  This paper is a survey devoted to the study of probability and infinite
ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We
focus mostly on the cases when a homeomorphism has either a unique ergodic
invariant measure or finitely many such measures (finitely ergodic
homeomorphisms). Since every Cantor dynamical system $(X,T)$ can be realized as
a Vershik map acting on the path space of a Bratteli diagram, we use
combinatorial methods developed in symbolic dynamics and Bratteli diagrams
during the last decade to study the simplex of invariant measures.
