Dynamics and the Godbillon-Vey class of C-1 foliations

Let F be a codimension-one, C-2-foliation on a manifold M without boundary. In this work we show that if the Godbillon-Vey class GV(F) is an element of H-3(M) is non-zero, then F has a hyperbolic resilient leaf. Our approach is based on methods of C-1-dynamical systems, and does not use the classification theory of C-2-foliations. We first prove that for a codimensionone C-1-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F) has positive Lebesgue measure. We then prove that if E(F) has positive measure for a C-1-foliation, then F must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive. The proof of this uses a pseudogroup version of the Pliss Lemma. The first statement then follows, as a C-2-foliation with non-zero Godbillon Vey class has non-trivial Godbillon measure. These results apply for both the case when M is compact, and when M is an open manifold.