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

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract:

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

equivalence respectively.

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

http://arxiv.org/abs/1504.00958.