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Lyapunov Exponents of Linear Cocycles

Continuity via Large Deviations

  • Book
  • © 2016

Overview

Authors:
  1. Pedro Duarte

    1. Universidade de Lisboa, Faculdade de Ciências da, Lisboa, Portugal

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  2. Silvius Klein

    1. Department of Mathematical Sciences, NTNU, Trondheim, Norway

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  • Unified approach to proving continuity of Lyapunov exponents for various types of linear cocycles
  • Uniform large deviation type estimates established for iterates of general quasi-periodic and random cocycles
  • A general Avalanche Principle for compositions of linear maps derived using a geometric approach
  • A list of related open problems
  • Includes supplementary material: sn.pub/extras

Part of the book series: Atlantis Studies in Dynamical Systems (ASDS, volume 3)

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About this book

The aim of this monograph is to present a general method of proving continuity of Lyapunov exponents of linear cocycles. The method uses an inductive procedure based on a general, geometric version of the Avalanche Principle. The main assumption required by this method is the availability of appropriate large deviation type estimates for quantities related to the iterates of the base and fiber dynamics associated with the linear cocycle. We establish such estimates for various models of random and quasi-periodic cocycles. Our method has its origins in a paper of M. Goldstein and W. Schlag. Our present work expands upon their approach in both depth and breadth. We conclude this monograph with a list of related open problems, some of which may be treated using a similar approach.

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction

    • Pedro Duarte, Silvius Klein
    Pages 1-21

  3. Estimates on Grassmann Manifolds

    • Pedro Duarte, Silvius Klein
    Pages 23-79

  4. Abstract Continuity of Lyapunov Exponents

    • Pedro Duarte, Silvius Klein
    Pages 81-111

  5. The Oseledets Filtration and Decomposition

    • Pedro Duarte, Silvius Klein
    Pages 113-160

  6. Large Deviations for Random Cocycles

    • Pedro Duarte, Silvius Klein
    Pages 161-205

  7. Large Deviations for Quasi-Periodic Cocycles

    • Pedro Duarte, Silvius Klein
    Pages 207-246

  8. Further Related Problems

    • Pedro Duarte, Silvius Klein
    Pages 247-260

  9. Back Matter

    Pages 261-263
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Reviews

“The effort of the authors to make this text self-contained and to make the exposition very clear and delightful was successful, making this monograph an excellent contribution for both graduate student and anyone interested in the continuity of Lyapunov Exponents.” (Paulo Varandas, Mathematical Reviews, June, 2018)

Authors and Affiliations

  • Universidade de Lisboa, Faculdade de Ciências da, Lisboa, Portugal

    Pedro Duarte

  • Department of Mathematical Sciences, NTNU, Trondheim, Norway

    Silvius Klein

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