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Ergodic Theory and LogicResearchAI-Generated

Does every free group action that is orbit equivalent to a totally weakly mixing action also admit a totally weakly mixing representative in its orbit equivalence class?

Related: Connes Embedding Problem, Zimmer Cocycle Superrigidity Theorem, Ornstein-Weiss Theorem on amenable group actions

The central problem asks whether total weak mixing is an orbit equivalence invariant for free group actions on standard probability spaces. Orbit equivalence is a coarse classification that identifies two group actions when there exists a measure-preserving bijection between the underlying spaces that matches up the orbits, without requiring the group structures themselves to be respected. Total weak mixing is a dynamical property capturing a strong form of ergodicity where no non-trivial compact factors exist. The question is whether this dynamical rigidity property is preserved or recoverable within the orbit equivalence class of a free group action.

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