# Fractals/Iterations in the complex plane/Mandelbrot set/chaotic part

Structure of Mandelbrot set's real slice
* Periodic part
* chaotic part = Main antenn
Chaotic part consist of chaotic bands deliminated by primary separators ( Misiurewicz points )

# ordering and sequences of hyperbolic componnets
## ordering

ordering of hyperbolic componnets
* Sharkovsky ordering shows
** ordering of hyperbolic componnets along real axis ( real slice
*** order of chaotic bands and inside periodic part
*** inside chaotic band
**** period of the biggest island inside chaotic band ( first number )
**** sequence of smaller islands periods ( next numbers)
* other orders inside chotic bands

## sequences
### Sharkovsky sequence inside chaotic bands
# Separators
## primary separator

* band-merging point = point merging 2 chaotic bands $B_{i-1}$ and $B_{i}$ are Misiurewicz points $M_{k,p}$
with
* $k=2^{i}$ = preperiod of critical value
* $p=2^{i-1}$ = period of critical value = period of critical point
Examples:
* $c=m_{0}=M_{1,1}\ \ =-2$ = tip of main antenna , external angle = 1/2
* $c=m_{1}=M_{2,1}\ \ =-1.543689012692076...$, external angles 5 and 7/12
* $c=m_{2}=M_{4,2}\ \ =-1.430357632451307...$, external angles 33 and 47/80
* $c=m_{3}=M_{8,4}\ \ =-1.407405118164702...$, external angles 1795 and 2557/4352
* $c=m_{4}=M_{16,8}\ =-1.402492176358564...$
* $c=m_{5}=M_{32,16}=-1.401441494253588...$
* $c=m_{6}=M_{64,32}=-1.401216504309415..$
* $c=m_{7}=M_{128,64}=-1.401168320839301..$
* $c=m_{8}=M_{256,128}=-1.401158001505211..$
* $c=m_{9}=M_{512,256}=-1.401155791424613..$
* $c=m_{10}=M_{1024,512}=-1.401155318093230..$
* $c=m_{11}=M_{2048,1024}=-1.401155216720152..$
* $c=m_{\infty }=M_{\infty ,\infty }=-1.4011551890......$ = Feigenbaum point = MF = the Myrberg-Feigenbaum point
So the Myrberg-Feigenbaum point is a limit of a series of band-merging points $m_{n}$:
    $\lim _{n\to \infty }m_{n}=m_{\infty }=c_{F}$
In the text form:
Differences $d=m_{i}-m_{i+1}$ or width of the band:
Ratio $r_{i-1}={\frac {m_{i-1}-m_{i-2}}{m_{i}-m_{i-1}}}$ tends to first Feigenbaum constant

## secondary separator

* tree separators ( tree is a subset of band )

# Chaotic bands
## B0

First chotic band = B0
primary separators ( Misiurewicz points )
* $c=m_{0}=M_{1,1}\ \ =-2$ = tip of main antenna , external angle = 1/2
* $c=m_{1}=M_{2,1}\ \ =-1.543689012692076...$, external angles 5 and 7/12
    $m_{0}\prec B_{0}\prec m_{1}$

### 0 Sharkovsky sequence

    $m_{0}\prec (2k+1)\prec m_{1}$
    $m_{0}\prec 3\prec 5\prec 7\prec ...\prec m_{1}$

## Bs

s-th chotic band = $B_{s}$
    $m_{s}\prec B_{s}\prec m_{s+1}$

### s-th Sharkovsky sequence

    $m_{s}\prec (2k+1)*2^{s}\prec m_{s+1}$
    $m_{s}\prec 3*2^{s}\prec 5*2^{s}\prec 7*2^{s}\prec ...\prec m_{s+1}$

# Islands

Islands

## real

real component centers of Mandelbrot Set (period less than 16) by Jay R. Hill
probably computed using :
* FINDPERC.EXE : uses 80-bit floating point arithmetic to achieve 19 digit results.
* FINDPERM : uses 57 digits with the extended arithmetic
* package MFLOAT by Friedrich Kaufmann. It displays results to 37 digits.

# See also

* real slice of Mandelbrot set
* JAKOBSON’S THEOREM
** proof by Jean Christophe Yoccoz
** math.stackexchange question: m-set-interior-point-probability-on-the-real-axis

# References

* ↑ fractalforums.org : primary-separators-misiurewicz-points-of-the-mandelbrot-sets-real-slice
* ↑ [ Pastor96 ] = G. Pastor, M. Romera and F. Montoya, "An approach to the ordering of one-dimensional quadratic maps", Chaos, Solitons and Fractals, 7/4 (1996), 565-584. Preprint
* ↑ A scaling constant equal to unity in 1D- quadratic maps by M. ROMERA, G. PASTOR and F. MONTOYA
* ↑ G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71
* ↑ G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71