This paper deals with a
property of group actions called approximate transitivity, defined by A.
Connes and E. J. Woods. Their definition arose from solving the following
interesting problem about von Neumann algebras: when is a von Neumann factor
ITPFI? Here we study the property strictly from the ergodic theory point of
view, and restate and prove part of their theorem in this context. Roughly
speaking, a group action is approximately transitive if any finite collection of
probability measures equivalent to the given measure on the space can be
approximated by the convex hull of a single (equivalent) probability measure
pushed around the space of measures under the action of the group. The main
question one might ask about approximate transitivity is whether it is a new
characterization of an already known property in ergodic theory, and if not, what are
its properties? For example, Connes and Woods have already shown that
approximate transitivity of a measure-preserving transformation implies zero
entropy.
One way to understand the property is to reinterpret the theorem of Connes and
Woods in the context of ergodic transformations. In this paper we prove that an
odometer of product type has an approximately transitive Poincaré flow. In the first
section of the paper we give the definitions, state a few known properties and prove
some new properties of approximate transitivity. For example we prove that any AT
flow is the factor action of a group action with simple spectrum. In the second section
we give a short ergodic theoretic proof of one direction of the theorem of Connes and
Woods.
