\ \end{align} $$

2/5
The $$\frac{p}{q} = \frac{2}{5}$$ bulb ( = period 5 hyperbolic component) has 2 external angles landing on it's root point (bond) :

$$\theta_-(2/5) = 0.(s_-) = 0.({\color{Blue}01001}) = \frac{9}{31} = 0.(290322580645161)$$ $$\theta_+(2/5) = 0.(s_+) = 0.({\color{Red} 01010}) = \frac{10}{31} = 0.(322580645161290)$$

Farey parents of 2/5 are 1/3 and 1/2

1/3 < 2/5 < 1/2 	 		  0.333333 < 0.400000 < 0.500000 $$ \frac {1}{3} \oplus \frac {1} {2}  = \frac{1+1}{2+3} = \frac {2} {5} $$

so denominator of smaller parent is b = 3.

Angles in the symbolic form (s-) s-(s+) s-(d^3(s+)) s-(d^1(s+)) s+(d^4(s+)) s+(s-) (s+)

The angle 289/992  or  01001p01010 has  preperiod = 5  and  period = 5. The corresponding parameter ray is landing at a Misiurewicz point of preperiod 5 and period dividing 5.

1/7
The wake 1/7 of main cardioid

$$M_{7,1} = c = 0.397391822296541 +0.133511204871878 i $$ = principal Misiurewicz c = 0.367375134418445 +0.147183763188559 i = root of the wake 1/7 c = 0.376008681846768 +0.144749371321633 i = period 7 center

External rays:
 * 1/127 =   0.(0000001)          = 0.0078740157480315  = wake
 * 129 /16256 = 0.0000001(0000010) = 0.00793553149606299 = pM_{7,1}
 * 131 /16256 = 0.0000001(0000100) = 0.00805856299212598 = pM
 * 135 /16256 = 0.0000001(0001000 = 0.00830462598425197 = pM
 * 143 /16256 = 0.0000001(0010000) = 0.00879675196850394 = pM
 * 159 /16256 = 0.0000001(0100000) = 0.00978100393700787 = pM
 * 191 /16256 = 0.0000001(1000000) = 0.01174950787401575 = pM
 * 255 /16256 = 0.0000010(0000001) = 0.0156865157480315 = pM
 * 1/64      = 0.000000(1)        = 0.015625            = M_{6,1} = longest tip
 * 2/127     = 0.(0000010)        = 0.01574803149606299 = wake

3/7


Wake 3/7 and its principal Misiurewicz point (hub)

$$M_{7,1} = c = -0.670209187903254 +0.458060975296946 i $$

Check with Mandel The angle 5249/16256  or  0101001p0101010 has preperiod = 7  and  period = 7. The corresponding parameter ray is landing at a Misiurewicz point of preperiod 7 and period dividing 7.

5/11


Check with Mandel: The 5/11-wake of the main cardioid is bounded by the parameter rays with the angles 681/2047 or  p01010101001  and 682/2047 or  p01010101010. Do you want to draw the rays and to shift c to the center of the satellite component?

The result is a center of period 11 satelite component c = -0.697838195122425 +0.279304134101366 i    period = 11

The angle 1394689/4192256  or  01010101001p01010101010 has  preperiod = 11  and  period = 11. The corresponding parameter ray lands at a Misiurewicz point of preperiod 11 and period dividing 11. Do you want to draw the ray and to shift c to the landing point?

The result is a principal Misiurewicz point of wake 5/11 M_{11,1} = c = -0.724112682973574 +0.286456567676711 i

12/25
$$\begin{cases} 0.(0101010101010101010101001)_2 = \frac{11184809}{33554431} =  0.333 333 29359690229883498844012583613770711832365 ... _{10} = wake    \\ 0101010101010101010101001(0101010101010101010101010) = 375299913023489 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1001010101010101010101010) = 375299921412097 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010010101010101010101010) = 375299923509249 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010100101010101010101010) = 375299924033537 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101001010101010101010) = 375299924164609 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101010010101010101010) = 375299924197377 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101010100101010101010) = 375299924205569 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101010101001010101010) = 375299924207617 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101010101010010101010) = 375299924208129 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101010101010100101010) = 375299924208257 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101010101010101001010) = 375299924208289 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101010101010101010010) = 375299924208297 % 1125899873288192 = principal Mis \\ 0101010101010101010101001(1010101010101010101010100) = 375299924208299 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0010101010101010101010101) = 375299940985515 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0100101010101010101010101) = 375299945179819 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101001010101010101010101) = 375299946228395 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010010101010101010101) = 375299946490539 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010100101010101010101) = 375299946556075 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010101001010101010101) = 375299946572459 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010101010010101010101) = 375299946576555 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010101010100101010101) = 375299946577579 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010101010101001010101) = 375299946577835 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010101010101010010101) = 375299946577899 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010101010101010100101) = 375299946577915 % 1125899873288192 = principal Mis \\ 0101010101010101010101010(0101010101010101010101001) = 375299946577919 % 1125899873288192 = principal Mis \\ 0.(0101010101010101010101010)_2 = \frac{11184810}{33554431} =  0.333 333 3233992255747087471100314590344267795809 = wake \\ \end{cases} $$

internal angle p/q = 12 / 25 internal angle in lowest terms = 12 % 25 rays of the bulb: (0101010101010101010101001) = 11184809 % 33554431 (0101010101010101010101010) = 11184810 % 33554431
 * Main> :main 12 25

rays of the principle hub: 0101010101010101010101001(0101010101010101010101010) = 375299913023489 % 1125899873288192 0101010101010101010101001(1001010101010101010101010) = 375299921412097 % 1125899873288192 0101010101010101010101001(1010010101010101010101010) = 375299923509249 % 1125899873288192 0101010101010101010101001(1010100101010101010101010) = 375299924033537 % 1125899873288192 0101010101010101010101001(1010101001010101010101010) = 375299924164609 % 1125899873288192 0101010101010101010101001(1010101010010101010101010) = 375299924197377 % 1125899873288192 0101010101010101010101001(1010101010100101010101010) = 375299924205569 % 1125899873288192 0101010101010101010101001(1010101010101001010101010) = 375299924207617 % 1125899873288192 0101010101010101010101001(1010101010101010010101010) = 375299924208129 % 1125899873288192 0101010101010101010101001(1010101010101010100101010) = 375299924208257 % 1125899873288192 0101010101010101010101001(1010101010101010101001010) = 375299924208289 % 1125899873288192 0101010101010101010101001(1010101010101010101010010) = 375299924208297 % 1125899873288192 0101010101010101010101001(1010101010101010101010100) = 375299924208299 % 1125899873288192 0101010101010101010101010(0010101010101010101010101) = 375299940985515 % 1125899873288192 0101010101010101010101010(0100101010101010101010101) = 375299945179819 % 1125899873288192 0101010101010101010101010(0101001010101010101010101) = 375299946228395 % 1125899873288192 0101010101010101010101010(0101010010101010101010101) = 375299946490539 % 1125899873288192 0101010101010101010101010(0101010100101010101010101) = 375299946556075 % 1125899873288192 0101010101010101010101010(0101010101001010101010101) = 375299946572459 % 1125899873288192 0101010101010101010101010(0101010101010010101010101) = 375299946576555 % 1125899873288192 0101010101010101010101010(0101010101010100101010101) = 375299946577579 % 1125899873288192 0101010101010101010101010(0101010101010101001010101) = 375299946577835 % 1125899873288192 0101010101010101010101010(0101010101010101010010101) = 375299946577899 % 1125899873288192 0101010101010101010101010(0101010101010101010100101) = 375299946577915 % 1125899873288192 0101010101010101010101010(0101010101010101010101001) = 375299946577919 % 1125899873288192

Landing point = principal Misiurewicz point

The angle 375299913023489/1125899873288192  or  0101010101010101010101001p0101010101010101010101010 has  preperiod = 25  and  period = 25. The corresponding parameter ray lands at a Misiurewicz point of preperiod 25 and period dividing 25. Do you want to draw the ray and to shift c to the landing point? c = -0.745846774741742 +0.124374904775875 i

m-describe 112 100 10000 -0.745846774741742 +0.124374904775875 4 the input point was -7.4584677474174200000000000000000001e-01 + 1.2437490477587499999999999999999999e-01 i nearby hyperbolic components to the input point:

- a period 1 cardioid with nucleus at 0.00000e+00 + 0.00000e+00 i the component has size 1.00000e+00 and is pointing west the atom domain has size 0.00000e+00 the atom domain coordinates of the input point are -nan + -nan i the atom domain coordinates in polar form are -nan to the east the atom coordinates of the input point are -0.74585 + 0.12437 i the atom coordinates in polar form are 0.75615 to the west the nucleus is 7.56146e-01 to the east of the input point

- a period 2 circle with nucleus at -1.00000e+00 + 0.00000e+00 i the component has size 5.00000e-01 and is pointing west the atom domain has size 1.00000e+00 the atom domain coordinates of the input point are 0.25415 + 0.12437 i the atom domain coordinates in polar form are 0.28295 to the east-north-east the atom coordinates of the input point are 0.50831 + 0.24875 i the atom coordinates in polar form are 0.56591 to the east-north-east the nucleus is 2.82954e-01 to the west-south-west of the input point external angles of this component are: .(01) .(10) the point escaped with dwell 4217.96435

nearby Misiurewicz points to the input point:

- 26p4 with center at -7.45846774741742277327028259457753e-01 + 1.24374904775875452739596099543026e-01 i the Misiurewicz domain has size 7.57002e-04 the Misiurewicz domain coordinate radius is 7.0135e-13 the center is 5.30927e-16 to the north-north-west of the input point the multiplier has radius 1.030029879100029796e+00 and angle -0.078808321127835692 (in turns)

1/31

 * wake -> limb 1/31

8/47 = 16/94
Haskell output

c output

15/94
First angle of the hub is:

6227271590044554501136183694529415329491604978647695361% 392318858461667547739736838930672110377831130880616169472

=Code=

Haskell code
Save it as a bh.hs and use it from console in an interactive way :

=Compare with=
 * program Mandel by Wolf Jung

=References=

=See also=
 * Repulsive Points by Faber
 * Newtons method for misiurewicz points by Claude Heiland-Allen
 * enumeration_of_misiurewicz_points by Claude Heiland-Allen