# Fractals/Iterations in the complex plane/tip misiurewicz

< Fractals

# introductionEdit

parameter lane, wake 1/3 with rays landing on the principal Misiurewicz points

## key words definitionsEdit

Parts of the parameter plane

- shrub
- wake
- limb
- Misiurewicz point
- tip (point) = end point = branch tip = the tip of the spoke = terminal point of the branche
^{[1]}= tip of the midget^{[2]}"A point in the Mandelbrot Set that is at the end of a filament (as opposed to a branch point); a point from which there is only one path to other points in the Mandelbrot Set."- The first tip = ftip
- the last tip = ltip

- tip (point) = end point = branch tip = the tip of the spoke = terminal point of the branche

## notationEdit

# taskEdit

- find Misiurewicz point
- preperiod and period
- c value

- find angles of external ray that land on it

# algorithmsEdit

## PastorEdit

"the external argument can be calculated as the limit of the arguments of the structural components of the branches 1, 11, 111,..., with periods 4, 5, 6,..., that is, the limit of .(0011), .(00111), .(001111),..., or the limit of .(0100), .(01000), .(010000), .... Hence, ftip(1/3) = .00(1) = .01(0), that are two equal values. "^{[3]}

## ClaudeEdit

Method by Claude

Steps of the algorithm:

- find angles of the wake
- find angles of principal Misiurewicz point M
- find angles of spoke's tips using: "The tip of each spoke is the longest matching prefix of neighbouring angles, with 1 appended"

### 1/3Edit

3 angles landing on M:

0.001(010) 0.001(100) 0.010(001)

The tip of each spoke is the longest matching prefix of neighbouring angles, with 1 appended

0.001(010) // 9/56 = 0.160(714285) 0.0011 // ltip = 3/16 = 0.1875 0.001(100) // 11/56 = 0.196(428571) 0.01 // ftip = 1/4 = 0.25 0.010(001) // 15/56 = 0.267(857142)

Check with program Mandel :

The angle 3/16 or 0011 has preperiod = 4 and period = 1. Entropy: e^h = 2^B = λ = 1.59898328 The corresponding parameter ray lands at a Misiurewicz point of preperiod 4 and period dividing 1. Do you want to draw the ray and to shift c to the landing point?

c = -0.017187977338350 +1.037652343793215 i period = 0

The angle 1/4 or 01 has preperiod = 2 and period = 1. Entropy: e^h = 2^B = λ = 1.69562077 The corresponding parameter ray lands at a Misiurewicz point of preperiod 2 and period dividing 1. Do you want to draw the ray and to shift c to the landing point?

M_{2,1) = c = -0.228155493653962 +1.115142508039937 i

The angle 1/6 or 0p01 has preperiod = 1 and period = 2. The corresponding parameter ray lands at a Misiurewicz point of preperiod 1 and period dividing 2. Do you want to draw the ray and to shift c to the landing point?

c = -0.000000000000000 +1.000000000000000 i period = 10000

# examplesEdit

## 1/2Edit

```
```

# ReferencesEdit

- ↑ Terminal Point by Robert P. Munafo, 2008 Mar 9.
- ↑ mathoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
- ↑ G. Pastor, M. Romera, G. Alvarez, J. Nunez, D. Arroyo, F. Montoya, "Operating with External Arguments of Douady and Hubbard", Discrete Dynamics in Nature and Society, vol. 2007, Article ID 045920, 17 pages, 2007. https://doi.org/10.1155/2007/45920