THE COMPLEXITY OF CLASSIFICATION PROBLEMS IN ERGODIC THEORY

Abstract

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 theory, 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 Γ, two free, measure-preserving, ergodic actions of Γ 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 space

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oai:CiteSeerX.psu:10.1.1.210.8072Last time updated on 10/22/2014

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