Karoly Simon, BME Budapest
Singularity of self-similar measures
Abstract: In this talk we study various one-parameter families of self-similar measures such that the similarity dimension of each measure in the family is greater than one.
The most famous family of this kind is the infinite Bernoulli convolutions.
That is the family of distributions of the infinite random sum
\sum_{n=0}^{\infty}\pm\lambda^n, where the plus and minus signs are chosen independently and with 1/2-- 1/2 probability, and to guarantee that the similarity dimension is greater than 1, we need to assume that \lambda\in(1/2,1).
Another such family is obtained if we take the natural (uniformly distributed) measure on the Sierpinski carpet and we consider its orthogonal projections to the angle-\alpha line for every \alpha\in(0,\pi). (Here \alpha is the parameter.)
Since the breakthrough result of B. Solomyak in 1995 about the absolute continuity of Lebesgue typical Bernoulli convolutions, there have been a number of results proving absolute continuity of homogeneous self-similar measures for almost all
or even for all but a set of zero Hausdorff dimension of parameters in families like the ones above.
With Lajos Vágó, we considered the same kind of one -parameter families of self-similar measures, but we asked what happens with the absolute continuity for parameters which are typical in the topological sense (dense and G_{\delta}), instead of measure-theoretical sense.
The surprising partial results we obtained are to the topic of the talk.