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

TL;DR

This paper develops a generalized Conley theory for closed cone fields, establishing links between causality conditions and the existence of Lyapunov and temporal functions across various geometric and dynamical settings.

Contribution

It introduces a unified framework for Lyapunov functions in cone fields, extending classical causality results to singular and semi-continuous cases.

Findings

Equivalence between stable causality and temporal functions.

Equivalence between global hyperbolicity and steep temporal functions.

Applicability to vector fields, differential inclusions, and Lorentzian metrics.

Abstract

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.