1. Dynamics and the Godbillon–Vey class of $C^1$ foliations
- Author
-
Steven Hurder, Rémi Langevin, Department of Mathematics, Statistics and Computer Science [Chicago] ( UIC ), University of Illinois at Chicago ( UIC ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics, Statistics and Computer Science [Chicago] (UIC), University of Illinois [Chicago] (UIC), University of Illinois System-University of Illinois System, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Hyperbolic sets ,Mathematics::Dynamical Systems ,General Mathematics ,Infinitesimal ,58H10 ,01 natural sciences ,Exponential growth ,Foliation dynamics ,Mathematics::K-Theory and Homology ,57R32 ,0103 physical sciences ,57R30 ,Direct proof ,Hyperbolic fixed-points ,0101 mathematics ,Mathematics ,Lebesgue measure ,37C40 ,Pliss Lemma ,37C85 ,010102 general mathematics ,Godbillon-Vey class ,57R30, 58H10, 37C40 ,[ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG] ,Godbillon measure ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Foliation (geology) ,010307 mathematical physics ,Mathematics::Differential Geometry ,Godbillon–Vey class ,MSC: 37C85, 57R30, 37C40, 57R32, 58H10 - 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., Comment: This manuscript is a revision of the section 3 material from the previous version, and includes edits to the pictures in the text
- Published
- 2018
- Full Text
- View/download PDF