Invariant measures for skew products and uniformly distributed sequences

摘要

‘Almost all” sequences (r 1, . . . , r n , . . . ) of positive integers have the following “universal” property: Whenever (X, μ) is a Borel probability compact metric space, and Φ 1, Φ 2, . . . , Φ n , . . . a sequence of commuting measure preserving continuous maps on (X, μ), such that the action (by composition) on (X, μ) of the semigroup with generators Φ 1, . . . ,Φ n , . . . is uniquely ergodic and equicontinuous, then for every ${x in X}$ the sequence w 1,w 2, . . . , w n , . . . where $$w_n:=varPhi_{r_n}(varPhi_{r_{n-1}}(ldots(varPhi_{r_2}(varPhi_{r_1}(x)))ldots))$$ is uniformly distributed for μ. This is a contribution to Problem 116 of Schreier and Ulam in the Scottish Book.