Fractals/Iterations in the complex plane/Koenigs coordinate

Animated periodic cycle

Koenigs coordinate are used in the basin of attraction of finite attracting (not superattracting) point (cycle),

Definition

• a rational map f of degree at least two $f(z)$
• a fixed point $z_{1}=0$
• multiplier of the fixed point is $\lambda$
• fixed point is attracting but not superattracting $0<\lambda <1$
• ${\mathcal {A}}$ = the attracting basin of the fixed point zero under function $f$. In other words interior of component containing fixed point = the open set consisting of all points whose orbits under f converge to 0.

$\phi _{\lambda }(z):{\mathcal {A}}\to \mathbb {C}$

It is approximated by normalized iterates :

$\phi _{n}(z)={\frac {f^{n}(z)}{\lambda ^{n}}}$

It can be defined by the formula :

$\phi _{\lambda }(z)=\lim _{n\to \infty }{\frac {f^{n}(z)}{\lambda ^{n}}}$

Function f is locally conjugate to the model linear map $z\to \lambda z$

Examples

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 attractive and sit inside the orange area Kc.
// 
// * The Koening coordinates can only been seen when c belongs to the main cariod{{typo help inline|reason=similar to cardiod|date=July 2023}} 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

• Koenigs function
• Kœnigs Linearization of Geometrically Attracting basins