Fractals/Iterations in the complex plane/Koenigs coordinate

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Description[edit]

Animated periodic cycle

Koenigs[1] coordinate[2] are used in the basin of attraction of finite attracting (not superattracting) point (cycle),[3][4]

Definition[edit]

  • a rational map f of degree at least two
  • a fixed point
  • multiplier of the fixed point is
  • fixed point is attracting but not superattracting
  • = the attracting basin of the fixed point zero under function . In other words interior of component containing fixed point = the open set consisting of all points whose orbits under f converge to 0.


It is aproximated by normalized iterates :

It can be defined by the formula :


Function f is locally conjugate to the model linear map[5]


Examples[edit]

Dynamics for quadratic 1D polynomials fc(z)=z²+c

// Created by inigo quilez - iq/2013
// License Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
// 
//-----------------https://www.shadertoy.com/view/MdX3zN-------------------------------------------
//
// Dynamics for quadratic 1D polynomials fc(z)=z²+c
//
// * Orange: the Fatou set Kc. 
// * Black: the Julia set Jc.
// * Checkerboard distortion: the Boettcher map phi(z). 
// * Checkerboard shadowing: the gradient of the Green's function, log|phi(z)|
// * Blue: the two fixed points. 
// * Green, the period 2 fixed points.
// * White: c
// * Yellow: the Koening coordinates
//
// Some theory:
//
// * c (white) belongs to Kc (orange), for these are all connected Julia sets. 
//
// * When both fixed points (blue) are in Jc but not in Kc, or in other words, when both points
//   are repeling (derivative of fc(z) is bigger than one), c does not belong to the Mandelbrot 
//   set's main cardioid, but to bulbs of higher period. In that case Kc (orange) is made of several 
//   branches (as many as the period of the bul)
//
// * When one of the two fixed points (blue dots) is inside Kc, meanins it is attractive (derivative
//   of fc(z) < 1), then c belongs to the main cardiod of the Mandelbrot set, and Kc is a single piece 
//   shape.
//
// * When the period 2 fixed points are always repelling (belong to Jc, not to Kc) except for the sets 
//   that have c belonging to the period-2 bulb of the Mandelbrot set. In those cases, the green dots
//   become attrative and sit inside the orange area Kc.
// 
// * The Koening coordinates can only been seen when c belongs to the main cariod of the Madelbrot set
//
//------------------------------------------------------------

// complex number operations
vec2 cadd( vec2 a, float s ) { return vec2( a.x+s, a.y ); }
vec2 cmul( vec2 a, vec2 b )  { return vec2( a.x*b.x - a.y*b.y, a.x*b.y + a.y*b.x ); }
vec2 cdiv( vec2 a, vec2 b )  { float d = dot(b,b); return vec2( dot(a,b), a.y*b.x - a.x*b.y ) / d; }
vec2 csqrt( vec2 z ) { float m = length(z); return sqrt( 0.5*vec2(m+z.x, m-z.x) ) * vec2( 1.0, sign(z.y) ); }
vec2 conj( vec2 z ) { return vec2(z.x,-z.y); }
vec2 cpow( vec2 z, float n ) { float r = length( z ); float a = atan( z.y, z.x ); return pow( r, n )*vec2( cos(a*n), sin(a*n) ); }

//------------------------------------------------------------

float argument( in vec2 p )
{
	float f = atan( p.y, p.x );
	if( f<0.0 ) f += 6.2831;
	f = f/6.2831;
	return f;
}

float grid( in vec2 p )
{
	vec2 q = 16.0*p;
	vec2 r = fract( q );
    float fx = smoothstep( 0.05, 0.06, r.x ) - smoothstep( 0.94, 0.95, r.x );
    float fy = smoothstep( 0.05, 0.06, r.y ) - smoothstep( 0.94, 0.95, r.y );
		
    return 0.5 + 0.5*mod( floor(q.x)+floor(q.y), 2.0 );
}

float cross( vec2 a, vec2 b )
{
    return a.x*b.y - a.y*b.x;
}

bool isInTriangle( in vec2 p, in vec2 a, in vec2 b, in vec2 c )
{
    vec3 di = vec3( cross( b - a, p - a ), 
				    cross( c - b, p - b ), 
				    cross( a - c, p - c ) );
			
    return all(greaterThan(di,vec3(0.0)));
}


float distanceToSegment( vec2 a, vec2 b, vec2 p )
{
	vec2 pa = p - a;
	vec2 ba = b - a;
	float h = clamp( dot(pa,ba)/dot(ba,ba), 0.0, 1.0 );
	
	return length( pa - ba*h );
}

vec3 circle( vec3 bcol, vec3 col, in vec2 a, in vec2 b )
{
	float rr = 0.04;
	
	vec3 res = mix( bcol, col, 1.0 - smoothstep( rr-0.01, rr, length(a-b) ) );
	
	float f = smoothstep( rr-0.01, rr, length(a-b) ) - smoothstep( rr, rr+0.01, length(a-b) );
		
	return mix( res, vec3(0.0), f );
}

//------------------------------------------------------------

void mainImage( out vec4 fragColor, in vec2 fragCoord )
{
	vec2 uv = fragCoord.xy/iResolution.xy;
	
	vec2 p = -1.0 + 2.0*uv;
	p.x *= iResolution.x/iResolution.y;
	
	float at = mod( (iGlobalTime+.5)/5.0, 8.0 );

	vec2 c =    vec2(-0.800, 0.100);
	c = mix( c, vec2( 0.280,-0.490), smoothstep(0.0,0.1,at) );
	c = mix( c, vec2(-0.500,-0.500), smoothstep(1.0,1.1,at) );
	c = mix( c, vec2(-0.160, 0.657), smoothstep(2.0,2.1,at) );
	c = mix( c, vec2(-0.650, 0.100), smoothstep(3.0,3.1,at) );
	c = mix( c, vec2(-0.114, 0.650), smoothstep(4.0,4.1,at) );
	c = mix( c, vec2(-0.731, 0.166), smoothstep(5.0,5.1,at) );
	c = mix( c, vec2(-0.100,-0.660), smoothstep(6.0,6.1,at) );
    c = mix( c, vec2(-0.800, 0.100), smoothstep(7.0,7.1,at) );
	
	// get the 2 fixed points
	vec2 one = vec2( 1.0, 0.0 );

    vec2 fix1_1 = 0.5*( one + csqrt( one - 4.0*c ) );
    vec2 fix1_2 = 0.5*( one - csqrt( one - 4.0*c ) );
	vec2 fix2_1 = -(csqrt(-4.0*c-3.0*one)+one)/2.0;
	vec2 fix2_2 =  (csqrt(-4.0*c-3.0*one)-one)/2.0;
	vec2 fix2_3 = -(csqrt( one-4.0*c)-one)/2.0;
	vec2 fix2_4 =  (csqrt( one-4.0*c)+one)/2.0;

		
	vec2 z = p;
	vec2 dz = vec2( 1.0, 0.0 );

	vec2 ph = z;
	vec2 gr = vec2( log(length(z)), atan(z.y,z.x) );
	float t = 0.0;

	for( int i=0; i<512; i++ )
	{
		if( dot(z,z)>10000.0 ) continue;

        t += 1.0;

        // derivative
        dz = 2.0*cmul( z, dz );

        // point
        z = cmul(z,z) + c;

        vec2 a = cdiv(z,z-c);
        float s = pow( 0.5, t );

        // phi
        ph = cmul( ph, cpow(a, s) );
		
        // green
        gr.x += log(length(a)) * s;
        float aa = atan(a.y,a.x);
        if( isInTriangle( z, vec2(0.0), fix1_2, c ) )
        {
            aa -= sign(aa)*2.0*3.14159;
        }
        gr.y += aa * s;
	}
	
	
	vec3 col = vec3(1.0,0.65,0.10);
	
	if( t<511.0 )
	{
        float s = pow( 0.5, t );
        vec2  phib = cpow( z, s );
        float phiR = length( phib );
        float greenR = log(length(z)) * s;
        float greenI = argument(z*s);
        float d = log( length(z) ) * length(z) / length(dz);
        vec2  gradG = -conj(cmul( dz, conj(z) ));
        float n = t/50.0;
        float sn = -log2(abs(greenR))/50.0;
	
        col = vec3( 0.6 + 0.4*dot(normalize(-gradG),vec2(0.707)) );

        col *= vec3( grid( ph ) );
        col *= vec3(1.0)*clamp(d*50.0,0.0,1.0);
	}
	else
	{
		z = p;

		float t = 0.0;
		for( int i=0; i<200; i++ )
		{
			if( length(z-fix1_2)>0.001 )
			{
			z = cmul(z,z) + c;
			t += 1.0;
			}
		}
		vec2 fix = fix1_2;
		if( length(2.0*fix1_1)<1.0 ) fix=fix1_1;
		if( length(2.0*fix)<1.0 )
		{
		    vec2 ph = cdiv( z - fix, cpow(2.0*fix,t) );
		    float g = log(length(ph));
		    float l = 1.0 - 0.1*smoothstep( 0.7, 0.71, sin(48.0*g) );
		    col += 0.1*(abs(g));
		    ph = 1.0*vec2( length(ph), atan(ph.y,ph.x)/3.14 );
			col *= l;
		}
		
	}


	// color depending of attractive/repulsive fixed point
	col = circle( col, vec3(1.0,1.0,1.0), p, c );

	vec3 col2 = vec3(0.0,1.0,0.0);
	col = circle( col, col2, p, fix2_1 );
	col = circle( col, col2, p, fix2_2 );
	col = circle( col, col2, p, fix2_3 );
	col = circle( col, col2, p, fix2_4 );

	vec3 col1 = vec3(0.0,0.7,1.0);
	col = circle( col, col1, p, fix1_1 );
	col = circle( col, col1, p, fix1_2 );

	fragColor = vec4( col, 1.0 );
}

Key words[edit]

  • Koenigs function [6][7]
  • Kœnigs Linearization of Geometrically Attracting basins

References[edit]

  1. Gabriel Koenigs biographie at The MacTutor History of Mathematics archive
  2. Shadertoy example
  3. G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Annales École Normale Supérieure, 1(3) (1884), Supplément, 3-41.
  4. Inigo Quilez images and tutuorial
  5. Classification and Structure of Periodic Fatou Components. Senior Honors Thesis in Mathematics, Harvard College By Benjamin Dozier. Adviser: Sarah Koch 3/19/2012
  6. Koenigs function in wikipedia
  7. Power series expansion of the Koenigs function