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Atom domains in the Mandelbrot set surround mini-Mandelbrot islands. So too in the Burning Ship fractal. These pictures are coloured using the period for hue, and distance estimation for value. Saturation is a simple switch on escaped vs unescaped pixels. Rendered with some Fragmentarium code.

The algorithm is simple: store the iteration count when |Z| reaches a new minimum. The last iteration count so stored is the atom domain. Better start checking after the first iteration if you initialize with 0. IEEE floating point has infinities so you can initialize the stored |Z| value to 1.0/0.0.

I was hoping to use atom domains for interior checking, by using Newton's method to find limit cycles and seeing if their maximal Lyapunov exponent is less than 1, but it didn't work. My guesses are that Newton's method doesn't converge to the limit cycle, but instead to some phantom attractor, or that the maximal Lyapunov exponent isn't an indicator of interiority as I had hoped (I tried with plain determinant too, no joy there either). The method marked some exterior points as interior.

One thing that is interesting to me is the grey region of unescaped pixels with chaotic atom domains (the region is that colour because the anti-aliasing blends subpixels scattered across the whole spectrum into a uniform grey). I'm not sure whether it is an artifact of rendering at a limited iteration count and should be exterior, or if it really is interior and chaotic.