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Stationary perturbations based on Bernoulli processes. (English) Zbl 0752.28009

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.

MSC:

28D05 Measure-preserving transformations
60A10 Probabilistic measure theory
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