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

Max distance from parabolic orbits to the fixed point = 0.7071067811865476

1/3
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 )

=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