College of Liberal Arts & Sciences

# Operator Theory Semimar

**Abstract**: Bratteli-Vershik representations (BV-models for short) have been used to study mainly minimal Cantor systems, where they proved to be extremely useful. In particular, this approach allows to describe the simplex of invariant measures and orbit equivalence classes. The talk is devoted to Bratteli-Vershik models of general compact zero-dimensional dynamical systems

generated by homeomorphisms. An ordered Bratteli diagram is called decisive if the corresponding Vershik map prolongs in a unique way to a homeomorphism of the whole path space of the Bratteli diagram. It is proved that a compact invertible zero-dimensional system has a decisive Bratteli-Vershik model if and only if the set of aperiodic points is either dense, or its closure misses one periodic orbit. Moreover, the notion of weak decisiveness (when the prolongations of the Vershik maps are numerous, but all yield mutually conjugate systems) is studied. It is shown that the systems admitting weakly decisive BV-models also admit decisive BV-models. As an example, a decisive BV-model of the full shift on two symbols is described.

This is a joint work with T. Downarowicz.