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

Description

"Near a non-degenerate 1-parabolic point z0, the orbits are attracted towards z0 on one side and repelled away on the other side. The parabolic basin of z0 is an open set containing z0 on the boundary and occupies most of  area near z0. So the local dynamics is relatively simple. However, once perturbed, it becomes the source of rich and delicate bifurcation phenomena. The points in the basin of unperturbed map can now escape through
the “gate” between the bifurcated fixed points, thus new recurrent orbits may be created. These “new” orbits depend extremely sensitively on the perturbation, and this causes a drastic change
of dynamics or the discontinuity of Julia sets. Also the perturbation into certain direction, such as z0 turning into irrationally indifferent fixed point (i.e. |λ| = 1 but λ is not a root of unity),
can create highly recurrent behavior, which leads into delicate questions, e.g. the linearizability problem or Cremer Julia sets which are not locally connected." 

Perturbation

Take a root point with rational internal argument $t={\frac {p}{q}}$. It has 2 equal simple continued fraction expansions ( representations):

$x=t=[a_{1},a_{2},a_{3},...,a_{n}]=[a_{1},a_{2},a_{3},...,a_{n}-1,1]$

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)

$x_{n}=[a_{1},a_{2},a_{3},...,a_{n}]$

is th n-th convergent of x. The convergents are ordered as follows:

$0<x_{2}<x_{4}<\ldots <x_{2m}<x<x_{2m+1}<\ldots <x_{3}<x_{1}\leq 1$

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):

$x=t=[a_{1},a_{2},a_{3},...,a_{n-1}]$

$x_{n}=t'(a)=[a_{1},a_{2},a_{3},...,a_{n-1},a]$

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):

(%i3) c:[0,3,5];
                            
(%i7) c5:cfdisrep(c);
                                       1
(%o7)                                -----
                                         1
                                     3 + -
                                         5
(%i8) ratsimp(c5);
                                      5
(%o8)                                 --
                                      16
(%i9) float(c5);
(%o9)                               0.3125
(%i10) 

type 2 on the main cardioid

• take second cf expansion ( even length)
• add one denominator a ( natural number):

$x=t=[a_{1},a_{2},a_{3},...,a_{n-1},1]$

$x_{k}=t''(a)=[a_{1},a_{2},a_{3},...,a_{n-1},1,a]$

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 then 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):

(%i4) x3:[0,2,1,5];
(%o4)                            [0, 2, 1, 5]
(%i5) cf:cfdisrep(x3);
                                       1
(%o5)                              ---------
                                         1
                                   2 + -----
                                           1
                                       1 + -
                                           5
(%i6) ratsimp(cf);
                                      6
(%o6)                                 --
                                      17
(%i7) 

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