We consider a descriptive set-theoretic analog of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly two aperiodic Borel automorphisms of uncountable Polish spaces. Using this, we classify all Borel ℝ-flows up to _C_∞-time-change isomorphism. We then extend the notion of descriptive Kakutani equivalence to all (not necessarily injective) Borel functions, and provide a variety of results leading towards a complete classification. The main technical tools are a series of Glimm–Effros and Dougherty–Jackson–Kechris-style embedding theorems.
Cite this article
Benjamin D. Miller, Christian Rosendal, Descriptive Kakutani equivalence. J. Eur. Math. Soc. 12 (2010), no. 1, pp. 179–219DOI 10.4171/JEMS/194