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.

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.

## Date:

Tue, 16/05/2017 - 14:00 to 15:00