Kowalski, Zbigniew S. Stationary perturbations based on Bernoulli processes. (English) Zbl 0752.28009 Stud. Math. 97, No. 1, 53-57 (1990). Let \(\sigma\) be a \((p_ 1,\dots,p_ s)\) Bernoulli shift, \(T\) an endomorphism of the Lebesgue space of \([0,1]\) and \(T^{-1}_ \varepsilon(y)=(1-\varepsilon)y+\varepsilon g(y)\) \((a<\varepsilon<b)\) be transformations of \([0,1]\) into itself with \(g\in C^ 2\) satisfying mild conditions. Define the perturbed transformation \[ \overline T(x,y)=\left(\sigma(x), T_{\varepsilon_{x(1)}}\circ T(y)\right), \] where \(\sum^ s_{i=1}\varepsilon_ i p_ i=0\). It is shown in this paper that \(\overline T\) is weakly mixing when \(s\geq 3\), while the weakly mixing is not true when \(s=2\). Some facts in S. Pelikan [Trans. Am. Math. Soc. 281, 813-825 (1984; Zbl 0532.58013)] and the author [Stud. Math. 87, 215-222 (1987; Zbl 0651.28013)] are related. Reviewer: Qian Minping (Beijing) Cited in 1 ReviewCited in 1 Document MSC: 28D05 Measure-preserving transformations 60A10 Probabilistic measure theory Keywords:Bernoulli shift; endomorphism; Lebesgue space; weakly mixing Citations:Zbl 0532.58013; Zbl 0651.28013 PDFBibTeX XMLCite \textit{Z. S. Kowalski}, Stud. Math. 97, No. 1, 53--57 (1990; Zbl 0752.28009) Full Text: DOI EuDML