Maciej P. Wojtkowski.
W -- Flows on Weyl Manifolds and Gaussian Thermostats.
(76K, AMS-TEX)
ABSTRACT. 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 in
\cite{Ch-E-L-S}, is uniformly hyperbolic, if only $r|E| < 1$, and this
condition is sharp.