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Twofold mixing implies threefold mixing for rank one transformations. (English) Zbl 0552.28016
In this paper V. A. Rokhlin’s problem [Izv. Akad. Nauk SSSR 13, 329-340 (1949; Zbl 0032.28402)] whether mixing implies mixing of all orders, is answered affirmatively in the case of a rank one transformation and order 3. the idea of proof is to show that for given measurable sets \(A_ i(1\leq i\leq 3)\) and large integers M, N for most of the points w and \(L\in {\mathbb{N}}\) large, \((1/L)| \{k<L: T^ kw\in T^{-M-N}A_ 1\cap T^{-N}A_ 2\cap A_ 3\}|\) is approximately \(m(A_ 1)m(A_ 2)m(A_ 3).\) This is achieved by subdividing [0,L) into subintervals most of them having this approximating property by construction.
Reviewer: M.Denker

MSC:
28D05 Measure-preserving transformations
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References:
[1] Halmos, Lectures on Ergodic Theory (1956)
[2] Rohlin, Izv. Acad. Nauk. SSSR 13 pp 329– (1949)
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