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Random Dynamical Systems

  • Book
  • © 1998

Overview

Authors:
  1. Ludwig Arnold

    1. Institute for Dynamical Systems, University of Bremen, Bremen, Germany

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  • This is the first comprehensive monograph on this active subject, dealing with the fundamentals through to current research, and written by one of the leaders in the field.
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-XV
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  2. Random Dynamical Systems and Their Generators

    1. Front Matter

      Pages 1-1
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    2. Basic Definitions. Invariant Measures

      • Ludwig Arnold
      Pages 3-47

    3. Generation

      • Ludwig Arnold
      Pages 49-107

  3. Multiplicative Ergodic Theory

    1. Front Matter

      Pages 109-109
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    2. The Multiplicative Ergodic Theorem in Euclidean Space

      • Ludwig Arnold
      Pages 111-162

    3. The Multiplicative Ergodic Theorem on Bundles and Manifolds

      • Ludwig Arnold
      Pages 163-199

    4. The MET for Related Linear and Affine RDS

      • Ludwig Arnold
      Pages 201-233

    5. RDS on Homogeneous Spaces of the General Linear Group

      • Ludwig Arnold
      Pages 235-301

  4. Smooth Random Dynamical Systems

    1. Front Matter

      Pages 303-303
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    2. Invariant Manifolds

      • Ludwig Arnold
      Pages 305-403

    3. Normal Forms

      • Ludwig Arnold
      Pages 405-463

    4. Bifurcation Theory

      • Ludwig Arnold
      Pages 465-531

  5. Back Matter

    Pages 533-588
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Keywords

About this book

Background and Scope of the Book This book continues, extends, and unites various developments in the intersection of probability theory and dynamical systems. I will briefly outline the background of the book, thus placing it in a systematic and historical context and tradition. Roughly speaking, a random dynamical system is a combination of a measure-preserving dynamical system in the sense of ergodic theory, (D,F,lP', (B(t))tE'lf), 'II'= JR+, IR, z+, Z, with a smooth (or topological) dy­ namical system, typically generated by a differential or difference equation :i: = f(x) or Xn+l = tp(x.,), to a random differential equation :i: = f(B(t)w,x) or random difference equation Xn+l = tp(B(n)w, Xn)· Both components have been very well investigated separately. However, a symbiosis of them leads to a new research program which has only partly been carried out. As we will see, it also leads to new problems which do not emerge if one only looks at ergodic theory and smooth or topological dynam­ ics separately. From a dynamical systems point of view this book just deals with those dynamical systems that have a measure-preserving dynamical system as a factor (or, the other way around, are extensions of such a factor). As there is an invariant measure on the factor, ergodic theory is always involved.

Reviews

"Ludwig Arnold's monograph is going to make a very big impact for many years to come."
DMV Jahresbericht, 103. Band, Heft 2, July 2001

Authors and Affiliations

  • Institute for Dynamical Systems, University of Bremen, Bremen, Germany

    Ludwig Arnold

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