Dynamics and the Godbillon–Vey class of 𝐶¹ foliations

抄録

Let ℱ be a codimension-one, 𝐶²-foliation on a manifold 𝑀 without boundary. In this work we show that if the Godbillon–Vey class 𝐺𝑉(ℱ) ∈ 𝐻³(𝑀) is non-zero, then ℱ has a hyperbolic resilient leaf. Our approach is based on methods of 𝐶¹-dynamical systems, and does not use the classification theory of 𝐶²-foliations. We first prove that for a codimension-one 𝐶¹-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 𝐶¹-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 𝐶²-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.