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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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