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

Abstract : Let $F$ be a codimension-one, $C^2$-foliation on a manifold $M$ without boundary. In this work we show that if the Godbillon--Vey class $GV(F) \in H^3(M)$ is non-zero, then $F$ 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 $E(F)$ 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. The proof of this uses a pseudogroup version of the Pliss Lemma. The theorem 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.
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https://hal.archives-ouvertes.fr/hal-01801745
Contributor : Imb - Université de Bourgogne <>
Submitted on : Monday, May 28, 2018 - 4:24:18 PM
Last modification on : Thursday, August 1, 2019 - 2:12:40 PM

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Steven Hurder, Rémi Langevin. Dynamics and the Godbillon–Vey class of $C^1$ foliations. Journal of the Mathematical Society of Japan, Maruzen Company Ltd, 2018, 70 (2), pp.423-462. ⟨10.2969/jmsj/07027485⟩. ⟨hal-01801745⟩

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