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.
Author(s): Joseph H. Silverman
Series: Graduate Texts in Mathematics
Publisher: Springer
Year: 2007
Language: English
Pages: 518
0387699031......Page 1
Graduate Texts in Mathematics 241......Page 2
The Arithmetic of\rDynamical Systems......Page 4
Preface......Page 6
Introduction......Page 11
Exercises......Page 17
1.1 Rational Maps and the Projective Line......Page 19
1.2 Critical Points and the Riemann-HurwitzFormula......Page 22
1.3 Periodic Points and Multipliers......Page 28
1.4 The Julia Set and the Fatou Set......Page 32
1.5 Properties of Periodic Points......Page 37
1.6 Dynamical Systems Associated toEndomorphisms of Algebraic Groups......Page 38
Exercises......Page 45
2.1 The Nonarchimedean Chordal Metric......Page 52
2.2 Periodic Points and Their Properties......Page 56
2.3 Reduction of Points and Maps Modulo p......Page 57
2.4 The Resultant of a Rational Map......Page 62
2.5 Rational Maps with Good Reduction......Page 67
2.6 Periodic Points and Good Reduction......Page 71
2.7 Periodic Points and Dynamical Units......Page 78
Exercises......Page 83
3.1 Height Functions......Page 89
3.2 Height Functions and Geometry......Page 97
3.3 The Uniform Boundedness Conjecture......Page 103
3.4 Canonical Heights and Dynamical Systems......Page 105
3.5 Local Canonical Heights......Page 110
3.6 Diophantine Approximation......Page 112
3.7 Integral Points in Orbits......Page 116
3.8 Integrality Estimates for Points in Orbits......Page 120
3.9 Periodic Points and Galois Groups......Page 130
3.10 Equidistribution and Preperiodic Points......Page 134
3.11 Ramification and Units in Dynatomic Fields......Page 137
Exercises......Page 143
4 Families of Dynamical Systems......Page 155
4.1 Dynatomic Polynomials......Page 156
4.2 Quadratic Polynomials and Dynatomic Modular Curves......Page 163
4.3 The Space Rat, of Rational Functions......Page 176
4.4 The Moduli Space M d of Dynamical Systems......Page 182
4.5 Periodic Points, Multipliers, and Multiplier \rSpectra......Page 187
4.6 The Moduli Space M 2 of Dynamical Systems ofDegree 2......Page 196
4.7 Automorphisms and Twists......Page 203
4.8 General Theory of Twists......Page 207
4.9 Twists of Rational Maps......Page 211
4.10 Fields of Definition and the Field of Moduli......Page 214
4.11 Minimal Resultants and Minimal Models......Page 226
Exercises......Page 232
5 Dynamics over Local Fields: Bad Reduction......Page 246
5.1 Absolute Values and Completions......Page 247
5.2 A Primer on Nonarchimedean Analysis......Page 249
5.3 Newton Polygons and the Maximum Modulus Principle......Page 255
5.4 The Nonarchimedean Julia and Fatou Sets......Page 261
5.5 The Dynamics of (z2 - z) / p......Page 264
5.6 A Nonarchimedean Montel Theorem......Page 270
5.7 Periodic Points and the Julia Set......Page 275
5.8 Nonarchimedean Wandering Domains......Page 283
5.9 Green Functions and Local Heights......Page 294
5.10 Dynamics on Berkovich Space......Page 301
Exercises......Page 319
6.1 Power Maps and the Multiplicative Group......Page 332
6.2 Chebyshev Polynomials......Page 335
6.3 A Primer on Elliptic Curves......Page 343
6.4 General Properties of Lattes Maps......Page 357
6.5 Flexible Lattes Maps......Page 362
6.6 Rigid Lattes Maps......Page 371
6.7 Uniform Bounds for Lattes Maps......Page 375
6.8 Affine Morphisms, Algebraic Groups, and Commuting Families of Rational Maps......Page 382
Exercises......Page 387
7 Dynamics in Dimension Greater Than One......Page 393
7.1 Dynamics of Rational Maps on Projective Space......Page 394
7.2 Primer on Algebraic Geometry......Page 408
7.3 The Weil Height Machine......Page 413
7.4 Dynamics on Surfaces with Noncommuting Involutions......Page 416
Exercises......Page 433
Notes on Exercises......Page 446
List of Notation......Page 450
References......Page 456
Index......Page 477
