W-flows on Weyl manifolds and Gaussian thermostats

We introduce W-flows, by modifying the geodesic flow on a Weyl manifold, and show that they coincide with the isokinetic dynamics. We establish some connections between negative curvature of the Weyl structure and the hyperbolicity of W-flows, generalizing in dimension 2 the classical result of Anosov on Riemannian geodesic flows. In higher dimensions we establish only weaker hyperbolic properties. We extend the theory to billiard W-flows and introduce the Weyl counterparts of Sinai billiards. We obtain that the isokinetic Lorentz gas with the constant external field E and scatterers of radius r, studied by Chernov, Eyink, Lebowitz and Sinai, is uniformly hyperbolic, if only r|E|<1, and this condition is sharp.