Lyapunov atom domains for negative multibrot sets

Multibrot for power -2

Back at the end of April last year I think I was futzing about on math.stackexchange.com answering a question about rendering negative multibrot sets, for example one produced by iterations of \(z \to z^{-2} + c\). I tried applying the atom domain colouring from the regular Mandelbrot set, but found it looked better if I accumulated all the partials with additive blending, not just the final domain. Here's a zoomed in view:

Multibrot for power -2 (zoomed in)

I implemented it as a GLSL fragment shader in Fragmentarium, here's the source code (which you can download too):

// Mandelbrot set for \( z \to z^{-n} + c \) coloured by Lyapunov atom domains
// Created: Thu Apr 30 15:10:00 2015
#include "Progressive2D.frag"
#include "Complex.frag"

const float pi = 3.141592653589793;
const float phi = (sqrt(5.0) + 1.0) / 2.0;

#group Lyapunov atom domains

uniform int Iterations; slider[10,200,5000]
uniform int Power; slider[-16,-2,16]

vec3 color(vec2 c) {
  // critical point is \( 0 \) for positive Power, and \( 0^Power + c = c \)
  // critical point is \( \infty \) for negative Power, and \( \infty^Power + c = c \)
  // so start iterating from \( c \)
  vec2 z = c;
  // Lyapunov exponent accumulator
  float le = 0.0;
  // atom domain accumulator
  float minle = 0.0;
  int mini = 1;
  // accumulated colour
  vec4 rgba = vec4(0.0);

  for (int i = 0; i < Iterations; ++i) {
    // \( zn1 \gets z^{Power - 1} \)
    vec2 zn1 = vec2(1.0, 0.0);
    for (int j = 0; j < abs(Power - 1); ++j) {
      zn1 = cMul(zn1, z);
    }
    if (Power < 0) {
      zn1 = cInverse(zn1);
    }
    // \( dz \gets Power z^{Power - 1} \)
    vec2 dz = float(Power) * zn1;
    // \( z \gets z^{Power} + c \)
    z = cMul(zn1,z) + c;
    // \( le \gets le + 2 log |dz| \)
    float dle = log(dot(dz, dz));
    le += dle;
    // if the delta is smaller than any previous, accumulate the atom domain domain
    if (dle < minle) {
      minle = dle;
      mini = i + 1;
      float hue = 2.0 * pi / (36.0 + 1.0/(phi*phi)) * float(mini);
      vec3 rainbow = 2.0 * pi / 3.0 * vec3(0.0, 1.0, 2.0);
      vec3 domain = clamp(vec3(0.5) + 0.5 * sin(vec3(hue) + rainbow), 0.0, 1.0);
      rgba += vec4(domain, 1.0);
    }
  }

  // accumulated 'iterations' logs of squared magnitudes
  // so divide by 2 iterations
  le /= 2.0 * float(Iterations);
  // scale accumulated colour and blacken interior
  return mix(rgba.rgb / rgba.a, vec3(le < 0.001 ? 0.0 : tanh(exp(le))), 0.5);
}

More negative powers than -2 don't look very good, though.