| Details: |
We say a number x in [0,1] is normal if for any positive integer D, all finite words of same length with letters from the alphabet {0, 1, ... , D-1} occurs with the same asymptotic frequency in the representation of x in base D, or in simple words, its digital expansion is uniformly random in any base. The famous Normal number theorem of E. Borel says that almost every number possesses this phenomenon.
It is generally believed that some naturally defined subsets of $\mathbb{R}$ also inherit the above property unless the set under consideration displays an obvious obstruction. This talk is about the study of Borel's theorem on fractals; Cantor type sets for instance. We show that for certain fractals how the property of being normal can be related to the behaviour of trajectories under some random walk on tori, and consequently can be settled studying measures which are `stationary' with respect to the random walk.
The talk is based on a joint work with Yiftach Dayan and Barak Weiss.
|