The Arithmetic of Dynamical Systems

by Silverman, Joseph H.

• ISBN13: 9780387699035
• ISBN10: 0387699031
• Format: Hardcover
• Copyright: 2007-06-01
• Publisher: Springer Nature

Summary

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.

Table of Contents

Preface	p. V
Introduction	p. 1
Exercises	p. 7
An Introduction to Classical Dynamics	p. 9
Rational Maps and the Projective Line	p. 9
Critical Points and the Riemann-Hurwitz Formula	p. 12
Periodic Points and Multipliers	p. 18
The Julia Set and the Fatou Set	p. 22
Properties of Periodic Points	p. 27
Dynamical Systems Associated to Algebraic Groups	p. 28
Exercises	p. 35
Dynamics over Local Fields: Good Reduction	p. 43
The Nonarchimedean Chordal Metric	p. 43
Periodic Points and Their Properties	p. 47
Reduction of Points and Maps Modulo p	p. 48
The Resultant of a Rational Map	p. 53
Rational Maps with Good Reduction	p. 58
Periodic Points and Good Reduction	p. 62
Periodic Points and Dynamical Units	p. 69
Exercises	p. 74
Dynamics over Global Fields	p. 81
Height Functions	p. 81
Height Functions and Geometry	p. 89
The Uniform Boundedness Conjecture	p. 95
Canonical Heights and Dynamical Systems	p. 97
Local Canonical Heights	p. 102
Diophantine Approximation	p. 104
Integral Points in Orbits	p. 108
Integrality Estimates for Points in Orbits	p. 112
Periodic Points and Galois Groups	p. 122
Equidistribution and Preperiodic Points	p. 126
Ramification and Units in Dynatomic Fields	p. 129
Exercises	p. 135
Families of Dynamical Systems	p. 147
Dynatomic Polynomials	p. 148
Quadratic Polynomials and Dynatomic Modular Curves	p. 155
The Space Rat[superscript d] of Rational Functions	p. 168
The Moduli Space M[subscript d] of Dynamical Systems	p. 174
Periodic Points, Multipliers, and Multiplier Spectra	p. 179
The Moduli Space M[subscript 2] of Dynamical Systems of Degree 2	p. 188
Automorphisms and Twists	p. 195
General Theory of Twists	p. 199
Twists of Rational Maps	p. 203
Fields of Definition and the Field of Moduli	p. 206
Minimal Resultants and Minimal Models	p. 218
Exercises	p. 224
Dynamics over Local Fields: Bad Reduction	p. 239
Absolute Values and Completions	p. 240
A Primer on Nonarchimedean Analysis	p. 242
Newton Polygons and the Maximum Modulus Principle	p. 248
The Nonarchimedean Julia and Fatou Sets	p. 254
The Dynamics of (z[superscript 2] - z)/p	p. 257
A Nonarchimedean Montel Theorem	p. 263
Periodic Points and the Julia Set	p. 268
Nonarchimedean Wandering Domains	p. 276
Green Functions and Local Heights	p. 287
Dynamics on Berkovich Space	p. 294
Exercises	p. 312
Dynamics Associated to Algebraic Groups	p. 325
Power Maps and the Multiplicative Group	p. 325
Chebyshev Polynomials	p. 328
A Primer on Elliptic Curves	p. 336
General Properties of Lattes Maps	p. 350
Flexible Lattes Maps	p. 355
Rigid Lattes Maps	p. 364
Uniform Bounds for Lattes Maps	p. 368
Affine Morphisms and Commuting Families	p. 375
Exercises	p. 380
Dynamics in Dimension Greater Than One	p. 387
Dynamics of Rational Maps on Projective Space	p. 388
Primer or Algebraic Geometry	p. 402
The Weil Height Machine	p. 407
Dynamics on Surfaces with Noncommuting Involutions	p. 410
Exercises	p. 427
Notes on Exercises	p. 441
List of Notation	p. 445
References	p. 451
Index	p. 473
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