Introduction to analysis of dynamical systems
This book seems to be the first book written in German on the theory of dynamical systems. Though there is a book by \textit{E. Hopf} [Ergodentheorie, Ergebn. Math. Grenzgeb. 5, Heft 2 (1937; JFM 63.0786.07)], it restricts within the ergodic theory. The book under review covers almost every parts of the dynamical systems. In fact, the contents are based on the author's lectures given during the past three decades, and the book treats topological and differentiable dynamical systems in the Chapters 3 and 4, where the author studies, among other things, the expansiveness, topological entropy, stable and unstable manifolds, structural stability, hyberbolic dynamical systems, as well as symbolic dynamics and geodesic flows. Chapter 5 studies the ergodic theory, including mixing and entropy. The title of Chapter 6 is thermodynamic formulation, in which the author treats the topological pressure, Gibbs-measure and Lyapunov exponents. In order to explain the contents of the book, the author, in Chapter 1, first develops a historical survey of several branches of mathematics which have relations to the theory of dynamical systems, next, the author begins with the definition of a dynamical system and then gives the fundamental notions such as the orbits, periodic fixed points, as well as ergodicity and hyperbolic fixed points and Markov decomposition with a number of related examples. In Chapter 2 ``Zero- and 1-dimensional dynamical systems'', the author studies interval selfmappings, topological Markov chains, homeomorphisms of the circle and rational mappings. Each chapter begins with an explanation of the content together with a brief historical survey of the subjects in the chapter. The selected subjects are wellbalanced in each chapter. All theorems, definitions and even the examples are numbered through the whole book. Chapters 1 to 5 can be essentially read independently to each other, while the contents of Chapters 1 and 2 seem for the reviewer to be not easier than those of the succeeding chapters. In the last chapter, the author gives outlines of the careers of some mathematicians whose works have relationships with dynamical systems, and then a number at interesting exercises, some of which will be solved after consulting the referred references.
scientific article; zbMATH DE number 2106907
scientific article