A time change of a flow $\{T_{t}\}$ , ${t\in \mathbb{R}}$ , is a reparametrization of the orbits of the flow such that each orbit is mapped to itself by an orientation-preserving homeomorphism of the parameter space. If a flow $\{S_{t}\}$ is isomorphic to a flow obtained by a reparametrization of a flow $\{T_{t}\}$ , then we say that $\{S_{t}\}$ and $\{T_{t}\}$ are isomorphic up to a time change. For ergodic flows $\{S_{t}\}$ and $\{T_{t}\}$ , Kakutani showed that this happens if and only if the two flows have Kakutani equivalent transformations as cross-sections. We prove that the Kakutani equivalence relation on ergodic invertible measure-preserving transformations of a standard non-atomic probability space is not a Borel set. This shows in a precise way that classification of ergodic transformations up to Kakutani equivalence is impossible. In particular, our results imply the non-classifiability of ergodic flows up to isomorphism after a time change. Moreover, we obtain anti-classification results under isomorphism for ergodic invertible transformations of a sigma-finite measure space. We also obtain anti-classification results under Kakutani equivalence for ergodic area-preserving smooth diffeomorphisms of the disk, annulus, and 2-torus, as well as real-analytic diffeomorphisms of the 2-torus. Our work generalizes the anti-classification results under isomorphism for ergodic transformations obtained by Foreman, Rudolph, and Weiss.
