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

This book is still under development. Please help us 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"Classification of diffeomorphisms and ε-neighborhoods of orbits Maja Resman, University of Zagreb, CroatiaThe resurgent character of the Fatou coordinates of a simple parabolic germ Artem Dudko (SUNY), David SauzinDynamics of Holomorphic Maps: Resurgence of Fatou coordinates, and Poly-time Computability of Julia Sets Author: Dudko, Artem "The first step in constructing Fatou coordinate for consists in lifting to a neighborhood of infinity by the coordinate change " LIOUVILLE THEOREM WITH PARAMETERS: ASYMPTOTICS OF CERTAIN RATIONAL INTEGRALS IN DIFFERENTIAL FIELDS by MALGORZATA STAWISKA 1/1 Here c =1/4 is a cusp of main cardioidOn the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at 1/4 by Ludwik Jaksztas f(z) = z^2+1/4 Max distance from parabolic orbits to the fixed point = 0.7071067811865476 1/2 thumb| 1/3 thumbnail|right|Orbits near fixed point thumbnail|right|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 :PARABOLIC IMPLOSION A MINI-COURSE by ARNAUD CHERITAT 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 ) 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