Many concrete deterministic dynamical systems exhibit apparently random
behaviour. This puzzle is studied my finding a time invariant probability
measure and discussing the statistical phenomenon using this measure. In
this way various systems (e.g. given by PDE's) can be said to be
"completely random" or "completely deterministic".

This leads to the project of classifying invariant probability measures.
The ergodic decomposition theorem shows that the basic building blocks of
these measures are the ergodic measures, which form a dense G_\delta set.
The equivalence relation of isomorphism is given by a Polish group action.
Thus the tools of descriptive set theory directly apply and one can show
that the action is "turbulent" and complete analytic. This precludes any
kind of recognizable classification.

This is joint work with Dan Rudolph and Benjy Weiss.