Universiteit Utrecht

Department of Mathematics

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

Grygoriy Torbin
(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