Descriptive Kakutani equivalence (Q845290)

Summary: 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 \(\mathbb R\)-flows up to \(C^{\infty }\)-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.