Dynamical systems by example (Q1738331)

While the suggestion ``in re mathematica ars proponendi quaestionem pluris facienda est quam solvendi'' that Cantor made in his thesis may be debatable, the pedagogical insight captured pithily as ``The only way to learn mathematics is to do mathematics'' by \textit{P. R. Halmos} [A Hilbert space problem book. Princeton, N.J.-Toronto- London: D. van Nostrand Comp., Inc. (1967; Zbl 0144.38704)] carries undoubted weight and permanent relevance. This interesting book applies the idea that ``students learn primarily by doing'' to the field of dynamical systems, to good effect. It is structured as a condensed introduction to key concepts in dynamical systems accompanied by a large number of problems, across a great variety of topics in dynamics. In the second part of the book, the problems are all repeated, accompanied by additional explanatory material and detailed solutions. The topics covered include basic topological dynamics, low-dimensional dynamics, hyperbolic maps, symbolic dynamics, and some aspects of the ergodic theory of measure-preserving transformations. Inevitably choices had to be made, as `dynamical systems' is so broad, and the main lacunæ are listed at the end of the preface. Despite these absences, in each case working through the problems in this book would be a useful grounding in some of the ideas used in more advanced areas. The problems build up rapidly, from elementary beginnings to quite sophisticated problems, and their careful selection suggests that most of them have been refined by exposure to generations of students. The intent being so clearly pedagogical, it is worth reflecting on how this text might be used. As a companion to a conventional course on dynamical systems, 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, but the student would need to be equipped with a certain level of background knowledge and mathematical maturity. While the technical prerequisites are formally modest, with additional material recalled as needed, the more difficult problems certainly call heavily on that maturity. Some of the brief introductions are extremely so, going from defining a $\sigma$-algebra to measure-theoretic entropy in a handful of pages, for example. That said, these introductions are not of course intended to be a substitute for a conventional text or course, but rather a quick summary and fixing of terminology for concepts fleshed out elsewhere. An independent reader might also need access to a curator or guide, as the problems vary in their level of difficulty considerably. In both cases, in the right conditions and alongside a standard text or course, this book provides an unconventional but carefully constructed route map that will give a reader practice in dealing with the fundamental concepts of dynamics systems and ergodic theory in an engaging and thoughtful way. A student who has gone through it will be well equipped for further study and research in dynamical systems. A third type of user might of course be 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.