Invitation to the Logic Seminar at the National University of Singapore
Date: Wednesday, 14 November 2018, 17:00 hrs
Room: S17#04-05, Department of Mathematics, NUS
Speaker: Konstantin Slutsky
Title: Orbit equivalences of Borel flows
The purpose of this talk is to provide an overview of the orbit
equivalence theory of Borel R^n-flows.
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 Henry Abel 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 R^n-flows taking the viewpoint of Borel dynamics.
For a free R^n-action, an orbit can be identified
with an “affine” copy of the Euclidean space, which allows us
to transfer any translation invariant structure from R^n
onto each orbit. The two structures of utmost importance will be
that of Lebesgue measure and standard Euclidean topology.
One may then consider orbit equivalence maps that furthermore
preserve these structures on orbits. Resulting orbit equivalences
are called Lebesgue orbit equivalence (LOE) and time-change
It turns out that properties of LOE maps correspond most closely to
those of orbit equivalence maps between their discrete
counterparts – free Z^n actions.
We illustrate this by outlining a proof of the analog for
R^n-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 cross-sections are often necessary. As a concrete example,
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.
We conclude the talk with an overview of time-change equivalence,
emphasizing the difference between Borel dynamics and ergodic theory
and mentioning several open problems.
The interest reader is referred to the technical report on