The Pugh-Shub conjectures, which have been verified in a number of cases, tell us that "most" volume-preserving partially hyperbolic systems are ergodic. It is an interesting question, then, to ask exactly what are the non-ergodic systems, and what kinds of ergodic decompositions do they have?
In this talk, I will show how, using the properties of solvable groups acting on subsets of the real line, one can give a classification of these non-ergodic systems for several large families of partially hyperbolic systems.