The topic of dynamical systems means different things depending on whether you are an in the field of engineering, mathematics, or physics. Engineers will tend to think of it as essentially Newtonian mechanics, whereas physicists will view it as a study of physical systems that are chaotic. Mathematicians have traditionally considered it to be a branch of differential geometry or global analysis. In recent decades however, the mathematical study of dynamical systems has been done in the context of algebraic geometry and number theory. This book elucidates some of this research, and although not entirely self-contained since many of the proofs are left to the references, it does introduce the reader to many of the ideas that have been put forward to study the "arithmetic" of dynamical systems.



In the usual study of dynamical systems, notions of complexity, such as Lyapunov exponents, topological entropy, basins of attraction, and strange attractors appear, as do constructions such as the invariant set, the Fatou and Julia sets, symbolic dynamics, and fixed periodic, and critical points. But in the arithmetic theory of dynamical systems, it is the `height' that plays the essential role as a measure of complexity. The theory of heights should be well known to those readers who come to the book with a strong background in algebraic number theory. But even if that is not the case the author does not make use of the general theory of arithmetic heights as outlined by the mathematician Andre Weil and discussed in detail in one of the author's earlier books on Diophantine geometry. The height is a measure of "arithmetic" complexity, and in the context of dynamical systems it is of natural interest to study how the height of a point varies under the iteration of a polynomial or rational map.



Even in physics where one studies chaotic maps such as the horseshoe map, it is frequently of interest to have p-adic versions of these maps. The latter are studied in this book in a more general context of maps over nonarchimedean fields (the p-adic numbers being an example of a nonarchimedean field), but before the author gets to these maps he spends the first few chapters reviewing "classical" dynamics and studying the case of rational maps over complete local fields that are "well-behaved" over the ring of integers of these fields.



The archimedean "classical" case where one studies the iteration of polynomial and rational maps defined over the complex numbers or one-dimensional complex projective space is reviewed in the first chapter of the book and the discussion is fairly standard. For those familiar with this theory the author is careful to point out some of the different terminology that is used when moving over to the context of algebraic geometry (such as calling critical points "ramification" points). Also interesting in his review of this theory is the use of `equicontinuity' to measure the "chaotic" behavior of a map. The Julia set and the Fatou set in fact are both defined in terms of equicontinuity, with the Fatou set being the largest open set on which a map is equicontinuous, and the Julia set is the complement of the Fatou set. This review prepares the reader for later discussions on the behavior of maps in the nonarchimedean context, where a straightforward symbolic dynamics can be defined on the Julia set. And for those readers who love elliptic curves (and anyone exposed to them will be) the author reviews how rational maps can arise from them. This review motivates a later chapter on dynamical systems associated with algebraic groups.



To obtain more insight into the behavior of maps defined over nonarchimedean fields the author first studies in chapter two the case of a rational maps that are defined over the residue field of a complete local field. So what is the first thing that must be dealt with in nonarchimedean dynamics? The field of rational p-adic numbers is not algebraically closed and totally disconnected, and if one takes its algebraic closure it will not be locally compact. These sticking points are dealt with later on in the book where the author defines Berkovich spaces. In this chapter attention is focused primarily on the behavior of maps between one-dimensional projective space that are reduced modulo a prime. As is the case in number theory, the primes are thought of as "irreducible" (or maximal ideals in the parlance of algebra), and mathematicians use them to "localize" problems with the goal of gaining insight into the global problem. The author first shows how to reduce points in (one-dimensional) projective space modulo a prime, and proves that this reduction is invariant under fractional linear transformations. Rational maps on projective space are reduced modulo a prime by reducing the coefficients of the pair of homogeneous polynomials that represent them. This reduction can cause these polynomials to have common roots in the resulting residue field, but after using the theory of resultants the author shows that a rational map always has an empty Julia set if the map has a "good reduction." Maps that have good reduction are those where the reduction modulo the prime does not change the degree, whose representative polynomials have no solutions in the projective space over the residue field, and whose resultant is non-zero.



In chapter 3 the author generalizes what is known about the arithmetic of elliptic curves to the dynamical setting. The theory of heights is outlined in fair detail, and the height of a point in projective space is viewed as a generalization of the notion of the size of a rational number: where in the latter case it is maximum(numerator, denominator). The author's goal is to relate the arithmetic information provided by the height to the geometric features of maps over projective space. For rational maps that are morphisms (there homogeneous polynomials have only zero in their zero sets), he shows that the height of a morphism of degree d is the d-th power of the height of the image. This multiplicative property naturally leads to a notion of logarithmic height and then one of "canonical" height, where in the latter the height of the image is the degree times the height. The canonical height is used to characterize the arithmetic properties of the preperiodic points of the map (these points have canonical height equal to zero). Also very interesting in this chapter is the discussion on the use of Galois theory in the study of rational maps, the author showing that the periodic points of a rational map are invariant under the Galois group.



As is typical in mathematics, one does not study single objects but instead collections of them to see what properties they have in common. This philosophy is readily apparent in chapter four of the book, wherein the author studies the set of rational maps. As the author shows, this set is actually an algebraic variety and readers will find an analog of modular curves in the guise of `dynatomic' curves in this chapter. The notion of a dynatomic curve is based on that of a dynatomic polynomial, which is defined so that its roots are the fixed points of the nth iterate of the rational map. The author studies in detail the dynatomic curves associated with maps based on quadratic polynomials, and he calculates their genera. This discussion motivates him to consider the moduli space of rational maps over projective space, which is defined as the quotient space where the action of the fractional linear transformations on the collection of rational maps of degree d is factored out. The discussion of the properties of this space is delegated to the references, as the concepts needed to prove them require geometric invariant theory.



The deepest result in the book is the construction of the Berkovich space in chapter 5. This occurs after the author discusses nonarchimedean dynamics for the case of "bad reduction". Interesting in this discussion is that the Julia set in this context splits into two pieces, allowing the methods of symbolic dynamics to be used to study the dynamics of the rational map: using it, it is shown that its periodic points are repelling and dense in the Julia set, and that there exists a dense orbit in the Julia set. Thus the dynamics is "chaotic" in the usual sense: periodic points are dense and the map is topologically transitive. That the Julia set is actually the closure of the repelling periodic points is an open conjecture. The construction of the Berkovich space is complicated but the author successfully leads the reader through it by describing explicitly its points and by using diagrams, the latter inclusion an example of a didactic strategy that is thankfully becoming more prevalent in the mathematical literature.