MG11 
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Vector Field Induced Chaos in Multi-dimensional Homogeneous Cosmologies

We show that in multidimensional gravity, vector fields completely determine the structure and properties of singularity. It turns out that, in presence of a vector field, the oscillatory regime exists for all homogeneous models in any number of space-time dimensions. By analyzing the Hamiltonian equations, we derive the Poincaré return map associated with the Kasner indices and fix the rules according to which the Kasner vectors rotate. In correspondence to a four-dimensional space-time, the oscillatory regime here constructed overlaps the usual Belinski-Khalatnikov-Liftshitz one.

Classical and Quantum Aspects of the Inhomogeneous Mixmaster Chaoticity

The dynamics of the generic inhomogeneous Mixmaster is analyzed via an ADM-reduction of the Hamiltonian constraints and adopting Misner-Chitre'-like variables. We establish that, near the Big-Bang, the evolution of the system is isomorphic to a billiard on a Lobachevski plane and the corresponding microcanonical distribution is calculated. The comparison between the continuity equation on the ensemble and the WKB semi-classical limit of the Schr\"odinger equation fixes the normal ordering procedure. Finally, the features of the full quantum dynamics, with particular attention to the ground state level, are outlined.

Covariant Description of the Inhomogeneous Mixmaster Chaos

We outline the covariant nature of the chaos characterizing the generic cosmological solution near the initial singularity, i.e. the so-called inhomogeneous Mixmaster model. Our analysis is based on a "gauge" independent ADM-reduction of the dynamics to the physical degrees of freedom. The out-coming picture shows how the inhomogeneous Mixmaster model is isomorphic point by point in space to a billiard on a Lobachevsky plane. The existence of an asymptotic (energy-like) constant of the motion allows us to construct the Jacobi metric associated to the geodesic flow and to calculate a non-zero Lyapunov exponent in each space point. The chaos covariance emerges from the independence of our scheme of the form of the lapse function and the shift vector; the origin of this result relies on the dynamical decoupling of space points, which takes place near the singularity, due to the asymptotic approach of the potential term to infinite walls.