Category and compact leaves

摘要

We prove that if F is a C~1 -foliation of a compact manifold M with finite transverse saturated LS category, cat_s~m (M, F) < ∞, then F has a compact leaf. In contrast, we show that if F is expansive on some non-trivial minimal set of F, then cat_s~m (M, F) = ∞. Examples of foliations are given to illustrate the main results of the paper.