Last modified on 25 May 2014, at 11:06

Fractals/Iterations in the complex plane/Fatou coordinate

DescriptionEdit

Fatou functionEdit

Fatou function  \Psi(z)  :[3]

  • is defined inside petal ( attracting petal or repelling ), not on the neighbourhood of the fixed point
  • is a conformal function which satifies Abel's equation[4][5]
  • transforms f(z) to unit translation z \to z+1
  • maps petal to right half of plane in u coordinate.
  • unrolls invariant curvs ( orbits ) : maps "circles" to straight lines

u = \Psi(z)

NormalizatioEdit

Fatou coordinate can be normalized :[6]

 \Psi(z_{cr}) = 0


Parabolic fixed point is mapped to point at infinity on Riemann sphere

 \Psi(z_f) = \infty

Fatou coordinateEdit

Fatou coordinate u :

u = \Psi(z)

Description at Hyperoperations Wiki

  • what we call "Abel function"[7], they call it "Fatou coordinates".[8]
  • Fatou coordinates [9][10]
  • Shishikura perturbed Fatou coordinates [11]

ComputingEdit

0/1Edit

Domains for Fatou coordinate for f(z) = z+a_2z^2+O(z^3)

1/3Edit

Orbits near fixed point
Critical orbit for f(z)=z^2 + mz where p/q=1/3 with attracting and repelling vectors

It is based on : "PARABOLIC IMPLOSION A MINI-COURSE" by ARNAUD CHERITAT.


Let's take lambda form of quadratic map :

f(z) =  \lambda z + z^2

where \lambda is a multiplier of fixed point ( here fixed point is a origin z= 0 )

\lambda  =  e^{2 \pi i p/q}

When numerator p and denominator q of internal angle are :

p=1

q=3

then internal angle in turns is :[12]

 \theta = \frac{p}{q} = \frac{1}{3}

and stability index of fixed point ( internal radius ) is :

 |\lambda| = 1

Note that Cheritat uses  \rho not  \lambda

Then q iteration of quadratic map :

f^q(z) = f^3(z) =z^8 +4 \lambda z^7 +6 \lambda^2 z^6 +2 \lambda z^6 +4 \lambda^3 z^5 +6 \lambda^2 z^5 +\lambda^4 z^4 +6 \lambda^3 z^4 +\lambda^2 z^4+\lambda z^4+2 \lambda^4 z^3+2 \lambda^3 z^3+2 \lambda^2 z^3+\lambda^4 z^2 +\lambda^3 z^2 +\lambda^2 z^2 +\lambda^3 z

Number k :

 k = m q + 1 for some m > 0

if m=1 then k = q+1 = 4

Take k term in the expansion of f^q denoted as Cz^k :

Cz^k = Cz^4 = (\lambda^4+6*\lambda^3+\lambda^2+\lambda) z^4

so

C=  \lambda^4+6*\lambda^3+\lambda^2+\lambda

Evaluate multiplier

 \lambda = 0.86602540378444*i-0.5

and C :

C = 0.86602540378444*i+4.499999999999998

Let :

r = k - 1

then prepared coordinate or pre-Fatou coordinate u are :

u = \Psi(z) = \frac{-1}{r C z^r}


Here is Maxima CAS session ( where m is used for multiplier ) :

(%i1) f(z):=m*z + z^2;
(%o1) f(z):=m*z+z^2
(%i2) z3:f(f(f(z)));
(%o2) ((z^2+m*z)^2+m*(z^2+m*z))^2+m*((z^2+m*z)^2+m*(z^2+m*z))
(%i3) z3:expand(z3);
(%o3) z^8+4*m*z^7+6*m^2*z^6+2*m*z^6+4*m^3*z^5+6*m^2*z^5+m^4*z^4+6*m^3*z^4+m^2*z^4+m*z^4+2*m^4*z^3+2*m^3*z^3+2*m^2*z^3+m^4*z^2+m^3*z^2+m^2*z^2+m^3*z
(%i4) k:4;
(%o4) 4
(%i5) C:coeff(z3,z,k);
(%o5) m^4+6*m^3+m^2+m
(%i14) m:exp(2*%pi*%i/3);
(%o14) (sqrt(3)*%i)/2-1/2
(%i15) m:float(rectform(m));
(%o15) 0.86602540378444*%i-0.5
(%i19) C:float(rectform(ev(C)));
(%o19) 0.86602540378444*%i+4.499999999999998

Next session :

(%i1) z:zx+zy*%i;
(%o1) %i*zy+zx
(%i3) C:Cx+Cy*%i;
(%o3) %i*Cy+Cx
(%i4) r:3;
(%o4) 3
(%i5) u:-1/(r*C*z^r);
(%o5) -1/(3*(%i*Cy+Cx)*(%i*zy+zx)^3)
(%i8) u:expand(u);
(%o8) -1/(3*Cy*zy^3-3*%i*Cx*zy^3-9*%i*Cy*zx*zy^2-9*Cx*zx*zy^2-9*Cy*zx^2*zy+9*%i*Cx*zx^2*zy+3*%i*Cy*zx^3+3*Cx*zx^3)
(%i9) realpart(u);
(%o9) -(3*Cy*zy^3-9*Cx*zx*zy^2-9*Cy*zx^2*zy+3*Cx*zx^3)/((3*Cy*zy^3-9*Cx*zx*zy^2-9*Cy*zx^2*zy+3*Cx*zx^3)^2+(-3*Cx*zy^3-9*Cy*zx*zy^2+9*Cx*zx^2*zy+3*Cy*zx^3)^2)
(%i10) imagpart(u);
(%o10) -(3*Cx*zy^3+9*Cy*zx*zy^2-9*Cx*zx^2*zy-3*Cy*zx^3)/((3*Cy*zy^3-9*Cx*zx*zy^2-9*Cy*zx^2*zy+3*Cx*zx^3)^2+(-3*Cx*zy^3-9*Cy*zx*zy^2+9*Cx*zx^2*zy+3*Cy*zx^3)^2)

... ( to do )

ProgramsEdit

QFractEdit

QFract by INOU Hiroyuki and pictures

To build from the source code, you need :

Download source files from this page  :

First unpack the archive as follows

tar zcvf qfract-110725_2-src.tar.gz

Go to the program directory :

cd qfract-110725_2

and edit files :

  • Makefile,
  • config.h,
  • plugins/Makefile

to adjust your environment. For example in config.h change :

#define PLUGIN_PATH "/Users/inou/prog/qfract4/plugins"
#define COLORMAP_PATH "/Users/inou/prog/qfract4/colormaps"

for your own settings. Then to compile everything run from console :

make

To run the program from console :

./qfract

ReferencesEdit

  1. Tetration Forum : Parabolic Iteration
  2. Tetration Forum : Parabolic Iteration, again
  3. stackexchange : half-iterate-of-x^2 + c
  4. S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda : Holomorphic Dynamic. January 13, 2000 | ISBN-10: 0521662583 | ISBN-13: 978-0521662581
  5. wiki : Abel%27s_equation
  6. Dynamics in one complex variable: introductory lectures by John W. Milnor, page 7-6
  7. wikipedia : Abel function
  8. new results from complex dynamics at Tetration Forum
  9. Minicourse "Analytic classification of germs of generic families unfolding a parabolic point
  10. Fatou coordinate at Hyperoperations Wiki
  11. Shishikura perturbed Fatou coordinates
  12. PARABOLIC IMPLOSION A MINI-COURSE by ARNAUD CHERITAT