Basin of Attraction : The set of points that evolve to an attractor is called its basin of attraction.
The phase space of a particular dynamical system is partitioned into one or many basins of attraction , perhaps intimately intertwined, each with its own attractor .
basins of attraction for a pendulum swinging over three magnets. (from Peitgen, Jürgens, and Saupe, Chaos and Fractals)
The boundaries between basins of attraction can themselves be fractal.
detail of boundary between the regions of the basins. They are fully entertwined.
(is this related to the "edge of chaos"..i.e. is the edge itself fractal?)
see Mandlebrot set for the extreme example of a complex domain of attraction.