An example of a measure-preserving flow with minimal self-joinings.

*(English)*Zbl 0533.28012A measure-preserving flow \(\{T_ t\}\) of a probability space (\(\Omega\),\(\mu)\) is said to have minimal self-joinings (MSJ) if for each \(k>0\) any ergodic \(\{T_ t\times...\times T_ t\}\)-invariant measure on \(\Omega^ k\) all of whose one-dimensional marginals are \(\mu\) must be a product of off-diagonals. An off-diagonal measure on \(\Omega^{\ell}\) (where \(\ell\) may be one) is any measure of the form \(T_{t_ 1}\times...\times T_{t_{\ell}}\mu_{\Delta}\), where \(\mu_{\Delta}\) is the diagonal measure over \(\mu\) on \(\Omega^{\ell}\). It is shown here that when \(\{T_ t\}\) is a simple 2-step flow over Chacón’s map [A. del Junco, M. Rahe and L. Swanson, ibid. 37, 276-284 (1980; Zbl 0445.28014)] \(\{T_ t\}\) is weakly mixing and has MSJ. It follows that the map \(T_ a\) is prime (has only the trivial factor algebras) when \(a\neq 0\). Moreover \(T_ a\) and \(T_ b\) are non- isomorphic for \(a\neq b\), whence any joining of non-zero times of \(\{T_ t\}\) (repetitions allowed) must also be a product of off-diagonals.

##### MSC:

28D10 | One-parameter continuous families of measure-preserving transformations |

28D05 | Measure-preserving transformations |

##### Keywords:

measure-preserving flow; minimal self-joinings; marginals; off-diagonal measure; weakly mixing
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\textit{A. Del Junco} and \textit{K. Park}, J. Anal. Math. 42, 199--209 (1983; Zbl 0533.28012)

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##### References:

[1] | R. V. Chacón,Weakly mixing transformations which are not strongly mixing, Proc. Am. Math. Soc. (1969). |

[2] | A. del Junco and D. Rudolph,Minimal self-joinings and related properties, to appear. · Zbl 0646.60010 |

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[10] | W. Veech,A criterion for a process to be prime, preprint. · Zbl 0499.28016 |

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