Fractals, Tilings, and Substitutions
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Part of Cambridge Mathematical Textbooks
- Author: Geoffrey R. Goodson, Towson State University, Maryland
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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.Read more
- Requires no prior knowledge of real analysis or metric spaces, making it an ideal transitional text between the calculus sequence and more advanced topics in real analysis and topology
- Offers an introduction to topological and symbolic dynamical systems that, unlike other available textbooks, is suitable for both senior undergraduates and beginning graduate students
- Illustrates concrete applications of real analysis and metric space theory to dynamical problems, showing students why advanced mathematics is important and useful
- Mathematica code is available online, so that students can see implementation of many of the dynamical aspects of the text
Reviews & endorsements
'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, MassachusettsSee more reviews
'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
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- Date Published: January 2017
- format: Adobe eBook Reader
- isbn: 9781316944356
- contains: 93 b/w illus. 435 exercises
- availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
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
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.
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