Title: Topological dynamical systems /
Authors: Vries, J. de
Year: 2014
Subjects: Topological dynamics; MATHEMATICS Calculus.; MATHEMATICS Mathematical Analysis.
Abstract: This book is an introduction into the theory of discrete dynamical systems with emphasis on the topological background. It is addressed primarily to graduate students. The prerequisites for understanding this book are modest: a certain mathematical maturity and a course in General Topology are sufficient. Students who have mastered this book will have a firm basis to start research on related topics. The theory is about the behaviour of points of a Hausdorff space under the iteration of a continuous self-map. The assumption that the space is metrizable is avoided as much as possible, but where the non-metric version of the theory would become unwieldy we do not hesitate to assume metrizability. A similar remark applies to the assumption of compactness of the space. Much attention is given to dynamical systems on intervals on the real line. A wide range of topics, such as asymptotic stability, shift systems, (chain-)recurrence, topological entropy and chaos, is discussed. Every chapter concludes with a set of exercises and a section of notes, with references to the literature.
Contents: Introduction -- Basic notions -- Dynamical systems on the real line -- Limit behaviour -- Recurrent behaviour -- Shift systems -- Symbolic representations -- Erratic behaviour -- Topological entropy -- Topology -- The cantor set -- Hints to the exercises.