C. Bandt et S. Graf, SELF-SIMILAR SETS .7. A CHARACTERIZATION OF SELF-SIMILAR FRACTALS WITH POSITIVE HAUSDORFF MEASURE, Proceedings of the American Mathematical Society, 114(4), 1992, pp. 995-1001
For self-similar sets with nonoverlapping pieces, Hausdorff dimension
and measure are easily determined. We express "absence of overlap" in
terms of discontinuous action of a family of similitudes, thus improvi
ng the usual "open set condition".