Random Dynamical Systems

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

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      Pages 111-162

   3. The Multiplicative Ergodic Theorem on Bundles and Manifolds

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      Pages 163-199

   4. The MET for Related Linear and Affine RDS

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      Pages 201-233

   5. RDS on Homogeneous Spaces of the General Linear Group

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      Pages 235-301

4. Smooth Random Dynamical Systems

   1. Front Matter

      Pages 303-303

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   2. Invariant Manifolds

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      Pages 305-403

   3. Normal Forms

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      Pages 405-463

   4. Bifurcation Theory

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      Pages 465-531

5. Back Matter

   Pages 533-588

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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.

• Kozykel
• Markov
• Measure
• Transformation
• cocycles
• glatte Ergodentheorie
• linear algebra
• multiplicative ergodic theory
• multiplikative Ergodentheorie
• random dynamical systems
• smooth ergodic theory
• stochastic bifurcation theory
• stochastische Bifurkationstheorie
• zufällige dynamische Systeme

"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

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

  Ludwig Arnold

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