Convergence of Dynamically Defined Upper Bound Sets

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.