Oseledec Multiplicative Ergodic Theorem for Laminations
Cover -- Title page -- Acknowledgement -- Chapter 1. Introduction -- Chapter 2. Background -- 2.1. (Riemannian) laminations and Laplacians -- 2.2. Covering laminations -- 2.3. Heat kernels, (weakly) harmonic measures and Standing Hypotheses -- 2.4. Brownian motion and Wiener measures without holonomy -- 2.5. Wiener measures with holonomy -- Chapter 3. Statement of the main results -- 3.1. Multiplicative cocycles -- 3.2. First Main Theorem and applications -- 3.3. Second Main Theorem and applica…
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- Full title :Oseledec Multiplicative Ergodic Theorem for Laminations
Summary
Cover -- Title page -- Acknowledgement -- Chapter 1. Introduction -- Chapter 2. Background -- 2.1. (Riemannian) laminations and Laplacians -- 2.2. Covering laminations -- 2.3. Heat kernels, (weakly) harmonic measures and Standing Hypotheses -- 2.4. Brownian motion and Wiener measures without holonomy -- 2.5. Wiener measures with holonomy -- Chapter 3. Statement of the main results -- 3.1. Multiplicative cocycles -- 3.2. First Main Theorem and applications -- 3.3. Second Main Theorem and applications -- 3.4. Plan of the proof -- Chapter 4. Preparatory results -- 4.1. Measurability issue -- 4.2. Markov property of Brownian motion -- Chapter 5. Leafwise Lyapunov exponents -- Chapter 6. Splitting subbundles -- Chapter 7. Lyapunov forward filtrations -- 7.1. Oseledec type theorems -- 7.2. Fibered laminations and totally invariant sets -- 7.3. Cylinder laminations and end of the proof -- Chapter 8. Lyapunov backward filtrations -- 8.1. Extended sample-path spaces -- 8.2. Leafwise Lyapunov backward exponents and Oseledec backward type theorem -- Chapter 9. Proof of the main results -- 9.1. Canonical cocycles and specializations -- 9.2.-weakly harmonic measures and splitting invariant bundles -- 9.3. First Main Theorem and Ledrappier type characterization of Lyapunov spectrum -- 9.4. Second Main Theorem and its corollaries -- APPENDICES -- Appendix A. Measure theory for sample-path spaces -- A.1. Multifunctions and measurable selections -- A.2.-algebras: approximations and measurability -- A.3.-algebra \Ac on a leaf -- A.4. Holonomy maps -- A.5. Metrizability and separability of sample-path spaces -- A.6. The leafwise diagonal is Borel measurable -- A.7.-algebra \Ac on a lamination -- A.8. The cylinder laminations are Riemannian continuous-like -- A.9. The extended sample-path space is of full outer measure
Appendix B. Harmonic measure theory and ergodic theory for sample-path spaces -- B.1. Fibered laminations -- B.2. Relation between a lamination and its covering lamination -- B.3. Invariance of Wiener measures with harmonic initial distribution -- B.4. Ergodicity of Wiener measures with ergodic harmonic initial distribution -- Bibliography -- Index -- \nomname -- Back Cover
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
Edition Statements
- Publication Date:2017
Publication Statements
- Publisher(s):American mathematical Society
- Place of Publication:Providence
- Language:eng
- Publication Date:2017
The Physical Object
- Description:1 Online-Ressource (190 Seiten)
Edition Identifiers
- ISBN:9781470436377, 147043637X, 9781470422530, 1470422530
- oclc:on1021121435
- Dewey Decimal:515.48
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