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This is the same Julia set as in Julia's Curls, with the same framing rectangle, but colored in this time.

This Julia set is typical of a class of this function's Julia sets that have a quadratic Julia set border (with the Julia seed not quite constant) and filigrees, lakes, or other internal structures. The outside points diverge. Inside points mostly converge to zero. Points reached from infinity by crossing an odd number of lines go to zero, others to infinity.

This Julia set does not contain any actual Herman rings. The only attractors are the two obligatory ones, zero and infinity, for this family of mappings.

The image uses five layers: one gives an iteration gradient; one for each of the two attracting basins tints that basin and adds "ribbons" via orbit traps, in both cases an off-centered ring around the attractor; and one for each basin modifies the luminance in high and low iteration areas.

The ribbon features would not render well without aggressive antialiasing, which, combined with the number of layers, is the cause of the fairly long render time of 30 minutes.

Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.

Detailed stats:
Name: Julia's Curls II
Date: January 30, 2009
Fractal: Herman Ring Julia set
Location: angle parameter = 0.21803398, c = 0.2291666667 + 3.291666667i
Depth: Very Shallow
Min Iterations: 1
Max Iterations: 136
Layers: 5
Anti-aliasing: 3x3, threshold 0, depth 2
Preparation time: 1 minute
Calculation time: 30 minutes (2GHz dual-core Athlon XP)