What are SRB measures, and which dynamical systems have them?

This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle and Bowen in the 1970s. SRB measures, as these objects are called, play an important role in the ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly speaking, • SRB measures are the invariant measures most compatible with volume when volume is not preserved; • they provide a mechanism for explaining how local instability on attractors can produce coherent statistics for orbits starting from large sets in the basin. An outline of this paper is as follows. The original work of Sinai, Ruelle and Bowen was carried out in the context of Anosov and Axiom A systems. For these dynamical systems they identified and constructed an invariant measure which is uniquely important from several different points of view. These pioneering works are reviewed in Section 1. Subsequently, a nonuniform, almost-everywhere notion of hyperbolicity expressed in terms of Lyapunov exponents was developed. This notion provided a new frame-work for the ideas in the last paragraph. While not all of the previous characteriza-tions are equivalent in this broader setting, the central ideas have remained intact, leading to a more general notion of SRB measures. This is discussed in Section 2. The extension above opened the door to the possibility that the dynamics on many attractors are described by SRB measures. Determining if this is (or is not) the case, however, let alone proving it, has turned out to be very challenging. No genuinely nonuniformly hyperbolic examples were known until the early 1990s, when SRB measures were constructed for certain Hénon maps. Today we still do not have a good understanding of which dynamical systems admit SRB measures, but some progress has been made; a sample of it is reported in Section 3