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This textbook is an introduction to dynamical systems and its applications to evolutionary game theory, mathematical ecology, and population genetics. This first English edition is a translation from the authors' successful German edition which has already made an enormous impact on the teaching and study of mathematical biology. The book's main theme is to discuss the solution of differential equations that arise from examples in evolutionary biology. Topics covered include the Hardy-Weinberg law, the Lotka-Volterra equations for ecological models, genetic evolution, aspects of sociobiology, and mutation and recombination. There are numerous examples and exercises throughout and the reader is led up to some of the most recent developments in the field. Thus the book will make an ideal introduction to the subject for graduate students in mathematics and biology coming to the subject for the first time. Research workers in evolutionary theory will also find much of interest here in the application of powerful mathematical techniques to the subject. ... Read more

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Dynamical systems theory is especially well-suited for determining the possible asymptotic states (at both early and late times) of cosmological models, particularly when the governing equations are a finite system of autonomous ordinary differential equations.
In this book we discuss cosmological models as dynamical systems, with particular emphasis on applications in the early Universe. We point out the important role of self-similar models. We review the asymptotic properties of spatially homogeneous perfect fluid models in general relativity. We then discuss results concerning scalar field models with an exponential potential (both with and without barotropic matter). Finally, we discuss the dynamical properties of cosmological models derived from the string effective action.
This book is a valuable source for all graduate students and professional astronomers who are interested in modern developments in cosmology. ... Read more

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A dynamical system refers to a set of elements that interact in complex, often nonlinear ways to form coherent patterns. Because of the complexity of these interactions, the system as a whole may evolve over time in seemingly unpredictable ways as new patterns of behavior emerge. This metatheory has proven useful in understanding diverse phenomena in meteorology, population biology, statistical mechanics, economics, and cosmology. The book demonstrates how the dynamical systems perspective can be applied to theory construction and research in social psychology, and in doing so, provides fresh insight into such complex phenomena as interpersonal behavior, social relations, attitudes, and social cognition. ... Read more

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A must read for those interested in human interaction
Dynamical systems in social psychology contains twelve independently written chapters on the dynamics of social interaction, perception and information. It centers on a the growing discipline of non-linear dynamics and it's application to human development and relations. Many of the authors are leaders within their respective disciplines (social psychology, non-linear dynamics, human movement and physics) and describe - discuss the topic with thought provoking insight and understanding. It clearly demonstrates the capabilities of dynamical systems theory for social psychologists and raises a number of questions to both broaden and direct future research and investigation in the area. Its only downfall is the complex material it deals with, making it difficult for those without a prior understanding of dynamical systems to follow. Nevertheless, it is an essential book for all those serious about the development of social psychology as a science and the application of non-linear dynamics to psychology as a whole. ... Read more

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CHAOS: An Introduction to Dynamical Systems was developed and class-tested by a distinguished team of authors at two universities through their teaching of courses based on the material. Intended for courses in nonlinear dynamics offered either in Mathematics or Physics, the text requires only calculus, differential equations, and linear algebra as prerequisites. Spanning the wide reach of nonlinear dynamics throughout mathematics, natural and physical science, CHAOS develops and explains the most intriguing and fundamental elements of the topic and examines their broad implications. Among the major topics included are: discrete dynamical systems, chaos, fractals, nonlinear differential equations, and bifurcations. The text also features Lab Visits, short reports that illustrate relevant concepts from the physical, chemical, and biological sciences, drawn from the scientific literature. There are Computer Experiments throughout the text that present opportunities to explore dynamics through computer simulation, designed to be used with any software package. And each chapter ends with a Challenge, which provides students a tour through an advanced topic in the form of an extended exercise. ... Read more

Customer Reviews (6)

Great Book
This book is both simple enough to understand, and sophisticated enough to provide further understanding.If you are an second or third year undergraduate planning on graduate work, this book is a great way to catch up on advanced mathematics.

A/S/Y strike a perfect balance between theory and applications!
It was about the mid 1990's, still assimilating the big hype caused by the eventual and much-publicized proof by Andrew Wiles of Fermat's Last Theorem, when my curiosity (bolstered more by having seen a movie such as The Jurassic Park!) finally led me to taking a first college course on Chaos and Fractals at a California State school. At that time, the funny, surcastic, and somewhat sloppy foreign professor (who happened to be a country-mate of mine, for better or worse), had chosen the brand-new text "Fractals Everywhere" by Michael F. Barnsely for teaching our mid-size class consisting mainly of senior and first-year graduate students in math and sciences. I recall the discussion starting out by covering the basics about the metric spaces and sequences, and I having a head-start over many others coming fresh on the heels of a heavy-duty general topology course just in the previous semester (so for example I could show off to others on the first instruction day what it meant for two metrics to be equivalent). Still, I admit the semester went by without many of us really absorbing the nuts and bolts of the subject, for example why exactly topological transitivity was needed for chaos in an Iterated Function System, and why exactly some known fractals had the given fractional dimensions (eventhough we could compute them). However the students were generally happy to have scratched the surface of this vast, engaging subject, and for the time being it seemed about enough exposure for most of us. Consequently for me, during the several ensuing years in the late 90's the subject leapt mostly into the background, but nearly a decade later since I first took the college course, somehow it came back to the foreground in the company of several other applied subjects such as control, game theory, and information/coding theory.

Now looking back, I find Barnsley's text a very good choice having gone through at the time, but the title by Alligood, Sauer, and Yorke (as a recommendation by a college professor at a different school who had taught his students from it) seemed like a more well-balanced introduction to the area of dynamical systems. In fact I also recall at the time there was a discussion as to whether yet another text by Robert Devaney would have made for a better first course. The aforementioned professor duely noted that Devaney only dealt with the discrete dynamical systems, while A/S/Y treated both the discrete and continuous, hence making the choice of the latter a more suitable one. In any event, the rundown of the topics discussed in the 13 chapters of A/S/Y include: one and two dimensional maps, fixed points, iterations, sinks, sources, saddles, Lyapunov exponents, chaotic orbits, conjugacy, fractals and their dimension, chaotic attractors, measure, Lotka-Volterra models, Poincare-Bendixson theorem, Lorentz and Roessler attractors, stable manifolds and crises, homoclinic and heteroclinic points, bifurcations, and cascades. There are answers and solutions to the selected exercises, as well as extensive references at the back, making up an ideal setting for self-study. The level and style of exposition is targeted towards an advanced undergraduate student who is into applied math or engineering fields. Therefore the authors emphasize concepts and applications instead of getting bogged down in too much mathematical rigor or heavy use of the abstract machinery (which is of course needed for a thorough treatment of the subject at an advanced level; there are in fact several newer titles which all occupy this niche). Notationally and stylistically also, A/S/Y is very accessible and attractive. All in all, an excellent first excursion/introduction to one of the most fascinating areas of applied math, whether for classroom use, or for self-study.

[Review updated and reposted on 08/08/08]

Exciting and Lucid Introduction to Chaos Theory
This book is a must-own for anyone interested in nonlinear dynamics and chaos -- I also highly recommend the "Nonlinear Dynamics and Chaos" text by Strogatz.

I especially like the numerous diagrams that clarify everything so well in this book. In addition, the writing includes just the right amount of informal discussion to truly explain the material without retreating into jargon.

A favorite moment in the book is a "challenge" exercise that explains the famous "Period Three Implies Chaos" result: the reader is gently guided through 10 steps resulting in a proof of Sharkovskii's Theorem, a more general result that includes the Period 3 thing as a special case.

Buy it! Simply phenomenal.

The definitive guide to dynamical systems!
When I purchased this book three years ago, I had only a rudimentary understanding of dynamical systems. Thankfully, all that was needed to get me started was some intermediate calculus and some basic college-level linear algebra. Since I had been doing both from the time I was a sophmore in high school, I had no trouble getting comfortable with it. The authors present dynamical systems in an easy-to-read style with tests that appear at the end of each chapter after you've had time to catch on.

If you're seriously thinking about getting started in dynamical systems, get this book!

great introduction to dynamical systems
I was enrolled in a course at GMU in which the draft version of this text was used.The math was not as difficult as some of the graduate texts, therefore it serves as a good intoduction for someone with as little as 2 years of undergraduate math.The challenges at the end of each chapter are more difficult than the regular problems, but they are meant to be.Many of the systems can be modeled on a spreadsheet.If you have any interest in Chaos, this book will only strengthen it. ... Read more

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Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines much of the material found in a traditional course on ordinary differential equations with an introduction to the more modern theory of dynamical systems. Applications of this theory to physics, biology, chemistry, and engineering are shown through examples in such areas as population modeling, fluid dynamics, electronics, and mechanics.

Differential Dynamical Systems begins with coverage of linear systems, including matrix algebra; the focus then shifts to foundational material on nonlinear differential equations, making heavy use of the contraction-mapping theorem. Subsequent chapters deal specifically with dynamical systems concepts flow, stability, invariant manifolds, the phase plane, bifurcation, chaos, and Hamiltonian dynamics.

Throughout the book, the author includes exercises to help students develop an analytical and geometrical understanding of dynamics. Many of the exercises and examples are based on applications and some involve computation; an appendix offers simple codes written in Maple®, Mathematica®, and MATLAB® software to give students practice with computation applied to dynamical systems problems.

Audience

This textbook is intended for senior undergraduates and first-year graduate students in pure and applied mathematics, engineering, and the physical sciences. Readers should be comfortable with elementary differential equations and linear algebra and should have had exposure to advanced calculus.

Contents: List of Figures; Preface; Acknowledgments; Chapter 1: Introduction; Chapter 2: Linear Systems; Chapter 3: Existence and Uniqueness; Chapter 4: Dynamical Systems; Chapter 5: Invariant Manifolds; Chapter 6: The Phase Plane; Chapter 7: Chaotic Dynamics; Chapter 8: Bifurcation Theory; Chapter 9: Hamiltonian Dynamics; Appendix:Mathematical Software; Bibliography; Index ... Read more

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Physical laws are for the most part expressed in terms of differential equations, and natural classes of these are in the form of conservation laws or of problems of the calculus of variations for an action functional. These problems can generally be posed as Hamiltonian systems, whether dynamical systems on finite dimensional phase space as in classical mechanics, or partial differential equations (PDE) which are naturally of infinitely many degrees of freedom. This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems as well as the theory of Hamiltonian systems in infinite dimensional phase space; these are described in depth in this volume. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations. These lecture notes cover many areas of recent mathematical progress in this field, including the new choreographies of many body orbits, the development of rigorous averaging methods which give hope for realistic long time stability results, the development of KAM theory for partial differential equations in one and in higher dimensions, and the new developments in the long outstanding problem of Arnold diffusion. It also includes other contributions to celestial mechanics, to control theory, to partial differential equations of fluid dynamics, and to the theory of adiabatic invariants. In particular the last several years has seen major progress on the problems of KAM theory and Arnold diffusion; accordingly, this volume includes lectures on recent developments of KAM theory in infinite dimensional phase space, and descriptions of Arnold diffusion using variational methods as well as geometrical approaches to the gap problem. The subjects in question involve by necessity some of the most technical aspects of analysis coming from a number of diverse fields. Before the present volume, there has not been one text nor one course of study in which advanced students or experienced researchers from other areas can obtain an overview and background to enter this research area. This volume offers this, in an unparalleled series of extended lectures encompassing this wide spectrum of topics in PDE and dynamical systems.

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This book is the second part of the text Differential Equations: A Dynamical Systems Approach written by John Hubbard and Beverly West. It is a continuation of the subject matter discussed in the first book, with an emphasis on systems of ordinary differential equations. This book will be most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, applied mathematics, as well as in the life sciences, physics, and economics. This book opens with an introduction, and follows with chapters on systems of differential equations, systems of linear differential equations, and systems of nonlinear differential equations. The book continues with structural stability, bifurcations, and an appendix on linear algebra. The authors also include an appendix containing important theorems from parts I and II, as well as answers to selected problems. ... Read more

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Excellent book!
Very clear and intuitive upper-undergrad/graduate level text. I highly recomend it. ... Read more

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Integrating feedforward control with feedback control can significantly improve the performance of control systems compared to using feedback control alone. Focusing on feedforward control techniques, Optimal Reference Shaping for Dynamical Systems: Theory and Applications lucidly covers the various algorithms for attenuating residual oscillations that are excited by reference inputs to dynamical systems. The reference shaping techniques presented in the book require the system to be stable or marginally stable, including systems where feedback control has been used to stabilize the system.

Illustrates Techniques through Benchmark Problems

After developing models for applications in which the dynamics are dominated by lightly damped poles, the book describes the time-delay filter (input shaper) design technique and reviews the calculus of variations. It then focuses on four control problems: time-optimal, fuel/time-optimal, fuel limited time-optimal, and jerk limited time-optimal control. The author explains how the minimax optimization problem can help in the design of robust time-delay filters and explores the input-constrained design of open-loop control profiles that account for friction in the design of point-to-point control profiles. The final chapter presents numerical techniques for solving the problem of designing shaped inputs.

Supplying MATLAB^® code and a suite of real-world problems, this book provides a rigorous yet accessible presentation of the theory and numerical techniques used to shape control system inputs for achieving precise control when modeling uncertainties exist. It includes up-to-date techniques for the design of command-shaped profiles for precise, robust, and rapid point-to-point control of underdamped systems.

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Written by one of the foremost experts in the field of neural networks, this is the first book to combine the theories and applications or neural networks and fuzzy systems. The book is divided into three sections: Neural Network Theory, Neural Network Applications, and Fuzzy Theory and Applications. It describes how neural networks can be used in applications such as: signal and image processing, function estimation, robotics and control, analog VLSI and optical hardware design; and concludes with a presentation of the new geometric theory of fuzzy sets, systems, and associative memories. ... Read more

Customer Reviews (3)

An Advanced Neural networks Book
Bart's is legendary known for his contribution to Neural networks and Fuzzy logic. This book though very good, is an advanced level book preferably for a graduate student. Thorough knowledge of Signal analysis and probability would be needed for most part of the book. Advantages of the book is it is, if not the deepest book, on Neural networks and Fuzzy logic, it will surely count as one of the most finest book dealing in the subjects.
Disadvantages: I dont know the reason why Bart has included so many theorem proving solutions like proving why BAM(Bilinear Associative memory) works. Though i commend that he has done a fine work with proving the stability of many learning algorithms. I wont recommend this book to a newbie if opting for Neural networks. Introduction to Neural Networks by Zurada ASIN-0314933913is better off in teaching all the different Algorithms in depth.

This book earns a 5 star from me because of the indepth coverage it has on the subjects.

This is a great book - elementary and mainly electrical engg
This book develops both the theories from the basics. It wonderfully describes the various theories and architectures logically developing the ideas. It has a practical approach and a generalised view without any sortof bias on computer programming alone. Professor Kosko also mathematicallydescribes the underlying mechanism of both neural and fuzzy system. Theinterdisciplinary examples from various fields show how machineintelligence can be used as proper alternative approaches effectively.I recomend every Electrical and computer engg. to read the bookeventhough he/she does not work in the area of m/c intelligence. A must forpeople in M/C intelligence.

Strong coverage of the relationship between fuzzy & neural
Kosko has two books, with similar titles published in 1991 and 1992(some copies) which together relate fuzzy and neural reasoning on both a mathematical/theoretical plane and the other on a practical plane.How to use neural networks to build fuzzy sets is one of many useful tools that can speed development of a fuzzy project.Also fuzzy rules can be derived using Kosko's methods with a neural net.If you can, acquire both.If neural practice or fuzzy practice is your goal this is the book to use. Strong on practical methods. ... Read more

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This work can be recommended as an extensive course onp-adic mathematics, treating subjects such as a p-adictheory of probability and stochastic processes; spectral theory ofoperators in non-Archimedean Hilbert spaces; dynamic systems;p-adic fractal dimension, infinite-dimensional analysis andFeynman integration based on theAlbeverio--Hoegh--Krohn approach; both linear andnonlinear differential and pseudo-differential equations; complexityof random sequences and a p-adic description of chaos. Also, the present volume explores the unique concept of using fieldsof p-adic numbers and their corresponding non-Archimedeananalysis, a p-adic solution of paradoxes in the foundations ofquantum mechanics, and especially the famousEinstein--Podolsky--Rosen paradox to create anepistemological framework for scientific use. Audience: This book will be valuable to postgraduate studentsand researchers with an interest in such diverse disciplines asmathematics, physics, biology and philosophy. ... Read more

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"A good book for a nice price!" Monatshefte fr Mathematik ". . . for lecture courses that cover the classical theory of nonlinear differential equations associated with Poincar and Lyapunov and introduce the student to the ideas of bifurcation theory and chaos this is an ideal text . . . " Mathematika "The pedagogical style is excellent, consisting typically of an insightful overview followed by theorems, illustrative examples and exercises" Choice ... Read more

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Springer dropped the ball
Over all, Verhulst does a ressonable job of conveying the central ideas of the topic.I found it to be slow going for a couple of reasons:First, it's clearly a translation and Springer should have done a better proofreading job ... many of the sentence constructions are unusual for English which is, at the very least, distracting.Second, there are substantial gaps in some of the arguments ... maybe ok for a book at this level, but it often creates considerable frustration.

In contrast, the book by Jordan and Smith (Nonlinear Ordinary Differential Equations) has few flaws and, in my view, should be read first.

This book is not even reasonable
This book was not written for an student of ODE, he was written to the author himself!

This book was used in a course of PhD here in Brazil and the results were very negative. I strongly don't recomend this book... Avoid it...

This book takes you by hand through dynamical systems theory
What I like more of this work is tha the autor explains all theconcepts he is using, so it is ideal for people who is a naturalscientist but not necesarilly knows all the formalism of modern mathematics. It is not hard to read and covers the basics for study recent research papers.END

This book takes you by hand through dynamical systems theory
What I like more of this work is tha the autor explains all theconcepts he is using, so it is ideal for people who is a naturalscientist but not necesarilly knows all the formalism of modern mathematics. It is not hard to read and covers the basics for study recent research papers.END ... Read more

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Winner of the 1998 Ferran Sunyer i Balaguer Prize

This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. In addition to the theory, the book also presents several important applications, including homogeneous potentials, Bianchi IX cosmological model, three-body problem, Hnon-Heiles system, etc. The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Sim, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed.

Series: Progress in Mathematics, Vol. 179 ... Read more

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Good book
This is a great book if you have already some notions on dynamics. I use it for two quarterly tutorial courses on hyperbolicity and found it veryusefull. Not for beginers, contains a lot of importants ideas in thesubject. ... Read more

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This volume contains nine refereed research papers in various areas from combinatorics to dynamical systems, with computer algebra as an underlying and unifying theme.

Topics covered include irregular connections, rank reduction and summability of solutions of differential systems, asymptotic behaviour of divergent series, integrability of Hamiltonian systems, multiple zeta values, quasi-polynomial formalism, Padé approximants related to analytic integrability, hybrid systems.

The interactions between computer algebra, dynamical systems and combinatorics discussed in this volume should be useful for both mathematicians and theoretical physicists who are interested in effective computation. ... Read more

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A compilation of research methods and reflecting the expertise of the major contributors to NDS psychology, this book examines the techniques that have proven to be most useful in the behavioral sciences. This book is designed to develop skill and expertise in framing hypotheses dynamically and in building viable analytic models to test them. It addresses topics and methods of current interest in an application driven manner, making the book useful to the behavioral sciences community, as well as those in engineering, medicine, and other fields who are interested in nonlinear dynamics.

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Based on a streamlined presentation of the authors' successful work Linear Systems, this textbook provides an introduction to systems theory with an emphasis on control. The material presented is broad enough to give the reader a clear picture of the dynamical behavior of linear systems as well as their advantages and limitations. Fundamental results and topics essential to linear systems theory are emphasized. The emphasis is on time-invariant systems, both continuous- and discrete-time.

Key features and topics:

* Notes, references, exercises, and a summary and highlights section at the end of each chapter.

* Comprehensive index and answers to selected exercises at the end of the book.

* Necessary mathematical background material included in an appendix.

* Helpful guidelines for the reader in the preface.

* Three core chapters guiding the reader to an excellent understanding of the dynamical behavior of systems.

* Detailed coverage of internal and external system descriptions, including state variable, impulse response and transfer function, polynomial matrix, and fractional representations.

* Explanation of stability, controllability, observability, and realizations with an emphasis on fundamental results.

* Detailed discussion of state-feedback, state-estimation, and eigenvalue assignment.

* Emphasis on time-invariant systems, both continuous- and discrete-time. For full coverage of time-variant systems, the reader is encouraged to refer to the companion book Linear Systems, which contains more detailed descriptions and additional material, including all the proofs of the results presented here.

* Solutions manual available to instructors upon adoption of the text.

A Linear Systems Primer is geared towards first-year graduate and senior undergraduate students in a typical one-semester introductory course on systems and control. It may also serve as an excellent reference or self-study guide for electrical, mechanical, chemical, and aerospace engineers, applied mathematicians, and researchers working in control, communications, and signal processing.

Also by the authors: Linear Systems, ISBN 978-0-8176-4434-5.

Customer Reviews (1)

Very good primer
I used this book for graduate course on linear systems theory. The book covers all important parts of linear systems theory. All the parts are very good explained. I think the style of the book is friendly to newcomers and certainly is suitable for graduate students. Every chapter contains a lot of examples that always help understanding the theory. I really liked studying with this book and I still use it as a reference.
... Read more

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This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.

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A fine overview of nonarchimedean dynamics

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. ... Read more