Journal article
Topologies on the group of Borel automorphisms of a standard Borel space
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, Vol.27(2), pp.333-385
06/01/2006
Abstract
The paper is devoted to the study of topologies on the group Aut(X, B) of all Borel automorphisms of a standard Borel space (X, B). Several topologies are introduced and all possible relations between them are found. One of these topologies, tau, is a direct analogue of the uniform topology widely used in ergodic theory. We consider the most natural subsets of Aut(X, B) and find their closures. In particular, we describe closures of subsets formed by odometers, periodic, aperiodic, incompressible, and smooth automorphisms with respect to the defined topologies. It is proved that the set of periodic Borel automorphisms is dense in Aut(X, B) (Rokhlin lemma) with respect to tau. It is shown that the tau-closure of odometers (and of rank 1 Borel automorphisms) coincides with the set of all aperiodic automorphisms. For every aperiodic automorphism T is an element of Aut(X, B), the concept. of a Borel-Bratteli diagram is defined and studied. It is proved that every aperiodic Borel automorphism T is isomorphic to the Vershik transformation acting on the space of infinite paths of an ordered Borel-Bratteli diagram. Several applications of this result are given.
Details
- Title: Subtitle
- Topologies on the group of Borel automorphisms of a standard Borel space
- Creators
- Sergey BezuglyiAnthony H. DooleyJan Kwiatkowski
- Resource Type
- Journal article
- Publication Details
- TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, Vol.27(2), pp.333-385
- ISSN
- 1230-3429
- Publisher
- Juliusz Schauder Ctr Nonlinear Studies
- Number of pages
- 53
- Language
- English
- Date published
- 06/01/2006
- Academic Unit
- Mathematics
- Record Identifier
- 9984936490402771
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