An Algorithmic Approach to Chain Recurrence

Abstract

In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley’s Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.