# Fractals/Iterations in the complex plane/pperturbation

Parabolic perturbation of a root point is a way of peturbating this root into certains other nearby roots

# Perturbation

Take a root point with rational internal argument $t={\frac {p}{q}}$. It has 2 equal simple continued fraction expansions ( representations):
where
* internal argument $t$ is a proper fraction: $t<1$ so first term $a_{0}$ is equal to zero: $a_{0}=0$
* when $b_{n}=1$ for all $n$ the expression is called a simple continued fraction
For any n smaller then then length of the expansion ( using one of the 2 equal expansions)
is th n-th convergent of x. The convergents are ordered as follows:

## First order

* type 1 and 2 = on the hyperbolic component ( parent component)
* type 3 and 4 = on the satellite ( child component)

### type 1 on the main cardioid

* take first ( canonical) cf expansion (with odd length) of t
* add one denominator a ( natural number):
Note:
* length of the expansion $x_{n}$ is even: n = 2*m where m is a positive natural number
* rotation number is a bit less then t: $t'(a)<t$
* $t'(a)\nearrow t\quad as\quad a\to +\infty$
Examples
Fat Basilica Julia set
* $x=t={\frac {1}{2}}=[2]=0.5$ and c = -0.75
* $x_{2}=t'(a)=[2,a]$
* $x_{2}=t'(5)=[2,5]={\frac {5}{11}}=0.4545454545454545$ and c = -0.690059870015044 +0.276026482784614 i. Root point of the wake 5/11
* $x_{2}=t'(10)=[2,10]={\frac {10}{21}}=0.4761904761904761$ and c = -0.733308614559099 +0.148209926690813 i
* $x_{2}=t'(100)=[2,100]={\frac {100}{201}}=0.4975124378109453$ and c = -0.749816792870443 +0.015628223336210 i
* $x_{2}=t'(1000)=[2,1000]={\frac {1000}{2001}}=0.4997501249375312$ and c = -0.749998151299478 +0.001570009708645 i
* $t={\frac {1}{2}}$
* $t'(5)=[2,5]={\frac {5}{11}}$
Fat Douady Rabbit
* $x=t={\frac {1}{3}}=[3]=0.33333...$ and c = -0.125000000000000 +0.649519052838329 i
* $x_{2}=t'(a)=[3,a]$
* $x_{2}=t'(5)=[3,5]={\frac {5}{16}}=0.3125$ and c = -0.014565020885908 +0.638716461552280 i
* $x_{2}=t'(10)=[3,10]={\frac {10}{31}}=0.3225806451612903$ and c = -0.067170580141901 +0.646596204019795 i
* $x_{2}=t'(100)=[3,100]={\frac {100}{301}}=0.3322259136212624$ and c = -0.118980261815329 +0.649487648552261 i
* $x_{2}=t'(1000)=[3,1000]={\frac {1000}{3001}}=0.3332222592469177$ and c = -0.124395662683559 +0.649518736524089 i
* $x=t={\frac {1}{3}}=[3]$
How to compute t in Maxima CAS ( here ona should add a0 term):

### type 2 on the main cardioid

* take second cf expansion ( even length)
* add one denominator a ( natural number):
Note:
* length of the expansion is odd: k = n+1 = 2*m+1 where m is a positive natural number
* rotation number $t''$ is a bit greater than t: $t<t''(a)$
* $t''(a)\nearrow t\quad as\quad a\to +\infty$
Examples
Fat Basilica Julia set
* $x=t={\frac {1}{2}}=[1,1]=0.5$ and c = -0.75
* $x_{3}=t''(a)=[1,1,a]$
* $x_{3}=t''(5)=[1,1,5]={\frac {6}{11}}=0.5454545454545454$ and c = -0.690059870015044 -0.276026482784614 i
* $x_{3}=t''(10)=[1,1,10]={\frac {11}{21}}=0.5238095238095238$ and c = -0.733308614559099 -0.148209926690813 i
* $x_{3}=t''(100)=[1,1,100]={\frac {101}{201}}=0.5024875621890548$ and c = -0.749816792870443 -0.015628223336210 i
* $x_{3}=t''(1000)=[1,1,1000]={\frac {1001}{2001}}=0.5002498750624688$ and c = -0.749998151299478 -0.001570009708645 i
Maxima CAS code ( here ona should add a0 term):
Fat Douady Rabbit
* $x=t={\frac {1}{3}}=[2,1]=0.33333...$ and c = -0.125000000000000 +0.649519052838329 i
* $x_{3}=t''(a)=[2,1,a]$
* $x_{3}=t''(5)=[2,1,5]={\frac {6}{17}}=$ and c = -0.232901570671607 +0.639465024433325 i
* $x_{3}=t''(10)=[2,1,10]={\frac {11}{32}}=$ and c = -0.182114258418529 +0.646704689279094 i
* $x_{3}=t''(100)=[2,1,100]={\frac {101}{302}}=$ and c = -0.131011849556424 +0.649487772656967 i
* $x_{3}=t''(1000)=[2,1,1000]={\frac {1001}{3002}}=$ and c = -0.125604257709865 +0.649518736649880 i

### type 3 on the period 2 satellite component

Fat Basilica Julia set
* on main cardioid $x=t={\frac {1}{2}}=0.5$ and c = -0.75
* on period 2 component ( internal ray 1/2)
** $t'''(a)={\frac {1}{a}}$ is a root point between period 2 and period 2*a
** $t'''(5)={\frac {1}{5}}=0.2$ and c = -0.922745751406263 +0.237764129073788 i
** $t'''(10)={\frac {1}{10}}=0.1$ and c = -0.797745751406263 +0.146946313073118 i
** $t'''(100)={\frac {1}{100}}=0.01$ and c = -0.750493317892932 +0.015697629882328 i
** $t'''(1000)={\frac {1}{1000}}=0.001$ and c = -0.750004934785966 +0.001570785991390 i

### type 3 on the Douady Rabbit satellite ( period 3 component)

Fat Douady Rabbit
* on main cardioid: $t={\frac {1}{3}}=[2,1]=0.33333...$ and c = -0.125000000000000 +0.649519052838329 i
* on period 3 component with root point on the internal angle = 1/3:
** $t'''(a)={\frac {1}{a}}$ is a root point between period 3 and period 3*a
** $t'''(5)={\frac {1}{5}}=0.2$ and c = -0.035468843775407 +0.713230932890222*I
** $t'''(10)={\frac {1}{10}}=0.1$ and c = -0.069357410041421 +0.667567542415601*I
** $t'''(100)={\frac {1}{100}}=0.01$ and c = -0.118968172732931 +0.649711213179649*I
** $t'''(1000)={\frac {1}{1000}}=0.001$ and c = -0.124395505045425 +0.649520981010889 i

### type 4 on the period 2 satellite component

Fat Basilica Julia set
* on main cardioid $x=t={\frac {1}{2}}=0.5$ and c = -0.75
* on period 2 component ( internal ray 1/2)
** $t''''(a)=-{\frac {1}{a}}={\frac {1}{1}}-{\frac {1}{a}}={\frac {a-1}{a}}$ where c is a root point between period 2 and period 2*a
** $t''''(5)=-{\frac {1}{5}}={\frac {4}{5}}$ and c = -0.922745751406263 -0.237764129073788 i
** $t''''(10)=-{\frac {1}{10}}={\frac {9}{10}}$ and c = -0.797745751406263 -0.146946313073118 i
** $t''''(100)=-{\frac {1}{100}}={\frac {99}{100}}$ and c = -0.750493317892932 -0.015697629882328 i
** $t''''(1000)=-{\frac {1}{1000}}={\frac {999}{1000}}$ and c = -0.750004934785966 -0.001570785991390 i

### type 4 on the Douady Rabbit satellite

Fat Douady Rabbit
* on main cardioid: $t={\frac {1}{3}}=0.33333...$ and c = -0.125000000000000 +0.649519052838329 i
* on period 3 component with root point on the internal angle = 1/3:
** $t''''(a)=-{\frac {1}{a}}={\frac {1}{1}}-{\frac {1}{a}}={\frac {a-1}{a}}$ where c is a root point between period 3 and period 3*a
** $t''''(5)=-{\frac {1}{5}}={\frac {4}{5}}$ and c = -0.216358795928715 +0.719846780290728 i
** $t''''(10)=-{\frac {1}{10}}={\frac {9}{10}}$ and c = -0.182180023389255 +0.668744570272412 i
** $t''''(100)=-{\frac {1}{100}}={\frac {99}{100}}$ and c = -0.131051918394844 +0.649712528934645 i
** $t''''(1000)=-{\frac {1}{1000}}={\frac {999}{1000}}$ and c = -0.125604696369978 +0.649520982328093 i

## Compare

* perturbation technique in the deep Mandelbrot zoom
* the parabolic Mandelbrot set M1 = Mandelbrot set -a^2 plane for function $f_{a}(z)=z+a+{\frac {1}{z}}$ having double fixed point of mutiplier = +1 at infinity and critical points at 1 and -1
** Carsten Lunde Petersen: Dynamics preserving homeomorphism between M1 and M

## References

* Nonconformal perturbations of z -> z 2 + c: the 1:3 resonance, Nonlinearity 17 (2004) 765 - 789 by Henk Bruin and Martijn van Noort
* ↑ The renormalization for parabolic fixed points and their perturbation by Hiroyuki Inou and Mitsuhiro Shishikura. May 5, 2006
* ↑ Dan Erik Krarup Sorensen: Complex Dynamical Systems: Rays and non-local connectivity. Ph. D. Thesis 1994, Mathematical Insitute TUD