  We propose a theory "a 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.
