11 research outputs found
Pseudo-Riemannian geodesics and billiards
Many classical facts in Riemannian geometry have their pseudo-Riemannian
analogs. For instance, the spaces of space-like and time-like geodesics on a
pseudo-Riemannian manifold have natural symplectic structures (just like in the
Riemannian case), while the space of light-like geodesics has a natural contact
structure. We discuss the geometry of these structures in detail, as well as
introduce and study pseudo-Euclidean billiards. In particular, we prove
pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the
integrability of the billiard in the ellipsoid and the geodesic flow on the
ellipsoid in a pseudo-Euclidean space.Comment: title abbreviated, text edited; to appear in Advances in Mathematic
Contact complete integrability
Complete integrability in a symplectic setting means the existence of a
Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we
describe complete integrability in a contact set-up as a more subtle structure:
a flag of two foliations, Legendrian and co-Legendrian, and a
holonomy-invariant transverse measure of the former in the latter. This turns
out to be equivalent to the existence of a canonical
structure on the leaves of the co-Legendrian foliation. Further, the above
structure implies the existence of contact fields preserving a special
contact 1-form, thus providing the geometric framework and establishing
equivalence with previously known definitions of contact integrability. We also
show that contact completely integrable systems are solvable in quadratures. We
present an example of contact complete integrability: the billiard system
inside an ellipsoid in pseudo-Euclidean space, restricted to the space of
oriented null geodesics. We describe a surprising acceleration mechanism for
closed light-like billiard trajectories
Arithmetical Chaos and Quantum Cosmology
In this note, we present the formalism to start a quantum analysis for the
recent billiard representation introduced by Damour, Henneaux and Nicolai in
the study of the cosmological singularity. In particular we use the theory of
Maass automorphic forms and recent mathematical results about arithmetical
dynamical systems. The predictions of the billiard model give precise
automorphic properties for the wave function (Maass-Hecke eigenform), the
asymptotic number of quantum states (Selberg asymptotics for PSL(2,Z)), the
distribution for the level spacing statistics (the Poissonian one) and the
absence of scarred states. The most interesting implication of this model is
perhaps that the discrete spectrum is fully embedded in the continuous one.Comment: 35 pages, 4 figures. to be published on Classical and Quantum Gravity
(scheduled for January 2009