Dynamics and the Godbillon-Vey class of $C^1$-foliations
by Steven Hurder and Remi Langevin

ABSTRACT:

We give a direct proof that a codimension one, $C^2$-foliation $\F$ with non-zero Godbillon--Vey class $GV(\F) \in H^3(M)$ has a hyperbolic resilient leaf. Our approach is based on methods of $C^1$-dynamics, 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. For a $C^2$-foliation, $GV(\F)$ non-zero implies the Godbillon measure is also non-zero, and the result follows. These results apply for both the case when $M$ is compact, and when $M$ is an open manifold.