Linda J. Moniz, University of Maryland, James A. Yorke, University of Maryland

We develop a rigorous formalism for a computational theory for finding

various sets important to the dynamics of a map. We develop a novel

approach to locating basin boundaries and arbitrary isolated

invariant sets. Our comprehensive theory also covers a number of

published results. Examples of these include periodic orbits,

chain-recurrent sets, and maximal invariant sets. We begin by

describing "upper-bound sets", that is, sets that contain the

desired set. In our theory, each upper-bound set is a collection

of grid boxes. We give conditions which guarantee that as precision

and resolution increase, the upper-bound sets converge to (shrink

down to) the desired set. We describe a measure of closeness of

particular upper-bound sets to the sets they contain. We prove

that our implementation of the theory for C^{2} maps achieves,

in the sense of the measure, the closest possible upper-bound

sets of this type.