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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.

##### MSC:
 28D10 One-parameter continuous families of measure-preserving transformations 28D05 Measure-preserving transformations
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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 [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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