Dynamical systems, filtrations and entropy

Introduction. I would like to try to pose some new directions in dynamical systems, at least in that part of the subject which deals with the qualitative behavior of the orbit structure of diffeomorphisms and flows of compact differentiable manifolds M without boundary. The birth of our first child, Alexander, to Beth and myself just two weeks before this talk has perhaps made me overly optimistic. But, I think that the subject can use some general approaches even if they be Pollyannic.