Dynamical Systems by Example

This book comprises an impressive collection of problems that cover a variety of carefully selected topics on the core of the theory of dynamical systems. Aimed at the graduate/upper undergraduate level, the emphasis is on dynamical systems with discrete time. In addition to the basic theory, the topics include topological, low-dimensional, hyperbolic and symbolic dynamics, as well as basic ergodic theory. As in other areas of mathematics, one can gain the first working knowledge of a topic by solving selected problems. It is rare to find large collections of problems in an advanced field of study much less to discover accompanying detailed solutions. This text fills a gap and can be used as a strong companion to an analogous dynamical systems textbook such as the authors' own Dynamical Systems (Universitext, Springer) or another text designed for a one- or two-semester advanced undergraduate/graduate course. The book is also intended for independent study.
Problems often begin with specific cases and then move on to general results, following a natural path of learning. They are also well-graded in terms of increasing the challenge to the reader. Anyone who works through the theory and problems in Part I will have acquired the background and techniques needed to do advanced studies in this area. Part II includes complete solutions to every problem given in Part I with each conveniently restated. Beyond basic prerequisites from linear algebra, differential and integral calculus, and complex analysis and topology, in each chapter the authors recall the notions and results (without proofs) that are necessary to treat the challenges set for that chapter, thus making the text self-contained.
 

List of contents

Part I. Theory and Problems.- I.1 Basic Theory.- I.2 Topological Dynamics.- I.3 Low-Dimensional Dynamics.- I.4 Hyperbolic Dynamics.- I.5 Symbolic Dynamics.- I.6 Ergodic Theory.- Part II. Problems and Solutions.- II.1 Basic Theory.- II.2 Topological Dynamics.- II.3 Low-Dimensional Dynamics.- II.4 Hyperbolic Dynamics.- II.5 Symbolic Dynamics.- II.6 Ergodic Theory.- References.- Index.

About the author

Luís Barreira is professor of mathematics at the University of Lisbon. His main research interests are in dynamical systems and ergodic theory. Prof. Barreira has authored several books published with Springer including Lyapunov Exponents, Thermodynamic Formalism and Applications to Dimension Theory (PM), (with C. Valls) Dynamical Systems (UTX) and Stability of Nonautonomous Differential Equations (LNM), (with C.  Valls and D. Dragicevic) Admissibility and Hyperbolicity (SBM).
Claudia Valls is professor of mathematics at the University of Lisbon. Her main research interest is in dynamical systems. Prof. Valls has authored several books published with Springer including (with L. Barreira) Dynamical Systems (UTX) and Stability of Nonautonomous Differential Equations (LNM), and (with L. Barreira and D. Dragicevic) Admissibility and Hyperbolicity (SBM).

Summary

This book comprises an impressive collection of problems that cover a variety of carefully selected topics on the core of the theory of dynamical systems. Aimed at the graduate/upper undergraduate level, the emphasis is on dynamical systems with discrete time. In addition to the basic theory, the topics include topological, low-dimensional, hyperbolic and symbolic dynamics, as well as basic ergodic theory. As in other areas of mathematics, one can gain the first working knowledge of a topic by solving selected problems. It is rare to find large collections of problems in an advanced field of study much less to discover accompanying detailed solutions. This text fills a gap and can be used as a strong companion to an analogous dynamical systems textbook such as the authors’ own Dynamical Systems (Universitext, Springer) or another text designed for a one- or two-semester advanced undergraduate/graduate course. The book is also intended for independent study.
Problems often begin with specific cases and then move on to general results, following a natural path of learning. They are also well-graded in terms of increasing the challenge to the reader. Anyone who works through the theory and problems in Part I will have acquired the background and techniques needed to do advanced studies in this area. Part II includes complete solutions to every problem given in Part I with each conveniently restated. Beyond basic prerequisites from linear algebra, differential and integral calculus, and complex analysis and topology, in each chapter the authors recall the notions and results (without proofs) that are necessary to treat the challenges set for that chapter, thus making the text self-contained.
 

Additional text

“This is a really nice problem book on the subject covering a large body of subjects … . All problems are presented with proper context (including definitions and relevant theorems) together with complete solutions. All solutions are provided with sufficient details such that the book is not only suitable for teachers but also for students who are looking for solved examples to better understand abstract topics. In summary, this is a most welcome addition to the present literature” (G. Teschl, Monatshefte für Mathematik, Vol. 196 (1), 2021)

“The book under review represents a great help for anyone who wants to study the theory of dynamical systems. … The book is very well written and may be also very useful for mathematicians teaching courses on dynamical systems.” (Krzysztof Ciesielski, Mathematical Reviews, September, 2020)
“It would quickly provide students with a good set of problems and an introduction to a good range of concepts and issues relevant to dynamics. The book could also be a valuable resource for individual study … . anyone lecturing on dynamical systems, who would find here an excellent and diverse selection of problems, carefully assembled and usefully ‘scaffolded’ in the educational terminology. I enjoyed it, and recommend it highly.” (Thomas B. Ward, zbMATH 1421.37001, 2019)

Report

"This is a really nice problem book on the subject covering a large body of subjects ... . All problems are presented with proper context (including definitions and relevant theorems) together with complete solutions. All solutions are provided with sufficient details such that the book is not only suitable for teachers but also for students who are looking for solved examples to better understand abstract topics. In summary, this is a most welcome addition to the present literature" (G. Teschl, Monatshefte für Mathematik, Vol. 196 (1), 2021)

"The book under review represents a great help for anyone who wants to study the theory of dynamical systems. ... The book is very well written and may be also very useful for mathematicians teaching courses on dynamical systems." (Krzysztof Ciesielski, Mathematical Reviews, September, 2020)
"It would quickly provide students with a good set of problems and an introduction to a good range of concepts and issues relevant to dynamics. The book could also be a valuable resource for individual study ... . anyone lecturing on dynamical systems, who would find here an excellent and diverse selection of problems, carefully assembled and usefully 'scaffolded' in the educational terminology. I enjoyed it, and recommend it highly." (Thomas B. Ward, zbMATH 1421.37001, 2019)

Product details

• Authors: Luís Barreira, Claudia Valls
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 03.05.2019
• EAN: 9783030159146
• ISBN: 978-3-0-3015914-6
• No. of pages: 223
• Dimensions: 167 mm x 247 mm x 20 mm
• Weight: 502 g
• Illustrations: IX, 223 p. 51 illus.
• Series: Problem Books in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; B, Dynamics, Mathematics and Statistics, Dynamical Systems and Ergodic Theory, Ergodic theory, Dynamical systems, basic theory dynamical systems, symbolic dynamics, supplement text dynamical systems, homework problems dynamical systems