Department of Mathematics

Probabilistic approach to fractal analysis of subsets of
non-normal numbers

(Institut for Applied Mathematics, Bonn,Germany/ NPU, Kiev, Ukraine)

6 February 2007

We plan to consider fractal and topological classification
of real numbers via asymptotic behavior of their digits in a fixed
system of numeration. Main attention will be paid to the case of
*s*-adic expansion and its generalizations (*Q*-representation,
*Q**-representation). The results we are going to discuss are based
on fractal analysis of singularly continuous probability measures.
So, we plan to discuss some new results in fractal analysis of such
measures and show how these results can be applied to study fractal
properties of sets of essentially non-normal and partially
non-normal real numbers. In particular we show that essentially
non-normal numbers are generic in the topological sense as well as
in the sense of fractal geometry. Possible applications to the
theory of transformations preserving the Hausdorff dimension are
also planned to be discussed.

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Yuri Yakubovich