Journal of the Mathematical Society of Japan

Dynamics and the Godbillon–Vey class of $C^1$ foliations

Steven HURDER and Rémi LANGEVIN

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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.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 2 (2018), 423-462.

Dates
First available in Project Euclid: 18 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1524038663

Digital Object Identifier
doi:10.2969/jmsj/07027485

Mathematical Reviews number (MathSciNet)
MR3788786

Zentralblatt MATH identifier
06902431

Subjects
Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] 57R30: Foliations; geometric theory
Secondary: 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx] 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology [See also 58H10] 58H10: Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.) [See also 57R32]

Keywords
Godbillon–Vey class Godbillon measure foliation dynamics hyperbolic sets exponential growth Pliss Lemma hyperbolic fixed-points

Citation

HURDER, Steven; LANGEVIN, Rémi. Dynamics and the Godbillon–Vey class of $C^1$ foliations. J. Math. Soc. Japan 70 (2018), no. 2, 423--462. doi:10.2969/jmsj/07027485. https://projecteuclid.org/euclid.jmsj/1524038663


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