Lyapounov functions of closed cone fields: from Conley theory to time functions. (English. French summary) Zbl 1391.83010
Summary: We propose a theory “à la Conley” [C. Conley, Isolated invariant sets and the Morse index. Providence, RI: American Mathematical Society (AMS) (1978; Zbl 0397.34056)] 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.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 53Z05 | Applications of differential geometry to physics |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 54H20 | Topological dynamics (MSC2010) |
| 34D08 | Characteristic and Lyapunov exponents of ordinary differential equations |
Keywords:
Lyapounov functions; closed cone fields; Conley theory; time functions; Lorentzian metrics; global hyperbolicityCitations:
Zbl 0397.34056References:
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