Translator Disclaimer
April, 2018 Dynamics and the Godbillon–Vey class of $C^1$ foliations
Steven HURDER, Rémi LANGEVIN
J. Math. Soc. Japan 70(2): 423-462 (April, 2018). DOI: 10.2969/jmsj/07027485

Abstract

Let $\mathcal{F}$ be a codimension-one, $C^2$-foliation on a manifold $M$ without boundary. In this work we show that if the Godbillon–Vey class $GV(\mathcal{F}) \in H^3(M)$ is non-zero, then $\mathcal{F}$ 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 $E(\mathcal{F})$ has positive Lebesgue measure. We then prove that if $E(\mathcal{F})$ has positive measure for a $C^1$-foliation, then $\mathcal{F}$ 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 $C^2$-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.

Citation

Download Citation

Steven HURDER. Rémi LANGEVIN. "Dynamics and the Godbillon–Vey class of $C^1$ foliations." J. Math. Soc. Japan 70 (2) 423 - 462, April, 2018. https://doi.org/10.2969/jmsj/07027485

Information

Published: April, 2018
First available in Project Euclid: 18 April 2018

zbMATH: 06902431
MathSciNet: MR3788786
Digital Object Identifier: 10.2969/jmsj/07027485

Subjects:
Primary: 37C85, 57R30
Secondary: 37C40, 57R32, 58H10

Rights: Copyright © 2018 Mathematical Society of Japan

JOURNAL ARTICLE
40 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.70 • No. 2 • April, 2018
Back to Top