Infinite measure preserving transformations with Radon MSJ.

*(English)*Zbl 1403.28015Minimal self-joinings (MSJ) was a concept introduced by D. J. Rudolph [J. Anal. Math. 35, 97–122 (1979; Zbl 0446.28018)] building on a remarkable example constructed by D. S. Ornstein [in: Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, 2, 347–356 (1972; Zbl 0262.28009)] of a measure-preserving transformation that commutes only with its powers and has no non-trivial invariant factor \(\sigma\)-algebras. Rudolph constructed a mixing measure-preserving transformation with MSJ, and used this as a ‘veritable counterexample machine’ in the words of one reviewer. Finding an appropriate analogue of MSJ for infinite measure-preserving systems is not straightforward for reasons essentially related to the possible presence of too many non-singular measures and the breakdown of the close link between joinings and ergodic components. Here the setting is joinings for ergodic measure-preserving homeomorphisms of locally compact non-compact Cantor spaces equipped with an infinite Radon measure (a setting which is a universal measurable model for ergodic infinite non-atomic \(\sigma\)-finite measure-preserving transformations on standard Borel spaces). By restricting to Radon measures the authors introduce Radon MSJ and Radon disjointness; in addition the underlying topological setting allows generic points to be defined using the Hopf ratio ergodic theorem in place of the usual ergodic theorem used in the finite measure-preserving setting. After a short development of the general theory of joinings in this setting, examples of transformations with Radon MSJ are introduced. For this example additional properties are explored using the mix of topological and measure-theoretic concepts made available by the setting. This builds on and further develops work on the infinite Chacon transformation introduced by T. Adams et al. [Isr. J. Math. 102, 269–281 (1997; Zbl 0896.58039)] and further studied by É. Janvresse et al. [Isr. J. Math. 224, 1–37 (2018; Zbl 1406.37011)].

Reviewer: Thomas B. Ward (Leeds)

##### MSC:

28D05 | Measure-preserving transformations |

##### Keywords:

infinite measure-preserving transformation; minimal self-joining; infinite Chacon transformation
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\textit{A. I. Danilenko}, Isr. J. Math. 228, No. 1, 21--51 (2018; Zbl 1403.28015)

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##### References:

[1] | Aaronson, J., The intrinsic normalizing constants of transformations preserving infinite measures, Journal d’Analyse Mathématique, 49, 239-270, (1987) · Zbl 0644.28013 |

[2] | J. Aaronson, An Introduction to Infinite Ergodic theory, Mathematical Surveys and Monographs, Vol. 50, American Mathematical Society, Providence, RI, 1997. · Zbl 0882.28013 |

[3] | Adams, T.; Friedman, N.; Silva, C. E., Rank-one weak mixing for nonsingular transformations, Israel Journal of Mathematics, 102, 269-281, (1997) · Zbl 0896.58039 |

[4] | Danilenko, A. I., Funny rank-one weak mixing for nonsingular Abelian actions, Israel Journal of Mathematics, 121, 29-54, (2001) · Zbl 1024.28014 |

[5] | Danilenko, A. I., On simplicity concepts for ergodic actions, Journal d’Analyse Mathématique, 102, 77-118, (2007) · Zbl 1145.37002 |

[6] | Danilenko, A. I., (C, F)-actions in ergodic theory, Geometry and Dynamics of Groups and Spaces, 265, 325-351, (2008) · Zbl 1229.37006 |

[7] | Danilenko, A. I., Finite ergodic index and asymmetry for infinite measure preserving actions, Proceedings of the American Mathematical Society, 144, 2521-2532, (2016) · Zbl 1383.37004 |

[8] | Junco, A.; Rahe, M.; Swanson, L., Chacon’s automorphism has minimal selfjoinings, Journal s’Analyse Mathématique, 37, 276-284, (1980) · Zbl 0445.28014 |

[9] | Junco, A.; Rudoplh, D., On ergodic actions whose self-joinings are graphs, Ergodic Theory and Dynamical Systems, 7, 531-557, (1987) · Zbl 0646.60010 |

[10] | Junco, A.; Silva, C. E., On factors of non-singular Cartesian products, Ergodic Theory and Dynamical Systems, 23, 1445-1465, (2003) · Zbl 1059.37004 |

[11] | Effros, E. G., Transformation groups and C*-algebras, Annals of Mathematics, 81, 38-55, (1965) · Zbl 0152.33203 |

[12] | N. Friedman, Introduction to Ergodic Theory, Van Nostrand Reinhold Mathematical Studies, Vol. 29, Van Nostrand Reinhold, New York-Totonto, ON-London, 1970. · Zbl 0212.40004 |

[13] | Furstenberg, H., Disjointness in ergodic theory, minimal sets and diophantine approximation, Mathematical Systems Theory, 1, 1-49, (1967) · Zbl 0146.28502 |

[14] | Glimm, J., Locally compact transformation groups, Transactions of the American Mathematical Society, 101, 124-138, (1961) · Zbl 0119.10802 |

[15] | Janvresse, E.; Roy, E.; Rue, T., Invariant measures for Cartesian powers of Chacon infinite transformation, Israel Journal of Mathematics, 224, 1-37, (2018) · Zbl 1406.37011 |

[16] | Kakutani, S., On equivalence of infinite product measures, Annals of Mathematics, 49, 214-224, (1948) · Zbl 0030.02303 |

[17] | Rudolph, D. J., An example of a measure preserving map with minimal self-joinings, and applications, Journal d’Analyse Mathématique, 35, 97-122, (1979) · Zbl 0446.28018 |

[18] | Rudolph, D. J.; Silva, S. E., Minimal self-joinings for nonsingular transformations, Ergodic Theory and Dynamical Systems, 9, 759-800, (1989) · Zbl 0677.28008 |

[19] | Rue, T., Joinings in ergodic theory, 5037-5051, (2009), New York |

[20] | Schmidt, K., Infinite invariant measures on the circle, Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, (Rome, 1975), 21, 37-43, (1977) |

[21] | Silva, C. E.; Witte, D., On quotients of nonsingular actions whose self-joinings are graphs, International Journal of Mathematics, 5, 219-237, (1994) · Zbl 0814.28008 |

[22] | Yuasa, H., Uniform sets for infinite measure-preserving systems, Journal d’Analyse Mathématique, 120, 333-356, (2013) · Zbl 1307.37003 |

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