I will talk about the extension to the setting of Banach space mappings a concept which has proven highly useful in the study of finite-dimensional dynamical systems exhibiting chaotic behavior, that of SRB measures. This extended notion of SRB measure and our results potentially apply to a large class of dissipative PDE, including dissipative parabolic and dispersive wave equations.

We generalize two results known in the finite-dimensional setting. The first is a geometric result, absolute continuity of the stable foliation, which in particular implies that an SRB measure with no zero exponents is visible, in the sense of time averages converging to spatial averages, with respect to a large subset of phase space. The second is the characterization of the SRB property in terms of the relationship between a priori different quantifications of chaotic behavior, Lyapunov exponents and metric entropy.

Complications of our infinite-dimensional environment include: (1) the absence of Lebesgue measure as a reference measure, not even k-dimensional volume elements (whereas the finite dimensional theory heavily involves the notion of volume growth along unstable leaves); and (2) mappings in our setting are not locally onto or differentiably invertible, possibly exhibit arbitrarily strong rates of contraction (even near attractors).

This work is joint with Lai-Sang Young.