Dynamical Systems and Semisimple Groups: An Introduction

Here is an introduction to dynamical systems and ergodic theory with an emphasis on smooth actions of noncompact Lie groups. The main goal is to serve as an entry into the current literature on the ergodic theory of measure preserving actions of semisimple Lie groups for students who have taken the standard first year graduate courses in mathematics. The author develops in a detailed and self-contained way the main results on Lie groups, Lie algebras, and semisimple groups, including basic facts normally covered in first courses on manifolds and Lie groups plus topics such as integration of infinitesimal actions of Lie groups. He then derives the basic structure theorems for the real semisimple Lie groups, such as the Cartan and Iwasawa decompositions and gives an extensive exposition of the general facts and concepts from topological dynamics and ergodic theory, including detailed proofs of the multiplicative ergodic theorem and Moore's ergodicity theorem. This book should appeal to anyone interested in Lie theory, differential geometry and dynamical systems.