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

TODO

Editor's note
This book is still under development. Please help us


on the boundary of main cardioidEdit

"constructing approximate Fatou coordinates for analytic maps f in a neighborhood of an f 0 (z) = z + z q+1 + ... with q > 1"[1][2][3]

  • "The first step in constructing Fatou coordinate for   consists in lifting   to a neighborhood of infinity by the coordinate change  " [4]

1/1Edit

Here c =1/4 is a cusp of main cardioid[5]

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


Max distance from parabolic orbits to the fixed point = 0.7071067811865476

1/2Edit

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 :

 

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

 

When numerator p and denominator q of internal angle are :

 

 

then internal angle in turns is :[7]

 

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

 

Note that Cheritat uses   not  

Then q iteration of quadratic map :

 

Number k :

  for some  

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

Take k term in the expansion of   denoted as   :

 

so

 

Evaluate multiplier

 

and C :

 

Let :

 

then prepared coordinate or pre-Fatou coordinate u are :

 

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 )

ReferencesEdit

  1. Classification of diffeomorphisms and ε-neighborhoods of orbits Maja Resman, University of Zagreb, Croatia
  2. The resurgent character of the Fatou coordinates of a simple parabolic germ Artem Dudko (SUNY), David Sauzin
  3. Dynamics of Holomorphic Maps: Resurgence of Fatou coordinates, and Poly-time Computability of Julia Sets Author: Dudko, Artem
  4. LIOUVILLE THEOREM WITH PARAMETERS: ASYMPTOTICS OF CERTAIN RATIONAL INTEGRALS IN DIFFERENTIAL FIELDS by MALGORZATA STAWISKA
  5. On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at 1/4 by Ludwik Jaksztas
  6. Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps ∗ Sebastian van Strien, University of Amsterdam, the Netherlands Tomasz Nowicki,
  7. PARABOLIC IMPLOSION A MINI-COURSE by ARNAUD CHERITAT