Fractals/Iterations in the complex plane/Fatou coordinate for f(z)=z^2 + c

on the boundary of main cardioid

"constructing approximate Fatou coordinates for analytic maps f in a neighborhood of an f 0 (z) = z + z q+1 + ... with q > 1"

• "The first step in constructing Fatou coordinate for $f_{0}$ consists in lifting $f_{0}$ to a neighborhood of infinity by the coordinate change $z\to {\frac {-1}{qz^{q}}}$"

1/1

Here c =1/4 is a cusp of main cardioid

    f(z) = z^2+1/4

• domains for Fatou coordinate for f(z)=z+az^2+O(z^3)
• 
  Circle and cardioid domains
• 
  two Euclidean discs which are symmetric to each other with respect to the real line and whose boundaries intersect the boundary points of J

Max distance from parabolic orbits to the fixed point = 0.7071067811865476

1/2

1/3

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 )

$\lambda =e^{2\pi ip/q}$

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 :

$k=mq+1$ 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 :

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

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 )

References

• On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at 1/4 by Ludwik Jaksztas
• On the derivative of the Hausdorff Dimension of the Julia sets for z^2+c Ludwik Jaksztas, Michel Zinsmeister