Abstract. 
(Joint with Lai-Sang Young) In this talk, we discuss the extension of the
characterization of SRB measures, as those for which volume growth on the
unstable foliation coincides with metric entropy, to the setting of C^2
Fréchet differentiable Banach space mappings leaving invariant compactly
supported Borel probability measures with finitely-many positive Lyapunov
exponents. The results discussed generalize previously known results, due
to Ledrappier and others, for diffeomorphisms of finite dimensional
Riemannian manifolds, and are applicable to dynamical systems defined by
large classes of dissipative parabolic PDEs.

There are two major differences in this setting. The first is that there is
no natural notion of d-dimensional volume element, and whatever notion
volume we use must not only be compatible with the MET, but should also be
regular enough to support distortion estimates. The second is that the map
need not be invertible away from its attractor, and contraction in stable
directions may be arbitrarily strong.