The Arithmetic of Dynamical Systems
(Sprache: Englisch)
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...
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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 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.
Klappentext zu „The Arithmetic of Dynamical Systems “
This book is designed to provide a path for the reader into an amalgamation oftwo venerable areas ofmathematics, Dynamical Systems and Number Theory. Many of the motivating theorems and conjectures in the new subject of Arithmetic Dynamics may be viewed as the transposition ofclassical results in the theory ofDiophantine equations to the setting of discrete dynamical systems, especially to the iteration theory ofmaps on the projective line and other algebraic varieties. Although there is no precise dictionary connecting the two areas, the reader will gain a flavor of the correspondence from the following associations: Diophantine Equations Dynamical Systems rational and integral rational and integral points on varieties points in orbits torsion points on periodic and preperiodic abelian varieties points ofrational maps There are a variety of topics covered in this volume, but inevitably the choice reflects the author's tastes and interests. Many related areas that also fall under the heading ofarithmetic or algebraic dynamics have been omitted in order to keep the book to a manageable length. A brief list of some of these omitted topics may be found in the introduction. Online Resources The reader will find additonal material, references and errata at http://www. math. brown. ectu/-jhs/ADSHome. html Acknowledgments The author has consulted a great many sources in writing this book. Every attempt has been made to give proper attribution for all but the most standard results.
Inhaltsverzeichnis zu „The Arithmetic of Dynamical Systems “
An Introduction to Classical Dynamics.- Dynamics over Local Fields: Good Reduction.- Dynamics over Global Fields.- Families of Dynamical Systems.- Dynamics over Local Fields: Bad Reduction.- Dynamics Associated to Algebraic Groups.- Dynamics in Dimension Greater Than One.
Bibliographische Angaben
- Autor: J.H. Silverman
- 2010, XVI, 511 Seiten, Maße: 15,5 x 23,5 cm, Kartoniert (TB), Englisch
- Verlag: Springer, Berlin
- ISBN-10: 1441924175
- ISBN-13: 9781441924179
Sprache:
Englisch
Pressezitat
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.
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... Thebook 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)
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