Maciej P. Wojtkowski

Principles for the design of billiards with nonvanishing Lyapunov exponents

We introduce a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents.

Invariant families of cones and Lyapunov exponents

We show that in several cases preservation of cones leads to non-vanishing of (some) Lyapunov exponents. It gives simple and effective criteria for nonvanishing of the exponents, which is demonstrated on the example of the billiards studied by Bunimovich. It is also shown that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view.

Ergodicity in Hamiltonian Systems

We discuss the Sinai method of proving ergodicity of a discontinuous Hamiltonian system with (nonuniform) hyperbolic behavior.

A system of one dimensional balls with gravity

We introduce a Hamiltonian system with many degrees of freedom for which the nonvanishing of (some) Lyapunov exponents almost everywhere can be established analytically.

W-flows on Weyl manifolds and Gaussian thermostats

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, is uniformly hyperbolic, if only r|E|

Linearly stable orbits in

We construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart. It shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing Lyapunov exponents cannot be easily extended to 3 dimensions.

3

We modify the system introduced in [W1] so that we can establish the nonvanishing ofall Lyapunov exponents easily.

-dimensional billiards

We describe ergodic properties of the system of two hard discs with arbitrary masses moving on the two dimensional torus.

The system of one-dimensional balls in an external field. II

We provide a list of all locally metric Weyl connections with nonpositive sectional curvatures on two types of manifolds, n-dimensional tori T and M = S 1 x S n-1 with the standard conformal structures. For M we prove that it carries no other Weyl connections with nonpositive sectional curvatures, locally metric or not. In the case of T we prove the same in the more narrow class of integrable connections.

Two-particle billiard system with arbitrary mass ratio

We study linear stability of a periodic orbit in the hamiltonian system with many degrees of freedom introduced in [W1].