Transitive Flows: An Algebraic Theory

Louisiana State University, Baton Rouge LA

Investigators

Abstract

Abstract Lisan The aim of this project is to pursue preliminary investigations into the study of dynamical systems or flows via algebraic (semigroup) approaches. It is anticipated that a regular proposal for an NSF grant will be submitted subsequent to this planning project. The study of dynamical systems has received considerable attention through out this century, and this interest has mushroomed in recent years with topics like chaos, turbulence, fractals and stability. One particular approach to the theory of dynamical systems is the field of topological dynamics. Although significant amount of literature exists on the subject, algebraic (semigroup) approaches have, however had limited usage in the area of topological dynamics. The proposer feels that these approaches give more insights into the field and that there is richer theory waiting to be developed. Some of the groundwork has already been laid out in this direction. The proposer and J. Lawson have investigated transitive flows and have shown that these flows reduce to the study of closed left congruences on the universal compactification of the acting semigroup and that the minimal flows are the corresponding quotients of a minimal left ideal in the universal compactification. Important notions of distality and proximality are also studied via factorization of these congruences.