Simplicial Models for the Global Dynamics of Attractors

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. In this paper, we use the techniques of the Conley index 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 A natural problem in dynamics is to seek situations in which a nite amount of data (produced either numerically or analytically) allows the topology and dynamics of a compact invariant set S to be recovered, at least partially. Before considering results of this type, it is necessary to clarify exactly what it means for information about S \to be recovered." One point of view is to give a known system M and show that either M embeds in S, or that S maps via a semi-conjugacy onto M. The existence of a periodic orbit can be viewed as an example of the former