Summary: THE COMPLEXITY OF CLASSIFICATION PROBLEMS
IN ERGODIC THEORY
ALEXANDER S. KECHRIS AND ROBIN D. TUCKER-DROB
The last two decades have seen the emergence of a theory of set
theoretic complexity of classification problems in mathematics. In these
lectures we will discuss recent developments concerning the application
of this theory to classification problems in ergodic theory.
The first lecture will be devoted to a general introduction to this
area. The next two lectures will give the basics of Hjorth's theory of
turbulence, a mixture of topological dynamics and descriptive set the-
ory, which is a basic tool for proving strong non-classification theorems
in various areas of mathematics.
In the last three lectures, we will show how these ideas can be applied
in proving a strong non-classification theorem for orbit equivalence.
Given a countable group , two free, measure-preserving, ergodic ac-
tions of on standard probability spaces are called orbit equivalent if,
roughly speaking, they have the same orbit spaces. More precisely this
means that there is an isomorphism of the underlying measure spaces
that takes the orbits of one action to the orbits of the other. A remark-
able result of Dye and Ornstein-Weiss asserts that any two such actions