Chaotic Dynamics: Fractals, Tilings, and Substitutions

by Geoffrey R. Goodson

Overview

This undergraduate textbook is a rigorous mathematical introduction to dynamical systems and an accessible guide for students transitioning from calculus to advanced mathematics. It has many student-friendly features, such as graded exercises that range from straightforward to more difficult with hints, and includes concrete applications of real analysis and metric space theory to dynamical problems. Proofs are complete and carefully explained, and there is opportunity to practice manipulating algebraic expressions in an applied context of dynamical problems. After presenting a foundation in one-dimensional dynamical systems, the text introduces students to advanced subjects in the latter chapters, such as topological and symbolic dynamics. It includes two-dimensional dynamics, Sharkovsky's theorem, and the theory of substitutions, and takes special care in covering Newton's method. Mathematica code is available online, so that students can see implementation of many of the dynamical aspects of the text.

Product Details

ISBN-13:	9781107112674
Publisher:	Cambridge University Press
Publication date:	12/28/2016
Series:	Cambridge Mathematical Textbooks
Pages:	416
Product dimensions:	7.20(w) x 10.24(h) x 0.91(d)

About the Author

Geoffrey R. Goodson is Professor of Mathematics at Towson University, Maryland. He previously served on the faculty of the University of Witwatersrand and the University of Cape Town. His research interests include dynamical systems, ergodic theory, matrix theory, and operator theory. He has published more than thirty papers, and taught numerous classes on dynamical systems.

Table of Contents

1. The orbits of one-dimensional maps; 2. Bifurcations and the logistic family; 3. Sharkovsky's theorem; 4. Dynamics on metric spaces; 5. Countability, sets of measure zero, and the Cantor set; 6. Devaney's definition of chaos; 7. Conjugacy of dynamical systems; 8. Singer's theorem; 9. Conjugacy, fundamental domains, and the tent family; 10. Fractals; 11. Newton's method for real quadratics and cubics; 12. Coppel's theorem and a proof of Sharkovsky's theorem; 13. Real linear transformations, the Hénon Map, and hyperbolic toral automorphisms; 14. Elementary complex dynamics; 15. Examples of substitutions; 16. Fractals arising from substitutions; 17. Compactness in metric spaces and an introduction to topological dynamics; 18. Substitution dynamical systems; 19. Sturmian sequences and irrational rotations; 20. The multiple recurrence theorem of Furstenberg and Weiss; Appendix A: theorems from calculus; Appendix B: the Baire category theorem; Appendix C: the complex numbers; Appendix D: Weyl's equidistribution theorem.

Editorial Reviews

'This remarkable book provides a thoroughly field-tested way of teaching analysis while introducing dynamical systems. Combining lightness with rigor, it motivates and applies a wide range of subjects in the theory of metric spaces as it explores a broad variety of topics in dynamics.' Boris Hasselblatt, Tufts University, Massachusetts

'This is a most impressive book. The author presents a range of topics which are not usually included in a book at this level (for example Sharkovsky's theorem, fractals, substitutions). The writing is clear and there are exercises of varying difficulty. A fine undergraduate text, which will also be of interest to graduate students and researchers in dynamics.' Joseph Auslander, Professor Emeritus of Mathematics, University of Maryland

'This carefully written book introduces the student to a wealth of examples in dynamical systems, including several modern topics such as complex dynamics, topological dynamics and substitutions.' Cesar E. Silva, Williams College, Massachusetts

'More rigorous than other undergraduate texts but less daunting than graduate books, this book is perfect for a core course on chaotic dynamic systems for undergraduates in their junior or senior year. Thoughtful, clear, and written with just the right amount of detail, Goodson develops the necessary tools required for an in-depth study of dynamical systems.' Alisa DeStefano, College of the Holy Cross, Massachusetts

'… readers familiar with the basics of calculus, linear algebra, topology, and some real analysis will find that the topics are presented in an interesting manner, making this a good treatment of discrete dynamical systems … Summing Up: Recommended. Upper-division undergraduates and above; faculty and professionals.' M. D. Sanford, CHOICE

'I think that this attractive textbook would be a welcome addition to the bookshelf of just about anyone with an interest in fractals, chaos, or dynamical systems. It presents most of the basic concepts in these fields at a level appropriate for senior math majors. Additional[ly], it has an extended treatment of substitution dynamical systems - the only undergraduate textbook I’m aware of that does so.' Christopher P. Grant, Mathematical Reviews

‘This book is a good example of what is possible as an introduction to this broad material of chaos, dynamical systems, fractals, tilings, substitutions, and many other related aspects. To bring all this in one volume and at a moderate mathematical level is an ambitious plan but these notes are the result of many years of teaching experience … The extraordinary combination of abstraction linked to simple yet appealing examples is the secret ingredient that is mastered wonderfully in this text.’ Adhemar Bultheel, European Mathematical Society

From the Publisher