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Orbit equivalences of Borel flows

We provide an overview of the orbit equivalence theory of Borel flows. In general, an orbit equivalence of
two group actions is a bijective map between phase spaces that maps orbits onto orbits. Such maps are
often further required to posses regularity properties depending on the category of group actions that
is being considered. For example, Borel dynamics deals with Borel orbit equivalences, ergodic theory
considers measure-preserving maps, topological dynamics assumes continuity, etc.

Since its origin in 1959 in the work of H. Dye, the concept of orbit equivalence has been studied quite
extensively. While traditionally larger emphasis is given to actions of discrete groups, in this talk we
concentrate on free actions of Rn-flows while taking the viewpoint of Borel dynamics.

For a free Rn-action, an orbit can be identified with an “affine” copy of the Euclidean space, which allows
us to transfer any translation invariant structure from Rn onto each orbit. The two structures of utmost
importance will be that of Lebesgue measure and the standard Euclidean
topology. One may than consider orbit equivalence maps that furthermore preserve these structures on
orbits. Resulting orbit equivalences are called Lebesgue orbit equivalence (LOE) and time-change
equivalence respectively.

Properties of LOE maps correspond most closely to those of orbit equivalence maps between their
discrete counterparts — free Zn actions. We illustrate this by discussing the analog for Rn-flows of
Dougherty-Jackson-Kechris classification of hyperfinite equivalence relations.

Orbit equivalences of flows often arise as extensions of maps between cross sections — Borel sets that
intersect each orbit in a non-empty countable set. Furthermore, strong geometric restrictions on crosssections
are often necessary. Following this path, we explain why one-dimensional R-flows posses cross
sections with only two distinct distances between adjacent points, and show how this implies classification
of R-flows up to LOE.

If time permits, we conclude the talk with an overview of time-change equivalence, emphasizing the
difference between Borel dynamics and ergodic theory.