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Silverman, J. H.

The Arithmetic of Dynamical Systems (Kartoniert / Broschiert / Paperback)

Springer, November 2010


Softcover reprint of hardcover 1st ed. 2007. - 528 S. - Sprache: Englisch - 235x155x28 mm Previously published in hardcover

ISBN: 1441924175 EAN: 9781441924179

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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Produktdetails

This book investigates the relatively new subject of Arithmetic Dynamics, which is the study of the number theoretic properties of algebraic numbers or points under repeated application of a polynomial or rational map. Classical discrete dynamics is the study of iteration of functions mapping the complex plane (or real line) to itself. Arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. The viewpoint of this book 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, an overarching theme is that at least qualitatively, the geometry determines the arithmetic.

Inhaltsverzeichnis

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.

Kritikerstimmen

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)

ISBN 1-441924-17-5, ISBN 1-44-192417-5, ISBN 1-44192-417-5, ISBN 1-441-92417-5, ISBN 1-4419-2417-5

ISBN 978-1-441924-17-9, ISBN 978-1-44-192417-9, ISBN 978-1-44192-417-9, ISBN 978-1-441-92417-9, ISBN 978-1-4419-2417-9

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