A discrete dynamical system is a map f of a measurable set S (configuration space) to itself, and a central question in this regard is the (asymptotic) behavior of sequences of iterations x0, f(x0), … , fn(x0), … in S. In recent years there has been a growing interest on the study of dynamical systems where the configuration space S has an underlying algebraic structure, a group, a ring, a field, or an algebraic variety, and the map f somehow takes into account this underlying structure, e.g., when f is a homomorphism, a polynomial map or a rational map. Thus, from the very beginning, dynamical systems, number theory, algebraic geometry and ergodic theory are closely linked in this new area. There are very few monographs devoted to this new field, the book by Silverman being the most comprehensive survey. Its emphasis is on the arithmetic and algebro-geometric aspects of the theory, which are barely touched upon by the authors of this new monograph.
The book under review starts by recalling the basic facts about rings, finite fields, the field of p-adic numbers, their finite extensions and algebraic closure, and the corresponding rings of integers, including a chapter on the elements of p-adic analysis. Then, the first part of the book develops from scratch the basic notions of dynamics on algebraic structures, putting special emphasis on the ergodicity and uniform distribution of sequences (orbits) x0, f(x0), …, fn(x0),.., obtained by iterated applications of the map f to a given initial point x0 of S, for the important case when S is the field of p-adic numbers equipped, of course, with the Haar measure, or a finite extension or algebraic closure of it, or the ring of p-adic integers. Of particular interest is the case when x0 is a periodic point, i.e., when there exists a positive integer n such that fn(x0)=x0 and therefore, by choosing a least integer with this property the corresponding orbit x0, f(x0), …, fn(x0)=x0 is a cycle. Then, if x0 is an n-periodic point and there exists an open ball around x0 such that for every point x in this ball the limit of iterations of fn(x) is x0, we say that x0 is an attractor, and in similar ways we define repellers, basin of attractions and Siegel disks for periodic points. Chapters 4 and 5 are devoted to the study of these points and sets and the measure-preserving and ergodic isometries on powers of the ring of p-adic integers, including the asymptotic distribution of cycles for monomial dynamical systems, i.e. functions of the form f(x)=xk.
Part 2 of the book consists of two short chapters generalizing some of the features of commutative dynamical systems, i.e., when the underlying algebraic structure is commutative, to some non-commutative structures. They consider polynomial transformations on finite or profinite non-commutative groups, where one of the main results is a characterization, by one of the authors, of finite solvable groups with operators having ergodic polynomials.
In the remaining (third) part of the book, using the relatively simple algebraic and number-theoretical tools developed so far, the authors study some specific dynamical systems and describe some applications to such varied topics as pseudo-random generators viewed as dynamical systems and a brief review of some applications to mathematical physics. The latter is mostly related to the work of the authors, with little or no reference to the work of other physicists or mathematicians that have done some work in this area when it was somehow trendy during the 1990s. In the remaining chapters the authors detail various attempts by Khrennikov and his collaborators to model mental process and genetics using p-adic dynamical systems, models that are, at best, only in the first stages, with no evidence whatsoever that the brain or the actual genes work as the proposed models suggest.
Written for non-specialists or graduate students interested on learning how some notions and results from classical dynamical systems are formulated or proven in the p-adic or non-Arquimedean realm, this monograph provides an accessible introduction to some topics of current research in this area by focusing on some specific dynamical systems where the algebraic and number-theoretical tools required are somehow simple. However, for the more interesting aspects of the theory, involving deeper results from number theory and algebraic geometry, the book by Silverman is a better reference, and even a better textbook, since Silverman’s book comes with exercises and the book under review does not.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com.