eBook ISBN:  9781470436377 
Product Code:  MEMO/246/1164.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470436377 
Product Code:  MEMO/246/1164.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 246; 2016; 174 ppMSC: Primary 37; 57; Secondary 58; 60
Given a \(n\)dimensional lamination endowed with a Riemannian metric, the author introduces the notion of a multiplicative cocycle of rank \(d\), where \(n\) and \(d\) 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, the author defines the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. He also proves 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 the author establishes 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 the author's 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.

Table of Contents

Chapters

Acknowledgement

1. Introduction

2. Background

3. Statement of the main results

4. Preparatory results

5. Leafwise Lyapunov exponents

6. Splitting subbundles

7. Lyapunov forward filtrations

8. Lyapunov backward filtrations

9. Proof of the main results

A. Measure theory for samplepath spaces

B. Harmonic measure theory and ergodic theory for samplepath spaces


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Given a \(n\)dimensional lamination endowed with a Riemannian metric, the author introduces the notion of a multiplicative cocycle of rank \(d\), where \(n\) and \(d\) 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, the author defines the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. He also proves 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 the author establishes 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 the author's 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.

Chapters

Acknowledgement

1. Introduction

2. Background

3. Statement of the main results

4. Preparatory results

5. Leafwise Lyapunov exponents

6. Splitting subbundles

7. Lyapunov forward filtrations

8. Lyapunov backward filtrations

9. Proof of the main results

A. Measure theory for samplepath spaces

B. Harmonic measure theory and ergodic theory for samplepath spaces