Bi-Invariant Sets And Measures Have Integer Hausdorff Dimension

. Let A; B be two diagonal endomorphisms of the d-dimensional torus with corresponding eigenvalues relatively prime. We show that for any A-invariant ergodic measure ¯, there exists a projection onto a torus T r of dimension r dim ¯, that maps ¯-almost every B-orbit to a uniformly distributed sequence in T r . As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer. 1. Introduction Let a; b ? 1 be relatively prime integers, and let ¯ be a measure on T = R=Z which is invariant and ergodic for multiplication mod 1 by a. Host (1995) proved that if ¯ has positive entropy, then the sequence fb n xg n is uniformly distributed mod 1 for ¯-almost every x 2 T. Here we extend this to higher dimensions, where the Hausdorff dimension of the measure plays a crucial role. All measures we consider are Borel probability measures. Denote by dimH S the Hausdorff dimension of a set S, and for any measure ¯, let dim..