The minimal set of Kuperberg's plug
Ana Rechtman (Universit´e de Strasbourg) joint work with Steven Hurder (University of Illinois at Chicago)
In 1993 K. Kuperberg constructed examples of Cand 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, for example, the minimal set is not specified in advance. In the talk, I will present a study of the minimal set. There are several choices that are made along the construction, I will start by presenting sufficient conditions that make the minimal set of topological dimension 2. In this situation, I will give a decomposition of the set into propellers. Propellers are surfaces with boundary tangent to the flow. This decomposition gives a laminated structure to the minimal set, whose laminated entropy is positive. In fact, the minimal set is a resilient leaf.
The flow is tangent to the lamination, and as a consequence of A. Katok's theorem, the flow has entropy zero. I will show that violating a condition in the construction (the radius inequality) gives rise to a plug whose flow has positive entropy and periodic orbits.