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Generic measure preserving transformations and the closed groups they generate. (English) Zbl 1514.37009

Summary: We show that, for a generic measure preserving transformation \(T\), the closed group generated by \(T\) is not isomorphic to the topological group \(L^0(\lambda, \mathbb{T})\) of all Lebesgue measurable functions from [0,1] to \(\mathbb{T}\) (taken with pointwise multiplication and the topology of convergence in measure). This result answers a question of E. Glasner and B. Weiss [Ergodic Theory Dyn. Syst. 25, No. 5, 1521–1538 (2005; Zbl 1085.54027)]. The main step in the proof consists of showing that Koopman representations of ergodic boolean actions of \(L^0(\lambda, \mathbb{T})\) possess a non-trivial spectral property not shared by all unitary representations of \(L^0(\lambda, \mathbb{T})\). The main tool underlying our arguments is a theorem on the form of unitary representations of \(L^0(\lambda, \mathbb{T})\) from our earlier work.

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
22F10 Measurable group actions
22A25 Representations of general topological groups and semigroups
03E15 Descriptive set theory
22D40 Ergodic theory on groups

Citations:

Zbl 1085.54027
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References:

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