xiv Introduction Prerequisites The reader should have basic knowledge of real analysis (including met- ric spaces) and linear algebra. No previous exposure to ordinary differential equations is assumed, although a first course in linear differential equations certainly is helpful. Multilinear algebra shows up in two places: in Section 5.2 we discuss how the volumes of parallelepipeds grow under the flow of a linear autonomous differential equation, which we relate to chain recur- rent sets of the induced flows on Grassmannians. The necessary elements of multilinear algebra are presented in Section 5.1. In Chapter 11 the proof of the multiplicative ergodic theorem requires further elements of multilinear algebra which are provided in Section 11.3. Understanding the proofs in Chapter 10 on ergodic theory and Chapter 11 on random linear dynamical systems also requires basic knowledge of σ-algebras and probability mea- sures (actually, a detailed knowledge of Lebesgue measure should suffice). The results and methods in the rest of the book are independent of these additional prerequisites. Acknowledgements The idea for this book grew out of the preparations for Chapter 79 in the Handbook of Linear Algebra [71]. Then WK gave a course “Dynamics and Linear Algebra” at the Simposio 2007 of the Sociedad de Matem´ atica de Chile. FC later taught a course on the same topic at Iowa State University within the 2008 Summer Program for Graduate Students of the Institute of Mathematics and Its Applications, Minneapolis. Parts of the manuscript were also used for courses at the University of Augsburg in the summer semesters 2010 and 2013 and at Iowa State University in Spring 2011. We gratefully acknowledge these opportunities to develop our thoughts, the feed- back from the audiences, and the financial support. Thanks for the preparation of figures are due to: Isabell Graf (Section 4.1), Patrick Roocks (Section 5.2) Florian Ecker and Julia Rudolph (Section 7.3.) and Humberto Verdejo (Section 11.6). Thanks are also due to Philipp uren, Julian Braun, and Justin Peters. We are particularly indebted to Christoph Kawan who has read the whole manuscript and provided us with long lists of errors and inaccuracies. Special thanks go to Ina Mette of the AMS for her interest in this project and her continuous support during the last few years, even when the text moved forward very slowly. The authors welcome any comments, suggestions, or corrections you may have. Fritz Colonius Wolfgang Kliemann Institut Mathematik Department of Mathematics Universit¨ at Augsburg Iowa State University
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