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Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 517-542.

Il est bien connu que l’espace-temps de Bianchi IX avec symétrie du groupe SO(3) montre, dans le voisinage du Big Bang, un comportement chaotique à trajectoires typiques dans le sens inverse du mouvement du temps. Ce comportement (modèle Mixmaster de l’univers) peut être codé par le décalage de fractions continues à deux côtés.

Exactement le même décalage code les suites d’intersections de géodésiques hyperboliques dont l’axe imaginaire pur se situe dans le demi-plan complexe supérieur, c’est-à-dire à flot géodésique dans une surface modulaire appropriée.

Une interprétation physique de cette coincidence a été suggérée dans [23] : en effet, le chaos Mixmaster est une description approchée du passage d’un univers quantique chaud au moment du Big Bang à l’univers classique refroidissant. Nous discutons et étayons cette suggestion ici, en regardant le modèle Mixmaster pour la deuxième classe d’espaces-temps de Bianchi IX : ceux avec une symétrie SU(2) (métriques d’Einstein auto-duales). Nous l’étendons aussi au contexte plus général relié aux équations de Painlevé VI.

It is well known that the so called Bianchi IX spacetimes with SO(3)-symmetry in a neighbourhood of the Big Bang exhibit a chaotic behaviour of typical trajectories in the backward movement of time. This behaviour (Mixmaster Model of the Universe) can be encoded by the shift of two-sided continued fractions. Exactly the same shift encodes the sequences of intersections of hyperbolic geodesics with purely imaginary axis in the upper complex half-plane, that is geodesic flow on an appropriate modular surface.

A physical interpretation of this coincidence was suggested in [MaMar14]: namely, that Mixmaster chaos is an approximate description of the passage from a hot quantum Universe at the Big Bang moment to the cooling classical Universe. Here we discuss and elaborate this suggestion, looking at the Mixmaster Model from the perspective of the second class of Bianchi IX spacetimes: those with SU(2)-symmetry (self-dual Einstein metrics). We also extend it to the more general context related to Painlevé VI equations.

Publié le : 2016-07-11
DOI : https://doi.org/10.5802/afst.1503
@article{AFST_2016_6_25_2-3_517_0,
     author = {Yuri Manin and Matilde Marcolli},
     title = {Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {2-3},
     year = {2016},
     pages = {517-542},
     doi = {10.5802/afst.1503},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2016_6_25_2-3_517_0/}
}
Yuri Manin; Matilde Marcolli. Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 517-542. doi : 10.5802/afst.1503. https://afst.centre-mersenne.org/item/AFST_2016_6_25_2-3_517_0/

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