Fractals/Iterations in the complex plane/Fatou coordinate

Description

• Parabolic iteration

Fatou function

Fatou function $\Psi(z)$ :

• is defined inside petal ( attracting petal or repelling )
• is a conformal function which satifies Abel's equation
• transforms f(z) to unit translation $z \to z+1$
• maps petal to right half of plane

$u = \Psi(z)$

Fatou coordinate

Fatou coordinate u :

$u = \Psi(z)$

Description at Hyperoperations Wiki

• what we call "Abel function", they call it "Fatou coordinates".
• Fatou coordinates
• Shishikura perturbed Fatou coordinates

Computing

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 :

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

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 )

Programs

QFract

QFract by INOU Hiroyuki and pictures

To build from the source code, you need :

• Qt 4.5 (or later?) from http://qt.nokia.com/ and install.
• Boost C++ Libraries (http://www.boost.org/).

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"

Then to compile everything :

make

To run program :

./qfract

References