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Oseledec multiplicative ergodic theorem for laminations


About this Title

Viêt-Anh Nguyên

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 246, Number 1164
ISBNs: 978-1-4704-2253-0 (print); 978-1-4704-3637-7 (online)
DOI: https://doi.org/10.1090/memo/1164
Published electronically: December 1, 2016
Keywords:Lamination, foliation, harmonic measure, Wiener measure, Brownian motion, Lyapunov exponents, multiplicative ergodic theorem, Oseledec decomposition, holonomy invariant

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Table of Contents


Chapters

  • Acknowledgement
  • Chapter 1. Introduction
  • Chapter 2. Background
  • Chapter 3. Statement of the main results
  • Chapter 4. Preparatory results
  • Chapter 5. Leafwise Lyapunov exponents
  • Chapter 6. Splitting subbundles
  • Chapter 7. Lyapunov forward filtrations
  • Chapter 8. Lyapunov backward filtrations
  • Chapter 9. Proof of the main results
  • Appendix A. Measure theory for sample-path spaces
  • Appendix B. Harmonic measure theory and ergodic theory for sample-path spaces

Abstract


Given a -dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank where and are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we define the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. We also prove an Oseledec multiplicative ergodic theorem in this context. This theorem implies the existence of an Oseledec decomposition almost everywhere which is holonomy invariant. Moreover, in the case of differentiable cocycles we establish effective integral estimates for the Lyapunov exponents. These results find applications in the geometric and dynamical theory of laminations. They are also applicable to (not necessarily closed) laminations with singularities. Interesting holonomy properties of a generic leaf of a foliation are obtained. The main ingredients of our method are the theory of Brownian motion, the analysis of the heat diffusions on Riemannian manifolds, the ergodic theory in discrete dynamics and a geometric study of laminations.

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