Does anyone know offhand if the Mandelbrot set has something like this? If I recall, in a certain system with multiple basins of attraction, a neighborhood (arbitrarily small open disc) around a point that goes to one will also contain points that go to the others. I'm thinking of bands of exit speeds (colors) as analogous to a basin. So the question is, pick a point of any (non-black) color, if you take an arbitrarily small neighborhood of points around it, must that neighborhood contain all colors? Sorry if stupid.