On the Ergodic Hypotheis and Some New Genericity Conjectures for Chaotic Dynamical Systems

Michael Shub

Boltzman's ergodic hypothesis underlies statistical mechanics and much of physical thinking. Yet in 1954, Kolmogorov announced that there are no ergodic Hamiltonian systems in a neighborhood of completely integrable ones. By contrast, in 1967, Anosov found the first open sets of ergodic systems i.e. stably ergodic diffeomorphisms and flows. These systems have the property that time averages converge to the same constant for almost every initial point, not only for the system itself but also for any sufficiently close perturbation. Anosov systems are uniformly hyperbolic while completely integrable systems have no hyperbolic behavior at all.

In this talk , we study the mixed situation which is only partially hyperbolic. Our main themes are first, that a little hyperbolicity goes a long way in guaranteeing stable ergodic behavior (which is more prevalent than one might have imagined) and secondly, that in fact it may be necessary for it. In both cases we use the accessibility property from control theory applied to the hyperbolic part of the derivative. We are led in this direction to formulate some new genericity conjectures.

In a sense, our main theorem may be interpreted to say that even for systems which are not uniformly hyperbolic, the same phenomenon which produces chaotic behavior i.e. some hyperbolicity may also guarantee robust statistics in the form of stable ergodicity.

Various parts of this talk represent joint work with Matt Grayson, Charles Pugh, Jonathan Brezin and Amie Wilkinson.