The equations/algorithms of these attractors (the link below) for given parameters and a number of iterations output a set of positions, the size of the set is the same as the number of iterations:
Take, for example, pickover some way down the page. $X,Y,Z$ start at some values. The $X, Y,Z$ with the dots over them mean $X, Y, Z$ in the next iteration, on the left is a picture of what this looks like after many iterations.
Can they or their 2D versions be rearranged to simulate an infinite number of iterations? The new equation/algorithm would take the parameters and additionally $X, Y$ and, if applicable, $Z$-coordinates but no number of iterations. It would return a single number instead of a set of positions, this number would represent the distance between the specified position and the nearest point in the attractor.
The number returned should never be negative but sometimes it may be exactly 0 where either a point has been chosen that is exactly on top of an existing point or the point in the attractor jumps chaotically, or in the case of some fractal attractors systematically possibly covering all the space in some part of the attractor.