Invariant measures for skew products and uniformly distributed sequences. II (Q500319)

Earlier work of the authors [Monatsh. Math. 167, No. 1, 81--103 (2012; Zbl 1273.28015)] is continued here in the following setting. Given a sequence \((r_1,r_2,\dots)\) of positive integers, a Borel probability space \((X,\mu)\), and a sequence of continuous \(\mu\)-preserving maps \(\phi_1,\phi_2,\dots\) which together have the property that the semi-group they generate is amenable, and that the action they generate on \(X\) is uniquely ergodic and non-sensitive on the support of \(\mu\), then for any \(x\in X\) there is an associated sequence of points \((x_n)\) in \(X\) given by \(x_n=\phi_{r_n}(\phi_{r_{n-1}}(\cdots (\phi_{r_2}(\phi_{r_1}(x)))\cdots))\) for \(n\geq1\). The main result obtained here is that for a typical initial sequence of integers (there is a precise definition of `typical'), any such system, and any \(x\in X\) the resulting sequence is equidistributed with respect to \(\mu\). The earlier work proved a similar result under the strong hypothesis that the semigroup generated by the sequence of maps is commutative and the action they generate is uniquely ergodic and equicontinuous.