On Conley's fundamental theorem of dynamical systems

To the memory of Charles C. Conley We generalize Conley's fundamental theorem of dynamical systems in Conley index theory. We also conclude the existence of a regular index filtration for every Morse decomposition. 1. Introduction. Conley is mostly known for his fundamental theorem of dynami-cal systems and his homotopy index theory [1]. In the former, he proved that every continuous flow on a compact metric space admits a Lyapunov function which strictly decreases along nonchain recurrent orbits. This result has been developed by Franks for homeomorphisms [3] and Hurley for noncompact metric spaces [5, 6, 7, 8]. In the latter, Conley defined a homotopy invariant for any isolated invariant set for a continuous flow. This invariant gives some valuable information about the behavior of the isolated invariant set. This paper concerns a combination of these two masterpieces. Indeed, we show the existence of Conley's Lyapunov function on every index pair in the sense of Conley index theory. We also conclude the existence of a regular index filtration for every Morse decomposition.