We consider self-similar  Iterated Function System (IFS) on the
line constructed with overlapping cylinders. Recently there have been a
number of outstanding results which have suggested that the overlap has
dramatic change in the most important properties of the IFS only if there is
an exact overlap between some of the cylinders.
In this talk, we point out that this is not always the case, at least as far
as the absolute continuity of self-similar measures is concerned.
Namely, we present some one-parameter families of homogeneous self-
similar measures on the line such that
• the similarity dimension is greater than 1 for all parameters and
• the singularity of some of the self-similar measures from this family is
not caused by exact overlaps between the cylinders.
We can obtain such a family as the family of orthogonal  projections of the
natural measure of the Sierpiński carpet.