Chaotic Dynamics
Fractals, Tilings, and Substitutions

This rigorous undergraduate introduction to dynamical systems is an accessible guide for mathematics students advancing from calculus.

Geoffrey R. Goodson (Author)

9781107112674, Cambridge University Press

Hardback, published 28 December 2016

416 pages, 93 b/w illus. 435 exercises
26 x 18.3 x 2.3 cm, 1.05 kg

'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

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.

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.

Subject Areas: Chaos theory [PBWS], Differential calculus & equations [PBKJ], Calculus & mathematical analysis [PBK], Mathematics [PB]