Iterated function systems with overlaps and self-similar measures

The paper considers the iterated function systems of similitudes which sati sfy a separation condition weaker than the open set condition, in that it a llows overlaps in the iteration. Such systems include the well-known Bernou lli convolutions associated with the PV numbers, and the contractive simili tudes associated with integral matrices. The latter appears frequently in w avelet analysis and the theory of tilings. One of the basic questions is st udied: the absolute continuity and singularity of the self-similar measures generated by such systems. Various conditions to determine the dichotomy a re given.