Mathematical Sciences: Dynamics and Topology of Invariant Plane Continua

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This research concerns the dynamics and topology of invariant plane continua. In particular, the research deals with the relationships between the topology of a set invariant under a plane homeomorphism and the dynamics of the homeomorphism restricted to the invariant set. The invariant sets considered are continua (compact and connected sets) and are either plane separating or non-separating. In the case that the invariant continuum separates the plane, the investigator has established connections between periodicity, rotation, and topology on the continuum. The research is in the general area of geometric analysis of dynamical systems. Two examples that have had a profound influence on the development of smooth dynamical systems are the twist maps of the annulus and the Poincare maps associated with the periodically forced van der Pol equation. In both of these situations there arise one-dimensional invariant plane- separating continua on which the dynamical systems has an extraordinarily rich behavior. These situations motivate this research.