Accession Number : ADA130575

Title :   A Generalization of Denjoy's Theorem on Diffeomorphisms of the Circle.

Descriptive Note : Technical summary rept.,


Personal Author(s) : Hall,Glen Richard

PDF Url : ADA130575

Report Date : May 1983

Pagination or Media Count : 16

Abstract : Discrete dynamical systems on manifolds, i.e., iterates of a map from a manifold to itself, can serve as simple qualitative models of real systems in physics, biology, and other sciences. Even maps on manifolds of low dimension (one and two) have been used as models since complicated dynamics often appears already in this setting. Often, when one is studying a model of a physical or biological system, one is particularly interested in the stable orbits, i.e. If a small error is made in setting the initial conditions, it would be hoped that this error would not grow with time, and, even better, that it would decay with time. Such stable orbits have long been observed, e.g. stable fixed points, periodic orbits form the most elementary examples. It was shown by Denjoy that for maps which are sufficiently smooth diffeomorphisms of the circle, these are the only examples of stable orbits, i.e. any stable orbit must be asymptotic to a periodic orbit. In this report the authors attempts to generalize this theorem to diffeomorphisms on two-dimensional manifolds. The theorem given requires the extra condition that the map be expansive off the stable set, and presented is an easy example which shows that some additional technical condition will be necessary if expansive is to be removed.

Descriptors :   *Mathematical models, *Maps, *Circles, *Set theory, Theorems, Orbits, Stability, Perturbations, Iterations, Two dimensional, Expansion

Subject Categories : Theoretical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE