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Infinite measure preserving transformations with Radon MSJ. (English) Zbl 1403.28015
Minimal 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)].

MSC:
28D05 Measure-preserving transformations
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