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Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space


About this Title

Zeng Lian, Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 and Kening Lu, Department of Mathematics, Brigham Young University, Provo, Utah 84602

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: Volume 206, Number 967
ISBNs: 978-0-8218-4656-8 (print); 978-1-4704-0581-6 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00574-0
Published electronically: January 22, 2010
MathSciNet review: 2674952
Keywords: Lyapunov exponents, Multiplicative Ergodic Theorem, infinite dimensional random dynamical systems, invariant manifolds.
MSC (2000): Primary 37H15; Secondary 34D08, 37A55, 37C45, 37D10, 37H10, 37L25

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Random dynamical systems and measures of noncompactness
  • Chapter 3. Main results
  • Chapter 4. Volume function in Banach spaces
  • Chapter 5. Gap and distance between closed linear subspaces
  • Chapter 6. Lyapunov exponents and Oseledets spaces
  • Chapter 7. Measurable random invariant complementary subspaces
  • Chapter 8. Proof of multiplicative ergodic theorem
  • Chapter 9. Stable and unstable manifolds
  • Appendix A. Subadditive ergodic theorem
  • Appendix B. Non-ergodic case

Abstract


We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

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