A first course in discrete dynamical systems / Richard A. Holmgren

This book is an introduction to discrete dynamical systems. It starts off by reviewing elementary properties of functions in one real variable and the topology of real numbers. After introducing the notions of periodic points and stable sets, the author formulates and proves a special case of Sharkovskiĭ's theorem. Some attention is paid to differentiability, including the mean-value theorem, and there is a brief introduction to parametrized families of functions and bifurcations. Cantor sets, which play an important role in chaotic dynamics, are presented for the logistic map. Other topics that are covered are symbolic dynamics and chaos, topological conjugacy and bifurcations for the logistic map, Newton's method for quadratic and cubic functions, numerical solutions of differential equations, and the dynamics of complex functions, including the quadratic family and the Mandelbrot set. Computer algorithms are presented in the appendix.
Generally, the book is elementary but well written. In particular, the presentation of the elementary material in the chapters on symbolic dynamics and the Newton method is nicely done. The author writes: "My text arose out of the desire to provide my students the mathematical background necessary to understand the text by R. L. Devaney [An introduction to chaotic dynamical systems, second edition, Addison-Wesley, Redwood City, CA, 1989; MR1046376 (91a:58114)].'' I think that the chapters on symbolic dynamics and the Newton method indeed may be of help. On the other hand, the text under review has an overlap with another book by Devaney [A first course in chaotic dynamical systems, Addison-Wesley, Reading, MA, 1992; MR1202237 (94a:58124)]. The texts are written from a different point of view. Holmgren's book has a more topological flavor than Devaney's text, while the latter has a more dynamical flavor. To conclude, I think that the level of the presentation is suitable not only for advanced undergraduates with a year of calculus behind them, but also for serious high school students who have interest in the mathematics behind some beautiful pictures of discrete dynamical systems. (MathSciNet)

Bibliogr. p. [203]-207. Index