# Intro

* Pariod doubling cascade in the Mandelbrot set ( 1/2 family) showed by the exponential mapping
* escape route 1/2
* period doubling
* Self-similarity in the Mandelbrot set shown by zooming on a round feature while panning in negative-X direction. The display center pans from (-1,0) to (-1.31,0) while the view magnifies from .5 x.5 to .12 x.12.
Periods of the period doubling cascade:
Angled internal address:
where:
* $\mathbf {MF} _{1/2}$ is the Myrberg-Feigenbaum point c = −1.401155 with external angles = (0.412454... , 0,58755...)
It is a part of Sharkovsky ordering

# External angles (combinatorial algorithms)
## External angle of the parameter rays landin on the root points of hyperbolic components from the 1/2 family
### Binary

The period also corresponds to the number of digits that make up the binary periodic fraction.
Note that :
* all angles are periodic binary fractions
* length of binary periodic part = period

### String Concatenation

MSS-harmonics ( Metropolis, Stein and Stein ):
$H_{MSS}(s_{1},s_{2})=(s_{1}s_{2},s_{2}s_{1})$
in the form of binary fraction:
* input : $0.(s_{1})$ and $0.(s_{2})$
* output : $0.(s_{1}s_{2})$ and $0.(s_{2}s_{1})$
I can be computed with c code :

# hyperbolic components ( numerical algorithms)
## Root points

    n Period = 2$^n$ Root point (c$_n$) 0 1 0.25 1 2 −0.75 2 4 −1.25 3 8 −1.3680989 4 16 −1.3940462 5 32 −1.3996312 6 64 −1.4008287 7 128 −1.4010853 8 256 −1.4011402 9 512 −1.401151982029 10 1024 −1.401154502237 ∞ −1.4011551890…

## Centers

Data by lkmitch:
Period 262144
x = -1.4011551890902510331817705605834440363471931682724714412452613678632418220750013209654238789188079407613058592287511922521315976742525764917468403339396930937 30785180509439998407177461884732043 f(x) = 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000001994 Approx Feigenbaum constant = 4.6692016090744525662279815203708867539460996466796182702147591041742162322183312521153913774503310728833354374225852275733454310057265948843688036767882792034 77482728794667534497622208785380761
Period 524288
-1.4011551890916651883071968100816546650318029614591678115404876967937578305271204258535401015561476261168939337667710591343229259782689769629427546629533666727 86737113473009338909783794873017929 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000008829391 4.6692016090968787947051350378647836776226665257418367260642987723027044088401013356743953120638886994058416817724607788627924785757422345903201225592627931049 69373245311653832559840230430927275
Period 1048576
-1.4011551890919680570294789310328705961503197610898601690317026045601400345356095836255726490446857492092529857212919383793658357679431376624985583329486967143 41390220891894934332628723842939248 -0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000022112864 4.6692016091016816811869601608458017299280888932440761709767910764158204329663666795299227205144754947240340906747557847995318754327409221893703408072446351537 83290121618647654581771235818384323

### c code from mandelbrot-numerics

m-feigenbaum program from mandelbrot-numerics library
It seems that double precision is not enough

### c++ code from Mandel

Centers of hyperbolic components are easier to compute then root points ( bifurcation points).
It is computed with cpp program using the code from Mandel

# Escape route 1/2

* escape route 1/2

# See also

* How to move on parameter plane ?
* real slice of the Mandelbrot set
** chaotic part main antenna is a shrub of $F_{1/2}$ family
* biaccessible points on the Mandelbrot set
* Feigenbaum_constants
* Commons Category:Period-doubling_bifurcation

# References

* Universal aspects of the period-doubling route to chaos | experiments, ODEs, and maps by Ross Dynamics Lab
* ↑ wikipedia: Period-doubling_bifurcation
* ↑ OPERATING WITH EXTERNAL ARGUMENTS IN THE MANDELBROT SET ANTENNA by G. Pastor, M. Romera, G. Alvarez, and F. Montoya
* ↑ On Finite Limit Sets for Transformations on the Unit interval by N. METROPOLIS, M. L. STEIN, AND P. R. STEIN
* ↑ M Nelson teaching. materials : part4.pdf
* ↑ External rays and the real slice of the Mandelbrot set by Saeed Zakeri ( paper in pdf from arxiv)
* ↑ ON BIACCESSIBLE POINTS OF THE MANDELBROT SET by SAEED ZAKERI
* ↑ wikipedia: Feigenbaum_constants