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The Arithmetic of Dynamical Systems de J. H. Silverman

The Arithmetic of Dynamical Systems: Graduate Texts in Mathematics, cartea 241

Autor J. H. Silverman
en Limba Engleză Hardback – 6 iun 2007

Publicată în prestigioasa serie Graduate Texts in Mathematics de la Springer, ediția de față semnată de J. H. Silverman reprezintă o sinteză riguroasă a unui domeniu hibrid: dinamica aritmetică. Ne-a atras atenția modul în care autorul reușește să creeze o punte între două arii matematice venerabile, Sistemele Dinamice și Teoria Numerelor. Considerăm că forța acestui volum rezidă în capacitatea de a transpune teoremele și conjecturile clasice din teoria ecuațiilor diofantiene în cadrul sistemelor dinamice discrete, în special pentru iterațiile pe linia proiectivă.

Din punct de vedere al conținutului, progresia materialului este una logică și exhaustivă pentru nivelul universitar (graduate). Primele capitole introduc dinamica clasică, urmate imediat de o analiză detaliată a dinamicii peste câmpuri locale, tratând atât cazurile de reducere bună, cât și cele de reducere rea. Un capitol central este dedicat familiilor de sisteme dinamice, oferind instrumentele necesare pentru a înțelege variația proprietăților aritmetice. Comparativ cu Heights of Polynomials and Entropy in Algebraic Dynamics de Graham Everest, care servește mai degrabă ca o invitație generală spre subiect, volumul lui J. H. Silverman adoptă o abordare mult mai tehnică și sistematică, fiind orientat către cercetarea aprofundată a punctelor raționale în orbite și a punctelor periodice.

Merită menționat că această lucrare se fundamentează pe experiența vastă a autorului în geometria aritmetică. Dacă în lucrările anterioare accentul cădea pe curbele eliptice, aici J. H. Silverman extinde perspectiva către dinamica asociată grupurilor algebrice și dimensiunilor superioare. Deși autorul recunoaște că selecția temelor reflectă interesele sale specifice, volumul rămâne o resursă fundamentală care unifică și clarifică conexiunile profunde dintre iterații și proprietățile structurale ale numerelor.

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Specificații

ISBN-13: 9780387699035
ISBN-10: 0387699031
Pagini: 528
Ilustrații: XVI, 511 p. 11 illus.
Dimensiuni: 160 x 241 x 34 mm
Greutate: 0.95 kg
Ediția:2007
Editura: Springer
Colecția Graduate Texts in Mathematics
Seria Graduate Texts in Mathematics

Locul publicării:New York, NY, United States

Public țintă

Graduate

De ce să citești această carte

Recomandăm această carte studenților de la masterat și doctorat care doresc să stăpânească fundamentele dinamicii aritmetice. Cititorul câștigă o înțelegere clară a modului în care metodele geometriei algebrice pot fi aplicate sistemelor dinamice. Este un manual esențial pentru oricine dorește să exploreze analogiile dintre punctele de torsiune pe varietăți abeliene și punctele preperiodice ale funcțiilor raționale.

Descriere scurtă

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.

Cuprins

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.

Recenzii

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. … 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)

Textul de pe ultima copertă

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.

Caracteristici

Provides an entry for graduate students into an active field of research Each chapter includes exercises, examples, and figures Will become a standard reference for researchers in the field Contains a description of many known results and conjectures, together with an extensive bibliography Includes supplementary material: sn.pub/extras