The Arithmetic of Dynamical Systems / Edition 1

by J. H. Silverman

Overview

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. As is typical in any subject combining Diophantine problems and geometry, a fundamental goal is to describe arithmetic properties, at least qualitatively, in terms of underlying geometric structures.

Key features:

- Provides an entry for graduate students into an active field of research

- Provides a standard reference source for researchers

- Includes numerous exercises and examples

- Contains a description of many known results and conjectures, as well as an extensive glossary, bibliography, and index

This graduate-level text assumes familiarity with basic algebraic number theory. Other topics, such as basic algebraic geometry, elliptic curves, nonarchimedean analysis, and the theory of Diophantine approximation, are introduced and referenced as needed. Mathematicians and graduate students will find this text to be an excellent reference.

Product Details

ISBN-13:	9781441924179
Publisher:	Springer New York
Publication date:	11/19/2010
Series:	Graduate Texts in Mathematics Series , #241
Edition description:	Softcover reprint of hardcover 1st ed. 2007
Pages:	511
Product dimensions:	1.06(w) x 6.14(h) x 9.21(d)

Table of Contents

Preface.- An Introduction to Classical Dynamics.- Dynamics over Local Fields: Good Reduction.- Dynamics over Global Fields.- Dynamics and Moduli.- Dynamics over Local Fields: Bad Reduction.- Dynamics with an Underlying Group.- Dynamics in Dimension Greater than One.

Editorial Reviews

From the reviews:

"The connections between dynamical systems and number theory arise in many different ways. … This remarkable book unifies and clarifies one of these connections, in the setting of what might be called ‘Arithmetic Dynamical Systems’. … suitable for many graduate students. … This book should be of great interest to anyone interested in dynamics or number theory, and will attract them into this fascinating field. Not for the first time, the mathematical community owes the author thanks for a wonderful book … ." (Thomas Ward, Mathematical Reviews, Issue 2008 c)

"The Arithmetic of Dynamical Systems is intended for an audience of researchers and graduate students in number theory. … The book could easily be used for a special-topics graduate course. … 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, MathDL, January, 2008)

"The Arithmetic of Dynamical Systems arrives with auspicious timing. ...[T]he field is young enough that there are few, if any, other such comprehensive introductions to the subject. With a growing number of graduate students and established researchers trying to learn the subject, such a clear exposition comes none too soon.

The book is well organized and well written.... Ideas and intuitions are conveyed clearly, but at the same time, the presntation is completely rigorous. ... Number theorists interested in studying dynamics will find this book to be both an excellent introduction and a valuable reference for the subject." (Rob Bendetto, Bulletins of the American Mathematical Society, December 2008)

“This textbook introduces the reader to the dynamics of a rational function acting on the projective line over a number field. … The book is aimed mostly at number theorists. … This volume will provide a generation of number theorists with a comprehensive introduction to an intriguing field, whose results might make possible advances in other parts of number theory.” (C. Baxa, Monatshefte für Mathematik, Vol. 158 (3), November, 2009)

From the Publisher