> 2. If they do converge, is the convergence unique? (I.e., no two sequences converge to the same point - the function f is injective).
The curve given in the paper doesn't satisfy that requirement. O_j(x1, x2) is defined as a mapping from the (j × j) square to integers smaller than j² by a case distinction that includes O_j(x1, 0) = x1. If you want to use that curve to fill the unit square, you need to rescale it as o_j(y1, y2) = O_j(j•y1, j•y2)/j² so that all values involved stay in [0, 1]. Then for all y1, we have o_j(y1, 0) = j•y1/j² = y1/j, which decays to 0 as j grows to infinity. Hence the limit function is not bijective.
The curve given in the paper doesn't satisfy that requirement. O_j(x1, x2) is defined as a mapping from the (j × j) square to integers smaller than j² by a case distinction that includes O_j(x1, 0) = x1. If you want to use that curve to fill the unit square, you need to rescale it as o_j(y1, y2) = O_j(j•y1, j•y2)/j² so that all values involved stay in [0, 1]. Then for all y1, we have o_j(y1, 0) = j•y1/j² = y1/j, which decays to 0 as j grows to infinity. Hence the limit function is not bijective.