A Theorem of Sell on Periodic Solutions
by
Joe Auslander
University of Maryland

Let j_t be a flow on a compact metric space X, and let x, y in X. Recall that x and y are said to be positively asymptotic if lim_{t --> \infty} d(j_t(x), j_t(y))=0. We say that x and y are shift asymptotic if there is a \tau in R such that x and j_\tau(y) are positively asymptotic.

Theorem Suppose x in X satisfies: there is a \delta > 0 such that if d(j_s (x), y) < \delta for some s in R then x and y are shift asymptotic. Then the omega limit set of x consists of a single periodic orbit.

This implies a theorem of George Sell on bounded solutions of autonomous systems of differential equations (Journal of Differential Equations, 1966, 143-157). Our hypotheses are weaker than Sell's. The proof depends on the properties of the equicontinuous structure relation for minimal flows.

Date received: June 18, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-97.