Mathematics

The dynamics of the minimal set of Kuperberg's plug

Ana Rechtman (Université de Strasbourg)

Abstract: In 1993 K. Kuperberg constructed examples of C∞ and real analytic flows without periodic orbits on any closed 3-manifold. These examples continue to be the only known examples with such properties. A plug is a manifold with boundary of the type D2 x [0, 1] endowed with a flow that enters through D2 x{0}, exits through D2 x{1} and is parallel to the rest of the boundary. Moreover, it has the particularity that there are orbits that enter the plug and never exit, that is there are trapped orbits. The closure of a trapped orbit limits to a compact invariant set contained entirely within the interior of the plug. This compact invariant set contains a minimal. The first construction of a plug without periodic orbits was done by P. Schweitzer. This plug is constructed from the minimal set that is the Denjoy flow on the torus, implying that the flow of the plug is only C1 . Kuperberg's construction is completely different in nature. I will present a study of the minimal set, its dynamics and topology. (Joint work with Steven Hurder (University of Illinois at Chicago).)