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 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 D^{2 }x [0, 1] endowed with a flow that enters through D^{2 }x{0}, exits through D^{2 }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 C^{1 }.
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.