# Logic Seminar: Orbit equivalences of multidimensional Borel flows.

**Author: Lona** | Image: Lona

**Author: Lona** | Image: Lona

We will concentrate on two instances of this paradigm and discuss Borel versions of two ergodic theoretical results: Katok’s representation theorem and Rudolph’s result on smooth orbit equivalence. The latter shows that any non-trivial free $R^m$-flow can be transformed into any other $R^m$-flow via an orbit equivalence that is a smooth orientation-preserving diffeomorphism on each orbit. Katok’s theorem provides a multidimensional generalization of the suspension flow construction and shows that all free $R^m$-flows emerge as special flows over $Z^m$-actions.