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

HISTORY:

Ever since the definition of the foliation geometric entropy by Ghys, Langevin and Walczak, one of the main open problems/conjectures has been to give a direct proof that vanishing entropy implies the secondary characteristic classes of the foliation must vanish. The reasoning why this must be true is simple: vanishing entropy intuitively implies that the foliation dynamics has only weak hyperbolicity, while non-vanishing secondary classes implies that there is a set of positive measure where the Godbillon/Weil measures of the foliation are positive. These measures vanish unless the derivative holonomy cocycle has a large algebraic hull, which was one of the conclusions of the work in this area of Hurder & Katok.

Duminy gave a celebrated proof that the Godbillon-Vey class non-vanishing for codimension one $C^2$-foliations implies there must be resilient leaves, but his proof used the structure theory of $C^2$-foliations in essential ways. This structure theory effectively hides the role of smooth ergodic theory in the proof. Hurder's later proof that the Godbillon-Vey classes vanish for foliations with subexponential growth used only smooth ergodic theory techniques, but did not recover the full strength of the Duminy Theorem.

During the first author's visit to the Univeriste of Bourgogne in Spring 1999, the authors took a new look at this old problem, and discovered that the advances in understanding of the foliation geometric entropy for codimension one foliations, along with two new ideas, yielded a purely ergodic theory proof of the announced theorem.