# Fractals/Iterations in the complex plane/island t

How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is **not accesible** from main cardioid ( M0) by a finite number of boundary crossing ?

# ExamplesEdit

## period 3 islandEdit

find angles of some child bulbs of period 3 component ( island) on main antenna with external rays (3/7,4/7)

Plane description :^{[1]}

-1.76733 +0.00002 i @ 0.05

One can check it using program Mandel by Wolf Jung :

The angle 3/7 or p011 has preperiod = 0 and period = 3. The conjugate angle is 4/7 or p100 . The kneading sequence is AB* and the internal address is 1-2-3 . The corresponding parameter rays are landing at the root of a primitive component of period 3.

## Period 5 islandsEdit

### on the main antennaEdit

There are 3 period 5 componenets on the main antenna ( checked with program Mandel by Wolf Jung ) :

- The angle 13/31 or p01101 has preperiod = 0 and period = 5. The conjugate angle is 18/31 or p10010 . The kneading sequence is ABAA* and the internal address is 1-2-4-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5.
- The angle 14/31 or p01110 has preperiod = 0 and period = 5. The conjugate angle is 17/31 or p10001 . The kneading sequence is ABBA* and the internal address is 1-2-3-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5.
- The angle 15/31 or p01111 has preperiod = 0 and period = 5. The conjugate angle is 16/31 or p10000 . The kneading sequence is ABBB* and the internal address is 1-2-3-4-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5. On the next root land 2 rays 16/33 i 17/33

#### 1-3-4-5Edit

Angled internal address in the form used by Claude Heiland-Allen:^{[2]}

1 1/2 2 1/2 3 1/2 4 1/2 5

or in the other form :

```
```

Where

- denotes Sharkovsky ordering which describes what is going on between period 1 and 3 on the real axis. It's first part is period doubling scenario from period 1 : denotes
- denotes period p component and infinite number of boundary crossing along 1/2 internal rays, for example denotes

So going from period 1 to period 5 on the main antenna means infinite number of boundary crossing ! It is to much so one has to start from main component of period 5 island.

External angles of this componnet can be computed by other algorithms.^{[3]}

##### 1/5Edit

Choose

```
```

First compute external angles for r/s wake :

and root of the island ( using program Mandel ) :

The angle 13/31 or p01101 has preperiod = 0 and period = 5. The conjugate angle is 18/31 or p10010 . The kneading sequence is ABAA* and the internal address is 1-2-4-5 . The corresponding parameter rays are landing at the root of a primitive component of period 5.

then in replace :

- digit 0 by block of length q from
- digit 1 by block of length q from

Result is :

theta_minus = 0.(0110101101011010110110010) theta_plus = 0.(0110101101011011001001101)

One can check it using program Mandel by Wolf Jung :

The angle 14071218/33554431 or p0110101101011010110110010 has preperiod = 0 and period = 25. The conjugate angle is 14071373/33554431 or p0110101101011011001001101 . The kneading sequence is ABAABABAABABAABABAABABAA* and the internal address is 1-2-4-5-25 . The corresponding parameter rays are landing at the root of a satellite component of period 25. It is bifurcating from period 5. Do you want to draw the rays and to shift c to the corresponding center?

## period 9 islandEdit

- the period 9 island in the antenna of the period 3 island

Check with Mandel:

The angle 228/511 or p011100100 has preperiod = 0 and period = 9. The conjugate angle is 283/511 or p100011011 . The kneading sequence is ABBABAAB* and the internal address is 1-2-3-6-9 . The corresponding parameter rays are landing at the root of a primitive component of period 9.

## period 18Edit

Period 18 island with angled internal address

```
```

whose:

- upper external angle is .(010101010101100101)
^{[4]} - center ( nucleus) c = -0.814158841137593 +0.189802029306573 i

Info from progrm Mandel :

The angle 87397/262143 or p010101010101100101 has preperiod = 0 and period = 18. The conjugate angle is 87386/262143 or p010101010101011010 . The kneading sequence is ABABABABABABABAAA* and the internal address is 1-2-16-18 . The corresponding parameter rays land at the root of a primitive component of period 18.

## period 32Edit

- +0.2925755 -0.0149977i @ +0.0005
^{[5]}

## period 44Edit

Plane parameters :^{[6]}

-0.63413421522307309166332840960 + 0.68661141963581069380394003021 i @ 3.35e-24

and external rays :

.(01001111100100100100011101010110011001100011) .(01001111100100100100011101010110011001100100)

One can check it with program Mandel by Wolf Jung :

The angle 5468105041507/17592186044415 or p01001111100100100100011101010110011001100011 has preperiod = 0 and period = 44. The conjugate angle is 5468105041508/17592186044415 or p01001111100100100100011101010110011001100100 . The kneading sequence is AAAABBBBABAABAABAABAABBBABABABAAABAAABABAAB* and the internal address is 1-5-6-7-8-10-13-16-19-22-23-24-26-28-30-34-38-40-43-44 . The corresponding parameter rays are landing at the root of a primitive component of period 44.

## period 52Edit

Plane parameters :^{[7]}

-0.22817920780250860271129306628202459167994 + 1.11515676722969926888221122588497247465766 i @ 2.22e-41

and external rays :

.(0011111111101010101010101011111111101010101010101011) .(0011111111101010101010101011111111101010101010101100)

One can check it with program Mandel by Wolf Jung :

The angle 1124433913621163/4503599627370495 or p0011111111101010101010101011111111101010101010101011 has preperiod = 0 and period = 52. The conjugate angle is 1124433913621164/4503599627370495 or p0011111111101010101010101011111111101010101010101100 . The kneading sequence is AABBBBBBBBBABABABABABABABABBBBBBBBBABABABABABABABAB* and the internal address is 1-3-4-5-6-7-8-9-10-11-13-15-17-19-21-23-25-27-28-29-30-31-32-33-34-35-37-39-41-43-45-47-49-51-52 . The corresponding parameter rays are landing at the root of a primitive component of period 52.

render using MPFR ( double precision is not enough)

## period 134Edit

# ReferencesEdit

- ↑ R2F(1/2B1)S by Robert P. Munafo, 2008 Feb 28.
- ↑ Patterns of periods in the Mandelbrot set by Claude Heiland-Allen
- ↑ Parameter rays of root points of period p components
- ↑ atom domains and newton basins by Claude Heiland-Allen
- ↑ R2.C(0) by Robert P. Munafo, 2012 Apr 16.
- ↑ Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen
- ↑ Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen