Invariant measures for skew products and uniformly distributed sequences

Abstract

‘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.