Reconstructing the Global Dynamics of Attractors via the Conley Index

Given an unknown attractor A in a continuous dy-namical system, how we can discover the topology and dynamics of A? As a practical matter, how can we do so from only a nite amount of information? One way of doing so is to produce a semi-conjugacy from A onto a model system M whose topology and dynamics are known. The complexity of M then provides a lower bound for the complexity of A. The Conley index can be used to construct a simplicial model and a surjective semi-conjugacy for a large class of attractors. The essential features of this construction are that the model M can be explicitly described; and that the nite amount of information needed to construct it is computable. 1. Introduction The Conley index theory has grown and matured substantially in recent years, in both its computational and theoretical aspects. The theoretical developments have deepened the dynamical information that can be extracted from the index; while the computational improvements have broadened the range of applications in which the index can be computed. In this note, I will consider a combination of recent theoretical and computational developments { a combination that I believe points the way to a rich new realm of applications of the index ideas. The computational developments I refer to are the ongoing eeorts to computerize the index computations. As several papers in this volume describe, it is becoming feasible to input a dynamical system (either continuous or discrete) to a computer, and obtain as the output