Workshop

The scope of the conference will be fairly wide, but it will emphasize specific problems in dynamical systems, smooth ergodic theory, and related topics, including differential geometry, group theory and number theory, to name a few, that have seen much progress, but where significant problems vital to the field remain open. Specific examples are the Katok entropy rigidity conjecture for geodesic flows of negatively curved manifolds, the Boltzmann ergodic hypothesis, Liouvillean phenomena, the construction of metrics with ergodic geodesic flow, smooth rigidity of actions of abelian groups of higher rank. This workshop will include a session to discuss open problems and to produce a list of these, with the intent of producing a concrete agenda for further research on several fronts. The following are some of the subjects related to dynamical systems that have seen outstanding advances recently, and that will be significant components of this workshop. Billiards, and the question of their hyperbolicity and ergodicity. The theory of partially hyperbolic systems and nonuniformly hyperbolic systems. The close relations between dynamical systems and differential (especially Riemannian) geometry. The significant interactions that have developed between dynamical systems and number theory. The extension of methods from the study of invariant measures for homogeneous and affine actions of higher rank abelian groups, initiated by Furstenberg's x2,x3-question, to other areas. The study of actions of lattices in semisimple Lie groups. The impact on statistical physics of Ruelle's work showing that a generalized Sinai--Ruelle--Bowen-state depends differentiably on the hyperbolic diffeomorphism for which it is defined. The workshop will be held on the occasion of the 60th birthday of Anatole Katok. We will take this occasion to honor Katok during the conference for his significant influence on almost all areas in dynamical systems through his research, collaboration, teaching, organization and exposition. This workshop is sponsored by the Clay Mathematical Institute and hosted by MSRI. It also benefits from generous contributions from Tufts University and the Pennsylvania State University.