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Rank one lightly mixing. (English) Zbl 0754.28010
Five successively stronger mixing conditions are (1) weakly mixing, (2) mildly mixing, (3) lightly mixing, (4) partially mixing, (5) mixing. Within the category of rank one transformations there are examples having property (i) but not \((i+1)\), for \(i=1,2,3,4\). In this paper the authors demonstrate this (for \(i=3\)) by showing that an example of rank one transformation constructed by Chacón is lightly mixing but not partially mixing. They also show that it is not lightly 2-mixing. This is the only such example (i.e., lightly mixing, but not partially mixing and not lightly 2-mixing) obtained by “ cutting and stacking” construction. Other such examples are obtained as products of transformations that are partially mixing but not lightly 2-mixing.

MSC:
28D05 Measure-preserving transformations
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[1] J. R. Blum, S. L. M. Christianson and D. Quiring,Sequence mixing and \(\alpha\)-mixing, Illinois J. Math.18 (1974), 131–135. · Zbl 0272.28011
[2] R. V. Chacón,Weakly mixing transformations which are not strongly mixing, Proc. Am. Math. Soc.22 (1969), 559–562. · Zbl 0186.37203 · doi:10.1090/S0002-9939-1969-0247028-5
[3] N. A. Friedman,Introduction to Ergodic Theory, Van Nostrand Reinhold, New York, 1970. · Zbl 0212.40004
[4] N. A. Friedman,Higher order partial mixing, Contemporary Math.26 (1984), 111–130. · Zbl 0557.28013
[5] N. A. Friedman and D. S. Ornstein,On partial mixing transformations, Indiana Univ. Math. J.20 (1971), 767–775. · Zbl 0213.07504 · doi:10.1512/iumj.1971.20.20061
[6] N. A. Friedman and D. S. Ornstein,On mixing and partial mixing, Illinois J. Math.16 (1972), 61–68. · Zbl 0224.28012
[7] N. A. Friedman and E. Thomas,Higher order sweeping out, Illinois J. Math.29 (1985), 401–417. · Zbl 0546.28012
[8] A. del Junco,A simple measure preserving transformation with trivial centralizer, Pacific J. Math.79 (1978), 357–362. · Zbl 0368.28019
[9] S. Kalikow,Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory and Dynamical Systems4 (1984), 237–259. · Zbl 0552.28016 · doi:10.1017/S014338570000242X
[10] J. King,Lightly mixing is closed under countable products, Isr. J. Math.62 (1988), 341–346. · Zbl 0653.28009 · doi:10.1007/BF02783302
[11] D. S. Ornstein,On the root problem in ergodic theory, Proc. Sixth Berkeley Symp. Math. Stat. Prob. (1970), 347–356.
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