Topological Methods in Nonlinear Analysis

Topologies on the group of Borel automorphisms of a standard Borel space

Abstract

The paper is devoted to the study of topologies on the group ${\rm Aut}(X,{\mathcal B})$ of all Borel automorphisms of a standard Borel space $(X, {\mathcal B})$. 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 ${\rm Aut}(X,{\mathcal B})$ 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 ${\rm Aut}(X,{\mathcal B})$ (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 $T\in {\rm Aut}(X,{\mathcal B})$, 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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 27, Number 2 (2006), 333-385.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463144526

Mathematical Reviews number (MathSciNet)
MR2237460

Zentralblatt MATH identifier
1136.37002

Citation

Bezuglyi, Sergey; Dooley, Anthony H.; Kwiatkowski, Jan. Topologies on the group of Borel automorphisms of a standard Borel space. Topol. Methods Nonlinear Anal. 27 (2006), no. 2, 333--385. https://projecteuclid.org/euclid.tmna/1463144526

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