Fractals/Iterations in the complex plane/wake

How to find the angles of the parametric external rays that land on the p/q root point on the boundary of Mandelbrot set's main cardioid ( period 1 component) ?

=Images=

= irreducible fraction =

First check of p/q is irreducible

=Wake and limb =

Wake is the region of parameter plane enclosed by two external rays landing on the same root point on the boundary of main cardioid (period 1 hyperbolic component).

p/q-limb is a part of Mandelbrot set contained inside p/q-wake

Limb:
 * start from from the root point
 * end with tip point

External angles of p/q-wake:
 * have period q under doubling map. It is the same as period of it's landing point ( c = root point ) and parent hyperbolic component
 * length of periodic part of binary expansion is q
 * preperiod under doubling map is zero

Points of of p/q-wake:
 * Roots are landing points of parameter rays with periodic angles
 * Misiurewicz points have preperiodic external angles

The size of the p/q limb is $$\frac{1}{2^q - 1}$$

The p/q bulb can be recognized by locating the smallest spoke ( branch) in the antenna and determining its relative location to the main spoke. =wake, principal Misiurewicz point and dendrite Julia set=

Notes from demo 3 (external rays) page 9/12 from program mandel by Wolf Jung

The parameter rays with the angles 1/7 and 2/7 land at the root of a period-3 component, which is of satellite type with rotation number 1/3.

For all parameters c in the wake between the rays
 * for c in the 1/3-limb of the Mandelbrot set,
 * for the principle Misiurewicz point of the wake

the dynamic rays with the angles 1/7, 2/7, and 4/7:
 * land together at the repelling fixed point $$z = \alpha_c$$
 * the critical value z = c is between the first two rays.

We shall compute the external angles of certain preimages of αc under fc(z). Note that an angle θ has two preimages under doubling modulo 1, θ/2 and (θ+1)/2.

$$z = -\alpha_c$$ is the only preimage of $$z = \alpha_c$$  different from the fixed point itself. The angle 1/7 has the preimages (1/7)/2 = 1/14 and (1/7 + 1)/2 = 4/7. The latter angle belongs to $$z = \alpha_c$$, so 1/14 is an external angle of $$z = -\alpha_c$$. In the same way, the other angles 9/14 and 11/14 are obtained. The rays are drawn blue. Move z to that preimage of $$z = -\alpha_c$$ between the rays for 2/7 and 4/7. The angle 1/14 has the preimages (1/14)/2 = 1/28 and (1/14 + 1)/2 = 15/28. Only the latter angle is in the chosen interval. The other two external angles of z are 9/28 and 11/28. The rays are drawn magenta. Now z is the preimage between the rays for 1/7 and 2/7. By taking preimages in this interval, the external angles 9/56, 11/56, and 15/56 are obtained. The rays are drawn red. Rays with preperiodic angles, i.e., even denominators, land at preperiodic points in the dynamic plane, or at Misiurewicz points in the parameter plane. For these parameters, the critcal value is preperiodic under the iteration of fc(z). =How to compute angles of the wake ? =
 * algorithm from demo 4 page 1 of program Mandel: When the parameter c of a center is given, read off the angles or the Hubbard tree.
 * draw the filled Julia set Kc ( the parameter c is the center of this hyperbolic component)
 * locate $$\mp \beta_c$$ ( the angles 0 and 1/2 are landing at the fixed point $$\beta_c$$ and at its preimage $$- \beta_c$$) and
 * locate the two accesses to the 5-periodic point
 * Follow the orbit of these accesses and note the digits 0 or 1 according to the upper or lower sides of the Julia set. Dynamic rays in the upper part between these two rays have an angle between 0 and 1/2, the first binary digit is 0. Angles in the lower part are between 1/2 and 1, with first digit 1.
 * You may need to magnify subsets carefully. Or use the inverse spider algorithm.
 * Angles of the wake from program mandel - code
 * Combinatorial algorithm = Devaney's method

The paritition of dynamic plane:
 * the partition used in the definition of the kneading sequence: divide open unit disc into two parts: $$\theta/2$$  and $$(\theta + 1)/2$$ (the two inverse images of $\theta$ under angle doubling);
 * the open part containing the angle 0 is labeled 0
 * the other open part is labeled 1
 * the boundary gets the label ⋆

=Combinatorial algorithm = Devaney's method =

Devaney's method for finding external angles of primary buds

Steps :
 * start with rational rotation angle,
 * orbit of rotation angle under circle map
 * translation of orbit into itinerary
 * conversion of itinerary into binary expansion with repeating binary fraction
 * conversion of binary expansion to binary fraction
 * conversion to decimal fraction

Input : rational rotation angle

Outpout : external angle ( decimal or binary fraction )

C++
Here is C++ code from the program Mandel by Wolf Jung :

haskell
Code and description by Claude Heiland-Allen
 * mandelbrot web interface

Primary Bulb The child bulb of the period $$1$$ cardioid at internal angle $$\frac{p}{q}$$ has external angles:


 * $$[(.\overline{b_0 b_1 \ldots b_{q-3} 0 1}, .\overline{b_0 b_1 \ldots b_{q-3} 1 0})]$$

where


 * $$[b_0 b_1 \ldots = \operatorname{map} \left(\in \left(1 - \frac{p}{q}, 1\right)\right) \circ

\operatorname{iterate} \left(+\frac{p}{q}\right) \$ \frac{p}{q}]$$

Examples
One can check the results with
 * program Mandel by Wolf Jung
 * Web interface by Claude Heiland-Allen

1/2
The 1/2-wake of the main cardioid is bounded by the parameter rays with the angles:
 * 1/3 =  p01 = 0.(01)
 * 2/3 =  p10 = 0.(10)

The 1/2-wake of the main cardioid is bounded by the parameter rays with the angles 1/3 or  p01  and 2/3  or  p10.

The angle 1/3  or  p01 has  preperiod = 0  and  period = 2. The conjugate angle is 2/3  or  p10. The kneading sequence is A*  and the internal address is  1-2. The corresponding parameter rays land at the root of a satellite component of period 2. It bifurcates from period 1.

Important points
 * root point between period 1 and 2 = c = -0.75 = -3/4 = birurcation point for internal angle 1/2. Landing point of 2 external rays 1/3 and 2/3
 * center of period 2 components c = -1
 * tip of main antenna c = -2 = $$M_{1,1}$$. It is landing point of externa ray for angle $$0.01 = \frac{1}{2} = 0.5 $$

1/3
Orbit of rational angle 3/7 ( and position in subintervals): 1 / 3 = 0  2 / 3  = 0  0 / 3  = 1

so intinerary = 001

first external angle = 001 = 1 / 7

The 1/3-wake of the main cardioid is bounded by the parameter rays with the angles
 * 1/7 =  p001 = 0.(001)
 * 2/7 =  p010 = 0.(010)

note that
 * 1/7 = 0.(142857)= 0.1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428...

but decimal expansion is not important here. Only decimal ratio and binary floating point is important

$$\begin{cases} 0.(s_-) = 0.(001)_2 = \frac{1}{7} =  0.(142857)_{10} = wake     \\ 0.(s_+) = 0.(010)_2 = \frac{2}{7} = 0.(285714) = wake     \\ \end{cases} $$

1/4
The 1/4-wake of the main cardioid is bounded by the parameter rays with the angles
 * 1/15 or  p0001 or $$0.(0001)$$
 * 2/15 or  p0010 or $$0.(0010)$$

n/5
There are 4 period 5 wakes:
 * 1/5
 * 2/5
 * 3/5
 * 4/5

The 1/5-wake of the main cardioid is bounded by the parameter rays with the angles :
 * 1/31 =  p00001 =  0.(00001)
 * 2/31 =  p00010 =  0.(00010)

The 4/5-wake of the main cardioid is bounded by the parameter rays with the angles
 * 29/31 = p11101 =  0.(11101)
 * 30/31 = p11110 =  0.(11110)

3/7
Divide interval ( circle):

$$I = \big( 0, 1\big] $$

into 2 subintervals ( lower partition) :

$$I_0 = \big( \tfrac{0}{7}, \tfrac{4}{7} \big] $$

$$I_1 = \big(\tfrac{4}{7}, \tfrac{7}{7} \big] $$

Orbit of rational angle 3/7 ( and position in subintervals): 3 / 7 = 0  6 / 7  = 1  2 / 7  = 0  5 / 7  = 1  1 / 7  = 0  4 / 7  = 0  0 / 7  = 1

So itinerary is :

$$0101001 $$

One can convert it to number :

$$0101001 \to 0.(0101001)_2 = \frac{0101001}{1111111}_2 = \frac{41}{127}_{10}$$

The 3/7-wake of the main cardioid is bounded by the parameter rays with the angles
 * 41/127 =  p0101001  = 0.(0101001)
 * 42/127 =  p0101010  = 0.(0101010)

root point :

c = -0.606356884415893 +0.412399740175787 i

Orbit of 41/127 under doubling map modulo 1 computed with this program ( exponent = 7 and mpz_init_set_ui(n, 41); :

41/127 82/127 37/127 74/127 21/127 42/127 84/127

5/11


The 5/11-wake of the main cardioid is bounded by the parameter rays with the angles:
 * 681/2047 =  p01010101001  = 0.(01010101001)
 * 682/2047 =  p01010101010  = 0.(01010101010)

Center of period 11: c = -0.697838195122425 +0.279304134101366 i root point ( bond): c = -0.690059870015044  +0.276026482784614 i

Angled internal address: $$1 \xrightarrow{5/11} 11$$

n/17
The 1/17-wake of the main cardioid is bounded by the parameter rays with the angles
 * 1/131071 =  p00000000000000001 = 0.(00000000000000001)
 * 2/131071 =  p00000000000000010 = .(00000000000000010)

10/21
Parameter plane:
 * root point c = -0.733308614559099 +0.148209926690813 i
 * The 10/21-wake of the main cardioid is bounded by the parameter rays with the angles:
 * 699049/2097151 or  p010101010101010101001
 * 699050/2097151 or  p010101010101010101010

Dynamical plane
 * external rays of the wake do not bound critical sector
 * 699049/2097151 or  p010101010101010101001
 * 699050/2097151 or  p010101010101010101010
 * alpha fixed point ( period 1) z = -0.494415413112564 +0.074521133088087i

1/25
uiIADenominator = 25 Using MPFR-3.1.5 with GMP-6.1.1 with precision = 200 bits internal angle = 1/25 first external angle : period = denominator of internal angle = 25 external angle as a decimal fraction = 1/33554431 = 1 /( 2^25 - 1) External Angle as a floating point decimal number = 2.9802323275873758669905622896719661257256902970579355078320375103410059138021557873907862143745145987726127630324761942815747600646073471636075786857847163330961076713939572483194617724677755177253857e-8 external angle as a binary rational (string) : 1/1111111111111111111111111 external angle as a binary floating number in exponential form =0.10000000000000000000000001000000000000000000000000100000000000000000000000010000000000000000000000001000000000000000000000000100000000000000000000000010000000000000000000000001000000000000000000000001*2^-24 external angle as a binary floating number in periodic form =0.(0000000000000000000000001)

So 1/25-wake of the main cardioid is bounded by the parameter rays with the angles :
 * 0.0000000298 = 1/33554431 = 1 /( 2^25 - 1) = 0.(0000000000000000000000001)
 * 0,0000000596 = 2/33554431 = 2 /( 2^25 - 1) = 0.(0000000000000000000000010)

One can check it with Mandel

The angle 1/33554431  or  p0000000000000000000000001 has preperiod = 0  and  period = 25. The conjugate angle is 2/33554431  or  p0000000000000000000000010. The kneading sequence is AAAAAAAAAAAAAAAAAAAAAAAA*  and the internal address is 1-25. The corresponding parameter rays are landing at the root of a satellite component of period 25. It is bifurcating from period 1. Do you want to draw the rays and to shift c to the corresponding center?

The center is :

c = 0.265278321904606 +0.003712059989878 i    period = 25

12/25
wake 12/25
 * root c = -0.738203140939397 +0.124839088573366 i
 * The 12/25-wake of the main cardioid is bounded by the parameter rays with the angles
 * 11184809/33554431 or  p0101010101010101010101001  and
 * 11184810/33554431 or  p0101010101010101010101010.
 * the center of the satellite component c = -0.739829393511579 +0.125072144080321 i    period = 25

1/31


The 1/31-wake of the main cardioid
 * is bounded by the parameter rays with the angles:
 * 1/2147483647 =  p0000000000000000000000000000001  = 0.(0000000000000000000000000000001)
 * 2/2147483647 =  p0000000000000000000000000000010  = 0.(0000000000000000000000000000010)
 * root point : c = 0.260025517721190 +0.002060296266000 i
 * center c = 0.260025517721190 +0.002060296266000 i
 * principal Misiurewicz point c = 0.259995759918769 +0.001610271381965*i
 * has preperiod = 31, period = 1
 * is a landing point for 31 external rays
 * 2147483649/4611686016279904256 = 0000000000000000000000000000001p0000000000000000000000000000010 = .0000000000000000000000000000001(0000000000000000000000000000010)
 * the biggest baby Mandelbrot set has the kneading sequence AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAB*  corresponds to the internal address  1-31-32 . The period is 32. The smallest angles are 3/4294967295 = 0.(00000000000000000000000000000011) and 4/4294967295 = 0.(00000000000000000000000000000100)

On the dynamical plane :
 * The angle

13/34


The 13/34-wake of the main cardioid is bounded by the parameter rays with the angles
 * 4985538889/17179869183 =  p0100101001001010010100100101001001  = 0.(0100101001001010010100100101001001)
 * 4985538890/17179869183 =  p0100101001001010010100100101001010  = 0.(0100101001001010010100100101001010)

s = 0 i = 0 internal angle = 13 / 34 ea = 0 / 17179869183 ; m = 0 s = 1 i = 1 internal angle = 26 / 34 ea = 4294967296 / 17179869183 ; m = 4294967296 s = 0 i = 2 internal angle = 5 / 34 ea = 4294967296 / 17179869183 ; m = 0 s = 0 i = 3 internal angle = 18 / 34 ea = 4294967296 / 17179869183 ; m = 0 s = 1 i = 4 internal angle = 31 / 34 ea = 4831838208 / 17179869183 ; m = 536870912 s = 0 i = 5 internal angle = 10 / 34 ea = 4831838208 / 17179869183 ; m = 0 s = 1 i = 6 internal angle = 23 / 34 ea = 4966055936 / 17179869183 ; m = 134217728 s = 0 i = 7 internal angle = 2 / 34 ea = 4966055936 / 17179869183 ; m = 0 s = 0 i = 8 internal angle = 15 / 34 ea = 4966055936 / 17179869183 ; m = 0 s = 1 i = 9 internal angle = 28 / 34 ea = 4982833152 / 17179869183 ; m = 16777216 s = 0 i = 10 internal angle = 7 / 34 ea = 4982833152 / 17179869183 ; m = 0 s = 0 i = 11 internal angle = 20 / 34 ea = 4982833152 / 17179869183 ; m = 0 s = 1 i = 12 internal angle = 33 / 34 ea = 4984930304 / 17179869183 ; m = 2097152 s = 0 i = 13 internal angle = 12 / 34 ea = 4984930304 / 17179869183 ; m = 0 s = 1 i = 14 internal angle = 25 / 34 ea = 4985454592 / 17179869183 ; m = 524288 s = 0 i = 15 internal angle = 4 / 34 ea = 4985454592 / 17179869183 ; m = 0 s = 0 i = 16 internal angle = 17 / 34 ea = 4985454592 / 17179869183 ; m = 0 s = 1 i = 17 internal angle = 30 / 34 ea = 4985520128 / 17179869183 ; m = 65536 s = 0 i = 18 internal angle = 9 / 34 ea = 4985520128 / 17179869183 ; m = 0 s = 1 i = 19 internal angle = 22 / 34 ea = 4985536512 / 17179869183 ; m = 16384 s = 0 i = 20 internal angle = 1 / 34 ea = 4985536512 / 17179869183 ; m = 0 s = 0 i = 21 internal angle = 14 / 34 ea = 4985536512 / 17179869183 ; m = 0 s = 1 i = 22 internal angle = 27 / 34 ea = 4985538560 / 17179869183 ; m = 2048 s = 0 i = 23 internal angle = 6 / 34 ea = 4985538560 / 17179869183 ; m = 0 s = 0 i = 24 internal angle = 19 / 34 ea = 4985538560 / 17179869183 ; m = 0 s = 1 i = 25 internal angle = 32 / 34 ea = 4985538816 / 17179869183 ; m = 256 s = 0 i = 26 internal angle = 11 / 34 ea = 4985538816 / 17179869183 ; m = 0 s = 1 i = 27 internal angle = 24 / 34 ea = 4985538880 / 17179869183 ; m = 64 s = 0 i = 28 internal angle = 3 / 34 ea = 4985538880 / 17179869183 ; m = 0 s = 0 i = 29 internal angle = 16 / 34 ea = 4985538880 / 17179869183 ; m = 0 s = 1 i = 30 internal angle = 29 / 34 ea = 4985538888 / 17179869183 ; m = 8 s = 0 i = 31 internal angle = 8 / 34 ea = 4985538888 / 17179869183 ; m = 0 s = 0 i = 32 internal angle = 21 / 34 ea = 4985538888 / 17179869183 ; m = 0 s = 1 i = 33 internal angle = 34 / 34 ea = 4985538889 / 17179869183 ; m = 1 internal angle = 13/34 period = denominator of internal angle = 34 external angle as a decimal fraction = 4985538889/17179869183 = 4985538889 /( 2^34 - 1) external angle as a binary rational (string) : 100101001001010010100100101001001/1111111111111111111111111111111111 external angle as a binary floating number in exponential form =0.1001010010010100101001001010010010100101001001010010100100101001*2^-1 external angle as a binary floating number in periodic form =0.(0100101001001010010100100101001)

34/89
Using GMP-5.1.3 with precision = 256 bits internal angle = 34/89 period = denominator of internal angle = 89 external angle as a decimal fraction = 179622968672387565806504265/618970019642690137449562111 external angle as a binary rational (string) : 1001010010010100101001001010010010100101001001010010100100101001001010010100100101001001/11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 external angle as a binary floating number in exponential form =0.10010100100101001010010010100100101001010010010100101001001010010010100101001001010010010100101001001010010100100101001001010010100100101001010010010100100101001010010010100100101001010010010100101001001010010010100101001001010010100100101001001010010100101*2^-1 external angle as a binary floating number in periodic form =0.(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001)

1/128
uiIADenominator = 128 Using MPFR-4.0.2 with GMP-6.2.0 with precision = 200 bits internal angle = 1/128 first external angle : period = denominator of internal angle = 128 external angle as a decimal fraction = 1/340282366920938463463374607431768211455 = 1 /( 2^128 - 1) External Angle as a floating point decimal number = 2.9387358770557187699218413430556141945553000604853132483972656175588435482079339324933425313850237034701685918031624270579715075034722882265605472939461496635969950989468319466936530037770580747746862e-39 external angle as a binary rational (string) : 1/11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 external angle as a binary floating number in exponential form =0.10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000*2^-127 external angle as a binary floating number in periodic form =0.(00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001)

89/268


Using GMP-5.1.3 with precision = 320 bits internal angle = 89/268 period = denominator of internal angle = 268 external angle as a decimal fraction = 67754913930863876636420964942226524366713408170066250043659752013773168429311121/474284397516047136454946754595585670566993857190463750305618264096412179005177855 external angle as a binary rational (string) : 0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001 /1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 external angle as a binary floating number in exponential form =0.10010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001 001001001001001001001001001001001001001001001001001001*2^-2 external angle as a binary floating number in periodic form = 0.(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001)

G Pastor gave an example of external rays for which the resolution of the IEEE 754 is not sufficient :

$$\theta_{268}^- = 0.((001)^{88}0001)_2 = \frac{67754913930863876636420964942226524366713408170066250043659752013773168429311121}{474284397516047136454946754595585670566993857190463750305618264096412179005177855}$$

$$\theta_{268}^+ = 0.((001)^{88}0010)_2 = \frac{67754913930863876636420964942226524366713408170066250043659752013773168429311122}{474284397516047136454946754595585670566993857190463750305618264096412179005177855}$$

lists of roots

 * madore.org/~david math/mandpoints.dat
 * List of the Misiurewicz points

=See also =
 * the biggest island of the wake
 * dynamic external rays that land on the parabolic fixed point
 * subwake
 * symbolic dynamics
 * Mandelbrot symbolics / combinatorics web interface by Claude
 * Ordered orbits of the shift, square roots, and the devil's staircase BY SHAUN BULLETT
 * core entropy
 * Continuity of core entropy of quadratic polynomials by Giulio Tiozzo
 * Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set by Giulio Tiozzo

=References=
 * Bifurcation in Complex Quadrat ic Polynomial and Some Folk Theorems Involving the Geometry of Bulbs of the Mandelbrot Set by Monzu Ara Begum