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An example of a measure-preserving flow with minimal self-joinings. (English) Zbl 0533.28012
A 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.

28D10 One-parameter continuous families of measure-preserving transformations
28D05 Measure-preserving transformations
Full Text: DOI
[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
[3] A. del Junco, M. Rahe and L. Swanson,Chacón’s automorphism has minimal self-joinings, J. Analyse Math.37 (1980), 276–284. · Zbl 0445.28014 · doi:10.1007/BF02797688
[4] H. B. Keynes and D. Newton,Real prime flows, Trans. Am. Math. Soc.217 (1976), 237–255. · Zbl 0341.54051 · doi:10.1090/S0002-9947-1976-0400189-5
[5] M. Ratner,Rigidity of horocycle flows, Ann. of Math., to appear. · Zbl 0506.58030
[6] M. Ratner,Factors of horocycle flows, ergodic theory and dynamical systems, to appear. · Zbl 0536.58029
[7] M. Ratner,Joinings of horocycle flows, to appear. · Zbl 0556.28020
[8] M. Ratner,Rigidity of products of horocycle flows, to appear. · Zbl 0556.28020
[9] D. Rudolph,An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math.35 (1979), 97–122. · Zbl 0446.28018 · doi:10.1007/BF02791063
[10] W. Veech,A criterion for a process to be prime, preprint. · Zbl 0499.28016
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