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Dynamical systems and semisimple groups: an introduction. (English) Zbl 1057.22001
Cambridge Tracts in Mathematics 126. Cambridge: Cambridge University Press (ISBN 0-521-59162-7/hbk). xvi, 245 p. (1998).
From the text: The theory of dynamical systems can be described as the study of the global properties of groups of transformations. The historical roots of the subject lie in celestial and statistical mechanics, for which the group is the time parameter. The more general modern theory treats the dynamical properties of the semisimple Lie groups. Some of the most fundamental discoveries in this area are due to the work of G. A. Margulis [see “Discrete subgroups of semisimple Lie groups”, Springer-Verlag (1991; Zbl 0732.22008)] and R. J. Zimmer[“Ergodic theory and semisimple groups”, Boston etc.: Birkhäuser (1984; Zbl 0571.58015)].
The book under review comprises a systematic, self-contained introduction to the Margulis-Zimmer theory, and provides an entry into current research. Assuming only a basic knowledge of manifolds, algebra, and measure theory, this book should appeal to anyone interested in Lie theory, differential geometry and dynamical systems.
Preface; 1. Topological dynamics; 2. Ergodic theory - part I; 3. Smooth actions and Lie theory; 4. Algebraic actions; 5. The classical groups; 6. Geometric structures; 7. Semisimple Lie groups; 8. Ergodic theory - part II; 9. Oseledec’s theorem; 10. Rigidity theorems; Appendix: Lattices in \(\mathrm{SL}(n,\mathbb R)\).

MSC:
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
22E40 Discrete subgroups of Lie groups
22F10 Measurable group actions
37A15 General groups of measure-preserving transformations and dynamical systems
53C24 Rigidity results
54H20 Topological dynamics (MSC2010)
57S20 Noncompact Lie groups of transformations
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