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We study random walks on a d-dimensional torus by affine expanding maps. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. From this, we deduce uniform distribution of almost every orbits modulo 1 in certain self-similar sets in R^d. As this conclusion amounts to normality of numbers in the one dimensional case, thus we obtain the version of Borel’s theorem on Normal
numbers for a class of fractals in R, for instance, Cantor type sets. The talk is based on a joint work with Yiftach Dayan and Barak Weiss. |