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

摘要

Let F be a codimension-one, F_1 -foliation on a manifold M without boundary. In this work we show that if the Godbillon-Vey class GV(F) ∈ 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 F_1 -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, then T 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_1 -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.