Category and compact leaves
by Steven Hurder

HISTORY:

The work of Colman and Hurder was presented at the conference "Foliations: Geometry and Dynamics" at the Banach Center, Warsaw on June 2, 2000. In the discussion afterwords, which included Elmar Vogt and Paul Schweitzer, the question was raised and answered as to whether there are examples besides compact Hausdorff foliations for which the transverse saturated category is finite. The "double Reeb" foliation of the 2-torus was proposed as an example with open leaves and finite category.

Study of this example immediately after the conference led to the main result of this paper, that the category is finite implies there are always compact leaves. However, further examples showed that such foliations can also have non-trivial minimal sets, but such minimal sets cannot be expansive.