Random Dynamical Systems

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.

I. Random Dynamical Systems and Their Generators. - 1. Basic Definitions. Invariant Measures. - 2. Generation. - II. Multiplicative Ergodic Theory. - 3. The Multiplicative Ergodic Theorem in Euclidean Space. - 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds. - 5. The MET for Related Linear and Affine RDS. - 6. RDS on Homogeneous Spaces of the General Linear Group. - III. Smooth Random Dynamical Systems. - 7. Invariant Manifolds. - 8. Normal Forms. - 9. Bifurcation Theory. - IV. Appendices. - Appendix A. Measurable Dynamical Systems. - A. 1 Ergodic Theory. - A. 2 Stochastic Processes and Dynamical Systems. - A. 3 Stationary Processes. - A. 4 Markov Processes. - Appendix B. Smooth Dynamical Systems. - B. 1 Two-Parameter Flows on a Manifold. - B. 4 Autonomous Case: Dynamical Systems. - B. 5 Vector Fields and Flows on Manifolds. - References.