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

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

2004-10-23

arxiv.org/abs/math/0410506v2

Mathematics

Dynamical Systems

53 pages

Scientific paper

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 $T\in Aut(X, B)$, 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.

Affiliated with

Also associated with

No associations

Rating

Topologies on the group of Borel automorphisms of a standard Borel space does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Topologies on the group of Borel automorphisms of a standard Borel space, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Topologies on the group of Borel automorphisms of a standard Borel space will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-485369