Mixing and sweeping out
Let T be an invertible measure-preserving transformation on the unit interval with Lebesgue measure \(\mu\). T is uniformly sweeping out on a sequence \(s=(k_ n)\) if for each set A, \(\mu (A)>0,\) and \(\epsilon >0,\) there exists \(N=N(A,\epsilon)\) such that \(k_ i\in s,\quad 1\leq i\leq N\) implies \(\mu (\cup^{N}_{i=1}T^{k_ i})>1-\epsilon.\) T is uniformly mixing on a sequence s if T is mixing on the sequence of differences, s- s. The author uses the stacking method to give an example of a weakly mixing transformation T and a sequence \((d_ n)\) such that T is uniformly mixing on \((d_ n)\), uniformly sweeping out on \(([\alpha d_ n])\) for all \(\alpha \in (0,1)\) and T is not mixing on \(([\alpha d_ n])\) for all rational \(\alpha \in (0,1).\) It was shown by the author in Can. J. Math. 35, 339-352 (1983; Zbl 0495.28013) that mixing implies uniform sweeping out. The converse is still open, but the example above gives a partial negative answer to this question. Various equivalences for different types of mixing are also given.
invertible measure-preserving transformation
scientific article; zbMATH DE number 4156838