The structure of hy-per nite Borel equivalence relations

  title={The structure of hy-per nite Borel equivalence relations},
  author={Randall L. Dougherty and Steve Jackson and Alexander S. Kechris},
We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete… 

The classification of hypersmooth Borel equivalence relations

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which

Borel homomorphisms of smooth -ideals

Given a countable Borel equivalence relation E on a Polish space, letIE denote the -ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism fromIE toIF if

Amenable versus hyperfinite Borel equivalence relations

  • A. Kechris
  • Mathematics
    Journal of Symbolic Logic
  • 1993
By a result of Feldman-Moore [FM], E is induced by the orbits of a Borel action of a countable group G on X so that every equivalence class [X]_E is countable.

Linear algebraic groups and countable Borel equivalence relations

This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with

Cardinal characteristics and countable Borel equivalence relations

A family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$.

Borel asymptotic dimension and hyperfinite equivalence relations

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In

Rigidity Theorems For Actions Of Product Groups And Countable Borel Equivalence Relations

This Memoir is both a contribution to the theory of Borel equivalence relations, considered up to Borel reducibility, and measure preserving group actions considered up to orbit equivalence. Here E

Countable Borel equivalence relations

These notes cover an introduction to countable Borel equivalence relations. The theory of countable Borel equivalence relation is a huge subject. It has a broad impact in several areas of mathematics

Realizations of countable Borel equivalence relations

We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples

Property τ and Countable Borel Equivalence Relations

For each prime q, it is proved that the orbit equivalence relations arising from the natural actions of SL2(ℤ[1/q]) on the projective lines over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.



An equivalence relation that is not freely generated

In this paper it is shown that there exists a Borel equivalence relation with countable equivalence classes that is not generated by a free Borel action of a countable discrete group. In Feldman and

On non-singular transformations of a measure space. I

We consider a Lebesgue measure space (M, ∇, m). By an automorphism of (M, ∇, m) we mean a bi-measurable transformation of (M, ∇, m) that together with its inverse is non-singular with respeot to m.

On quasi-invariant measures in uniquely ergodic systems

Let X be a compact metrizable space, ~ (X) its G-algebra of Borel sets, and let T be a homeomorphism of X. We denote for a Borel measure # on X by T/2 the measure that is obtained by setting

On orbit equivalence of borel automorphisms

LetE andF be two Borel sets of the countable productZ of the two point space {0,1}. Assume thatE andF are invariant sets for the odometer transformationR and thatE andF are of measure zero with

Countable sections for locally compact group actions

  • A. Kechris
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1992
Abstract It has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete

A Glimm-Effros dichotomy for Borel equivalence relations

A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in

The Structure of Borel Equivalence Relations in Polish Spaces

An exposition of recent work on Borel equivalence relations in Polish spaces is presented. This includes a general Glimm-Effros dichotomy for Borel equivalence relations and a study of countable

Amenable equivalence relations and Turing degrees

In [12] Slaman and Steel posed the following problem: Assume ZF + DC + AD. Suppose we have a function assigning to each Turing degree d a linear order <d of d. Then must the rationals embed order

Essays in Commutative Harmonic Analysis

This book considers various spaces and algebras made up of functions, measures, and other objects-situated always on one or another locally compact abelian group, and studied in the light of the

An amenable equivalence relation is generated by a single transformation

Abstract We prove that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a non-singular transformation T of X such that, up to a null set: It follows that any two