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Chaotic dynamics of spatially homogeneous spacetimes

Abstract : In 1963, Belinsky, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map’s dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright-Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright-Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows. The Wainwright-Hsu equations turn out to be quite interesting and challenging from a purely dynamical system viewpoint. In order to understand the asymptotic behaviour of (many of) the solutions of these equations, we will in particular be led to: • carry a detailed analysis of the local dynamics of a vector field in the neighborhood of degenerate non- linearizable partially hyperbolic singularities, • deal with non-uniformly hyperbolic maps with singularities for which the usual theory (due to Pesin and Katok-Strelcyn) is not relevant due to the poor regularity of the maps, • consider some unusual arithmetic conditions expressed in terms of continued fractions and use some rather sophisticated ergodic properties of the Gauss map to prove that these properties are generic.
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Tom Dutilleul. Chaotic dynamics of spatially homogeneous spacetimes. Mathematics [math]. Université Paris 13 - Sorbonne Paris Cité, 2019. English. ⟨tel-02488655⟩

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