Lyapounov Functions of closed Cone Fields: from Conley Theory to Time Functions.

Patrick Bernard (1, 2) , Stefan Suhr (1, 2, 3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Fachbereich Mathematik [Hamburg]

We propose a theory " à la Conley " for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independent such as the existence of Lyapounov functions) the case of continuous vector-fields on manifolds, of differential inclusions, of Lorentzian metrics, and of continuous cone fields. We generalize to this setting the equivalence between stable causality and the existence of temporal functions. We also generalize the equivalence between global hyperbolicity and the existence of a steep temporal functions.

We propose a theory " à la Conley " for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independent such as the existence of Lyapounov functions) the case of continuous vector-fields on manifolds, of differential inclusions, of Lorentzian metrics, and of continuous cone fields. We generalize to this setting the equivalence between stable causality and the existence of temporal functions. We also generalize the equivalence between global hyperbolicity and the existence of a steep temporal function. —– Résumé. On développe une théorie à la Conley pour les champs de cones, qui utilise une notion d'orbites relaxées basée sur les élargissements de cones dans l'esprit de la géométrie des espaces temps. On travaille dans le contexte des champs de cones fermés (ou, ce qui est équivalent, semi-continus), avec des singularités. Ce contexte contient (pour les questions indépendantes de la paramétrisation, comme l'existence de fonctions de Lyapounov) le cas des champs de vecteurs continus, celui des inclusions différentielles, des métriques Lorentziennes, et des champs de cones continus. On généralise à ce contexte l'équivalence entre la causalité stable et l'existence d'une fonction temporale. On généralise aussi l'équivalence entre l'hyperbolicité globale et l'existence d'une fonction temporale uniforme.

Domaines: Systèmes dynamiques [math.DS]; Relativité Générale et Cosmologie Quantique [gr-qc]

Patrick Bernard, Stefan Suhr. Lyapounov Functions of closed Cone Fields: from Conley Theory to Time Functions.. 2017. ⟨hal-01487859v2⟩