Dynamics and the Godbillon-Vey Class of C^1 Foliations

Abstract

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) \in 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 codimension--one 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 F, 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 theorem 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.