Convergence under ×q of ×p Invariant Measures on the Circle

AbstractConsider the semigroup of maps {Tn}n∈ N where Tn(x) = nx mod 1. Suppose p and q are multiplicatively independent integers in that they are not both powers of the same integer. Further suppose that μ is a Borel probability measure, invariant, ergodic, and of positive entropy for Tp. We show that the sequence of measures Tiq(μ) converges weak* to Lebesgue measure on a subsequence of values i of uniform full density