Invariant measures for skew products and uniformly distributed sequences II

Abstract

“Almost all” sequences $(r_1,\ldots ,r_n,\ldots )$ of positive integers have the following “universal” property: whenever $(X,\mu )$ is a Borel probability measure, compact metric space and $\varPhi _1,\varPhi _2,\ldots ,\varPhi _n,\ldots$ a sequence of continuous, measure preserving maps on $(X,\mu )$, such that the action (by composition) on $(X,\mu )$ of the semigroup with generators $\varPhi _1,\ldots ,\varPhi _n,\ldots$ is amenable (as discrete), uniquely ergodic and non-sensitive on $\text {supp}\mu$, then for every $x\in X$ the sequence $w_1,w_2,\ldots ,w_n,\ldots$ where

$$\begin{aligned} w_n:=\varPhi _{r_n}(\varPhi _{r_{n-1}}(\ldots (\varPhi _{r_2}(\varPhi _{r_1}(x)))\ldots )) \end{aligned}$$

is uniformly distributed for $\mu$. This continues investigation of [8].