Abstract
For a family of dynamical properties, knowing that the condition holds for order 4 implies that it holds for all orders. Here we establish this for the properties minimal self-joinings, simplicity and for cartesiandisjointness.
An application of the first yields an analog to Kalikow’s celebrated result that for rank-1 transformations, 2-fold mixing implies 3-fold mixing. Via a joining argument we show that for any rank-1 ℤD-action, 4-fold mixing implies mixing of all orders. Indeed, the rank need only be sufficiently close to 1 for the implication to hold and so this result is new even when the acting group is ℤ.
By means of limit-joinings, we settle affirmatively an old open question by establishing, for anyM, thatM-fold Rényi-mixing impliesM-fold mixing.
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Partially supported by National Science Foundation Postdoctoral Research Fellowship.
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King, J.L. Ergodic properties where order 4 implies infinite order. Israel J. Math. 80, 65–86 (1992). https://doi.org/10.1007/BF02808154
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DOI: https://doi.org/10.1007/BF02808154