The talk is devoted to Bratteli-Vershik models of general
compact zero-dimensional
systems with the action of a homeomorphism. 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 a Bratteli diagram. We prove that a compact
invertible
zero-dimensional system has a decisive Bratteli-Vershik model if and
only if the set of
aperiodic points is dense, or its closure misses one periodic orbit.
This is a joint work
with T. Downarowicz.