Logic Seminar

An orbit equivalence between Borel actions of Polish groups is a Borel bijection between phase spaces that preserves orbit partitions. We are interested in free R^m-actions which are known as multidimensional flows. In this case, orbit equivalence is a coarse invariant collapsing all non-trivial flows into one class. Since any translation-invariant structure can be transferred from the acting group onto individual orbits, it is natural to consider strengthenings of orbit equivalence that respect these structures. Notable examples of such structures include measure, topology, and metric. 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.