ITERATED FUNCTION SYSTEMS WITH OVERLAPS AND SELF-SIMILAR MEASURES KA-SING LAU, SZE-MAN NGAI  HUI RAO A The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the well-known Bernoulli convolutions associated with the PV numbers, and the contractive similitudes associated with integral matrices. The latter appears frequently in wavelet analysis and the theory of tilings. One of the basic questions is studied : the absolute continuity and singularity of the self-similar measures generated by such systems. Various conditions to determine the dichotomy are given. 1. Introduction We will call a family of contractive maps S j  N j= on  d an iterated function system (IFS). An iterated function system will generate an invariant compact subset K   N j= S j K, and if, further, the system is associated with a set of probability weights w j  N j= , then it will generate an invariant measure μ   N i= w j μ  S - j . (1.1) In order to obtain sharp results on the invariant set K or the invariant measure μ, it is often assumed that the maps are similitudes (or the extension to self-conformal maps). The corresponding K and μ are called the self-similar set and the self-similar measure respectively. For the iteration, it is often assumed that the iterated function system satisfies the open set condition (OSC) [10]. One of the advantages of the open set condition is that the ‘generic’ points of the set K can be uniquely represented in a symbolic space, and the dynamics of the iterated function system can be identified with the shift operation in the symbolic space. Without the open set condition, the iteration has overlaps ; then such a representation will break down, and it is more difficult to handle the situation. The simplest case of an iterated function system with overlaps is provided by the maps S  x  ρx, S  x  ρx1, x   with    ρ  1. (1.2) The invariant measure μ associated with the weights   has been studied for a long time in the context of Bernoulli convolutions [27]. Recently Solomyak [23, 28] proved Received 18 September 1999 ; revised 18 February 2000. 2000 Mathematics Subject Classification 28A80 (primary), 42B10 (secondary). The first author’s research was partially supported by an HK RGC grant. The second author’s research was partially supported by a postdoctoral fellowship from CUHK and by the NSF grant DMS-96-32032. The third author’s research was partially supported by a postdoctoral fellowship from CUHK. J. London Math. Soc. (2) 63 (2001) 99–116.  London Mathematical Society 2001.