Number theory and dynamics lie at nearly opposite ends of the mathematical spectrum. The study of chaotic dynamical systems dates back barely a century and is intimately linked to applications, while number theory is ancient and largely theoretical. In the past fifteen or twenty years, however, a body of research has sprung up linking the two fields. After a plethora of papers but very few expositions, The Arithmetic of Dynamical Systems now provides a fantastic introduction to this active area of research.
The book's central object of study is the following type of dynamical system. Let f(z) be a polynomial in one variable with coefficients in Q, the field of rational numbers. Then f maps Q to itself, giving us a dynamical system; that is, we can try to understand what happens when we apply f repeatedly to a rational input. The only obvious features are periodic points, which are points that land back on themselves after some positive number of iterations, or, more generally, preperiodic points, which eventually land on periodic points. For example, if f(z)=z2–1, then 0 and –1 are periodic, because each maps to the other; and all three of 0, –1, and 1 are preperiodic, because 1 maps to the periodic point 0.
More generally, if f is a rational function with coefficients in Q, then f maps the projective line over Q (i.e., Q along with a point at infinity) to itself. Generalizing further, we can replace Q by an arbitrary number field K (i.e., a finite extension of Q); more generally still, we can consider morphisms of varieties defined over K in place of rational functions acting on the projective line. In all such settings, one can study the set of K-rational preperiodic points; that is, preperiodic points with coordinates in K .
Douglas Northcott proved in 1950 that if f is a morphism of projective N-space of degree at least two, then there are only finitely many such K-rational preperiodic points. For example, it is not difficult to show that the only Q-rational preperiodic points of the function f(z)=z2–1 are the three we already noted: 0, –1, and 1; if we work on the projective line, then we get the point at infinity as a fourth, but that is all. Of course, over the complex numbers there are infinitely many preperiodic points of this function; indeed, the finiteness is due to the properties of Q, and more generally, of number fields.
Northcott's result suggests several difficult and open questions, even for the one-dimensional case of maps of the projective line. For example, if we fix the degree of f and the number field K, is there an upper bound on how many K-rational preperiodic points such a map f can have? On the other hand, if we fix a function f but let the number field K grow, how fast does the set of K-rational preperiodic points grow? Turning our attention to integers, if f is a rational function that is not a polynomial (and that is written in minimal form in a certain natural sense), is there an upper bound for how many consecutive iterates of a non-preperiodic point can be integers? These and other questions are motivated by parallels with the theory of elliptic curves, and such connections are of course not lost in a book penned by the author of some of the standard references on elliptic curves. For example, as Silverman notes on the second page of the book, the torsion elements of an abelian group G are precisely the preperiodic points of the doubling map on G.
After a discussion of some basic terminology and motivating problems in the brief introduction, The Arithmetic of Dynamical Systems begins with a chapter on the fundamentals of complex dynamics. Topics here include multipliers of periodic points, coordinate change by linear fractional transformations, and Fatou and Julia sets. Here and throughout the book, most results are proven from scratch, but proofs of some of the deeper theorems are referred to the extensive bibliography. Then, in the second chapter, Silverman turns to the study of dynamics at non-archimedean places in the simple (but also the most general) case of good reduction; that is, he describes dynamical systems that still makes sense if we work modulo a prime p. (The case of bad reduction, for which something goes wrong modulo p, is studied in Chapter 5.) With the so-called local field cases mostly discussed, the third chapter turns to dynamics over a number field. It is here that Northcott's Theorem and the questions posed above appear. Thus, aspects of arithmetic heights, integrality, Diophantine approximation, and Galois twists are all discussed, although little or no background on these topics is assumed beyond the basics of algebraic number theory.
The remaining four chapters are devoted to more advanced topics. Chapter 4 considers families of dynamical systems, such as the famous one-parameter family of quadratic polynomials fc(z)=z2+c. The chapter also introduces dynatomic polynomials (a sort of dynamical analogue of cyclotomic polynomials), which can be used to define certain dynamical moduli spaces analogous to the modular curves X0(N) and X1(N) from the theory of elliptic curves. Next, the previously deferred case of bad reduction is considered in Chapter 5. The topics presented include non-archimedean Fatou components, wandering domains, and local canonical height functions. The chapter ends with a short introduction to Berkovich spaces and their use in p -adic dynamics.
Chapter 6 concerns maps induced by algebraic groups, especially elliptic curves. (Again, no prior background is assumed; Section 6.3 presents a lightning-quick summary of the relevant aspects of the theory of elliptic curves.) Finally, after six chapters about dynamics on the projective line, Chapter 7 discusses a few special number-theoretic aspects of dynamics on higher-dimensional varieties.
There are, of course, many other types of problems and questions that reside on the interface between dynamics and algebra. For example, the ergodic theory of Zd-actions falls into this category but is not touched by Silverman's book. (The reader interested in that theory might instead turn to Klaus Schmidt's Dynamical Systems of Algebraic Origin.) Similarly, Brjuno numbers and Diophantine approximation arise naturally in the study of linearization at indifferent fixed points in complex dynamics, but again, they are not considered here. In fact, in his introduction, Silverman lists no fewer than ten topics that could be construed as some form of algebraic dynamics but that are not discussed in the book.
The Arithmetic of Dynamical Systems is intended for an audience of researchers and graduate students in number theory. Silverman assumes the reader is familiar with Q and its finite extensions, is comfortable with Galois theory, and knows a little algebraic number theory and perhaps some algebraic geometry. On the other hand, he makes no assumption that the reader knows anything at all about dynamical systems.
The book could easily be used for a special-topics graduate course. Dozens of exercises, grouped by section, appear at the end of each chapter. The variety of exercises, which include general proofs and special examples alike, is a great strength of the book. There is quite a range of difficulty; some problems are fairly straightforward but most are moderately challenging. Especially hard exercises, often drawn from the literature, bear a star, and several are doubly-starred, indicating that they are in fact unsolved. (The narrative itself also states quite a few conjectures and open questions, undoubtedly providing a wealth of PhD problems for years to come.) Hints and references for some exercises are included at the end of the book.
Silverman's presentation of the material is well-organized and clear, with numerous helpful examples and plenty of intuitive explanations to accompany the completely rigorous proofs. Fans of his elliptic curves texts will be pleased to find the qualities that made those books so successful on display here as well. With its combination of readability and precision, The Arithmetic of Dynamical Systems will serve not only as an excellent introduction to the Diophantine aspects of dynamics for the uninitiated, but also as a valuable reference for experts in the field. It is certain to be an essential resource for anyone interested in this active and growing area of research.
Rob Benedetto is Associate Professor of Mathematics at Amherst College.