Iterated function systems with a weak separation condition
Nonoverlapping contractive self-similar iterated function systems (IFS) have been studied in great detail via the open set condition. On the other hand much less is known about IFS with overlaps. To deal with such systems, a weak separation condition (WSC) has been introduced recently [LN1]; it is weaker than the open set condition and it includes many important overlapping cases. This paper has two purposes. First, we consider the class of self-similar measures generated by such IFS; we give a necessary and sufficient condition for the self-similar measures to be absolutely continuous with respect to Hausdorff measures. This extends a result in [LNR]. As most of the known examples of the WSC involve algebraic integers (e.g., the golden ratio, integral dilation matrices) and the contraction ratios are equal, our second goal is to give new examples of self-similar IFS with the WSC and with more arbitrary contraction ratios.