# Fractals/Iterations in the complex plane/p misiurewicz

How to compute external angles of principal Misiurewicz point of wake p/q using Devaney's algorithm ?

# names

* a principal Misiurewicz points of the wake ( or the limb or the shrub )
* the main node of the shrub
* the hub = center part of shrub ( Pastor notation), the point where spokes join
* a junction point of q spokes which is attached directly to the p/q bulb ( Devaney notation )
* "the first dominating α-Misiurewicz point in M p/q , i.e., the one of lowest pre-period"
* Eye of elephant resting on internal angle 1/4 of main cardioid ( Curtis McMullen)

# notes

Principal misiurewicz point of p/q-wake is $c=M_{q,1}$
* it has q arms ( spokes, branches) numbered from 0 to q-1 in a clockwise direction
* it is a landing point for q external angles
* critical point has preperiod q and period p = 1 under complex quadratic map for $c=M_{q,1}$
External angles of q rays landing on $c=M_{q,p}$
* in the binary expansion length of preperiodic and periodic part is q
* period and preperiod of angle under doubling map is q
Important differences:
* Romero-Pastor notation uses q/p not p/q
* Preperiod: the usual convention is to use the preperiod of the critical value $z_{cv}=c$, not preperiod of critical point $z_{cr}=0$. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane

# introduction

How to work with the shift map ?
If length of string s is q then

## shifting q digits in blocks of b digits

Note that

### q=5 b=4
# Algorithm

Algorithm is based on the Theorem 5.3 in: Geometry of the Antennas in the Mandelbrot Set by R L Devaney and M Moreno-Rocha, April 11, 2000
External Angles of Hub ( see section 3.9 of the Book by Claude) or spoke
The $p/q$ bulb ( = hyperbolic component) has 2 external angles landing on it's root point (bond) :
such that :
These angles have :
* repeating binary expansion denoted by round brackets or overline
* length of repeating ( periodic ) part is $q$
Other names of these angles are angles of the wake.
The junction point of its hub ( principal Misiurewicz point) $M$ has external angles in increasing order
where
* s is a finite string of q binary digits = s consist of q binary digits = length(s)= q
* $\sigma$ is the shift map
* ${\frac {p}{q}}$ fraction has Farey parents a/b and r/s
* b is a denominator of lower Farey parent
${\begin{cases}{\frac {a}{b}}<{\frac {p}{q}}<{\frac {r}{s}}\\{\frac {a}{b}}\oplus {\frac {r}{s}}={\frac {p}{q}}\\\end{cases}}$
Implementation:

## input and output

* input : 2 external angles of the wake $p/q$
* output : $q$ external angles of principal Misiurewicz point ( hub)

## steps

* input = $p/q$
** check input
*** both p and q are:
**** integers
**** > 0
*** proper fraction : p < q
*** irreducible fraction = in lowest terms ( An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 A fraction is in lowest terms when the greatest common factor (GCF) of the numerator and denominator is 1 )
** if input is good then there are $q$ angles to compute
* compute 2 angles of the wake : $s_{-}(p/q)$ and $s_{+}(p/q)$
* compute first 2 of q angles : $0.s_{-}(s_{+})$ and $0.s_{+}(s_{-})$
* compute last $q-2$ angles
** compute Farey parents of $p/q$
** compute $q-p$
** ( to do )

# Examples by wake
## 1/2

So here are 4 angles (q+2) in increasing order :
* 2 rays landing on the root point ( s+ and s- )
* q=2 rays landing on the Misiurewicz point
Farey parents of 1/2 are 0/1 and 1/1
The denominator of smaller parent :
$M_{2,1}=c=-1.543689012692076+0.000000000000000i$
Compare with
* $\mathbf {MF} _{1/2}$ is the Myrberg-Feigenbaum point c = −1.401155 with external angles = (0.412454... , 0,58755...)
${\begin{cases}0.(s_{-})=0.(01)={\frac {1}{3}}={\frac {4}{12}}=0.(3)=wake\\0.s_{-}(s_{+})=0.01(10)={\frac {5}{12}}=0.41(6)=PrincipalMis=M_{2,2}\\0.0(1)={\frac {1}{2}}={\frac {6}{12}}=0.5=tip=M_{1,1}=c=-2\\0.s_{+}(s_{-})=0.10(01)={\frac {7}{12}}=0.58(3)=PrincipalMis=M_{2,2}\\0.(s_{+})=0.(10)={\frac {2}{3}}={\frac {8}{12}}=0.(6)=wake\\\end{cases}}$

## 1/3

The ${\frac {p}{q}}={\frac {1}{3}}$ bulb ( = period 3 hyperbolic component) has 2 external angles landing on it's root point (bond) :
such that :
Principal Misiurewicz point $M_{3,1}$ of ${\frac {1}{3}}$ wake is a landing point for $q=3$ external angles. It is denoted by
where :
* first number denotes preperiod
* second number denotes period
Two of them one can easly compute from angles the wake :
such that :
So the problem is to compute only 1 ray.
First find Farey parents of ${\frac {1}{3}}$
such that :
Take denominator of smaller parent :
and compute last fraction.
First find periodic part :
* remember that shift map works on the infinite sequence
* take only first q digits from result of shift map
then last angle is :
So here are 5 angles (q+2) in increasing order :
One can check it with Mandel:

## 1/4

The ${\frac {p}{q}}={\frac {1}{4}}$ bulb ( = period 4 hyperbolic component) has 2 external angles landing on it's root point (bond) :
Principal Misiurewicz point $M$ of ${\frac {1}{4}}$ wake is a landing point for $q=4$ external angles.
Two of them one can easly compute from angles the wake :
So the problem is to compute only $q-2=2$ rays.
First find Farey parents of ${\frac {1}{4}}$
Take denominator of lower parent :
and compute last fractions.
First find periodic parts for n :
then 2 last angles are :
So here are $q+2=6$ angles in increasing order :

## 2/5

The ${\frac {p}{q}}={\frac {2}{5}}$ bulb ( = period 5 hyperbolic component) has 2 external angles landing on it's root point (bond) :
Farey parents of 2/5 are 1/3 and 1/2
${\frac {1}{3}}\oplus {\frac {1}{2}}={\frac {1+1}{2+3}}={\frac {2}{5}}$
so denominator of smaller parent is b = 3.
Angles in the symbolic form

## 1/7

The wake 1/7 of main cardioid
External rays:
* 1/127 = 0.(0000001) = 0.0078740157480315 = wake
* 129 /16256 = 0.0000001(0000010) = 0.00793553149606299 = pM_{7,1}
* 131 /16256 = 0.0000001(0000100) = 0.00805856299212598 = pM
* 135 /16256 = 0.0000001(0001000 = 0.00830462598425197 = pM
* 143 /16256 = 0.0000001(0010000) = 0.00879675196850394 = pM
* 159 /16256 = 0.0000001(0100000) = 0.00978100393700787 = pM
* 191 /16256 = 0.0000001(1000000) = 0.01174950787401575 = pM
* 255 /16256 = 0.0000010(0000001) = 0.0156865157480315 = pM
* 1/64 = 0.000000(1) = 0.015625 = M_{6,1} = longest tip
* 2/127 = 0.(0000010) = 0.01574803149606299 = wake

## 3/7

Wake 3/7 and its principal Misiurewicz point (hub)
Check with Mandel

## 5/11

Check with Mandel:
The result is a center of period 11 satelite component c = -0.697838195122425 +0.279304134101366 i period = 11
The result is a principal Misiurewicz point of wake 5/11 M_{11,1} = c = -0.724112682973574 +0.286456567676711 i

## 12/25

Landing point = principal Misiurewicz point

## 1/31

* wake -> limb 1/31

## 8/47 = 16/94

Haskell output
c output

## 15/94

First angle of the hub is:

# Code
## Haskell code

Save it as a bh.hs and use it from console in an interactive way :

# Compare with

* program Mandel by Wolf Jung

# References

* ↑ wikipedia : Misiurewicz point
* ↑ wikipedia : Misiurewicz point
* ↑ Operating with External Arguments of Douady and Hubbard by G. Pastor, M. Romera, G. Alvarez, J. Nunez, D. Arroyo, and F. Montoya
* ↑ Geometry of the Antennas in the Mandelbrot Set by RL Devaney and M Moreno-Rocha
* ↑ EXTENSIONS OF HOMEOMORPHISMS BETWEEN LIMBS OF THE MANDELBROT SET BODIL BRANNER AND NURIA FAGELLA
* ↑ Geometry of the Antennas in the Mandelbrot Set (2000) by R. L. Devaney , M. Moreno-rocha
* ↑ Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen
* ↑ finding_parents_in_the_farey_tree by Claude Heiland-Allen

# See also

* Repulsive Points by Faber
* Newtons method for misiurewicz points by Claude Heiland-Allen
* enumeration_of_misiurewicz_points by Claude Heiland-Allen