Abstract
This paper deals with countable products of countable Borel equivalence relations and equivalence relations “just above” those in the Borel reducibility hierarchy. We show that if E is strongly ergodic with respect to μ then Eℕ is strongly ergodic with respect to μℕ. We answer questions of Clemens and Coskey regarding their recently defined Γ-jump operations, in particular showing that the ℤk+1-jump of E∞ is strictly above the ℤk-jump of E∞. We study a notion of equivalence relations which can be classified by infinite sequences of “definably countable sets”. In particular, we define an interesting example of such an equivalence relation which is strictly above E ℕ∞ , strictly below =+, and is incomparable with the Γ-jumps of countable equivalence relations.
We establish a characterization of strong ergodicity between Borel equivalence relations in terms of symmetric models, using results from [Sha21]. The proofs then rely on a fine analysis of the very weak choice principles “every sequence of E-classes admits a choice sequence”, for various countable Borel equivalence relations E.
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Acknowledgements
The results in this paper are partially from my PhD thesis. I would like to express my sincere gratitude to my advisor, Andrew Marks, for his guidance and encouragement, and for numerous informative discussions.
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Shani, A. Strong ergodicity around countable products of countable equivalence relations. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2654-5
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DOI: https://doi.org/10.1007/s11856-024-2654-5