Weyl geometry and dynamics of Gaussian thermostats

A Weyl structure on a manifold generalizes a Riemannian metric. It is a torsion free connection with the parallel transport preserving a given conformal class of Riemannian metrics. We established recently that the geodesic flows of Weyl connections (W-flows) coincide with the Gaussian thermostats, introduced 15 years ago by Hoover and Posch as a model thermostatting mechanism in computational nonequilibrium statistical mechanics. Two kinds of problems are raised by this discovery. What methods and results of the theory of Riemannian geodesic flows apply to these nonconservative deformations ? What does the Weyl geometry tell us about the dynamical models of nonequilibrium statistical mechanics ? In particular we will address the validity of the Conjugate Pairing Rule of Cohen and Morriss and the Fluctuation Theorem of Gallavotti and Cohen.