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Summary
Summary
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.
Table of Contents
Preface | p. V |
Introduction | p. 1 |
Exercises | p. 7 |
1 An Introduction to Classical Dynamics | p. 9 |
1.1 Rational Maps and the Projective Line | p. 9 |
1.2 Critical Points and the Riemann-Hurwitz Formula | p. 12 |
1.3 Periodic Points and Multipliers | p. 18 |
1.4 The Julia Set and the Fatou Set | p. 22 |
1.5 Properties of Periodic Points | p. 27 |
1.6 Dynamical Systems Associated to Algebraic Groups | p. 28 |
Exercises | p. 35 |
2 Dynamics over Local Fields: Good Reduction | p. 43 |
2.1 The Nonarchimedean Chordal Metric | p. 43 |
2.2 Periodic Points and Their Properties | p. 47 |
2.3 Reduction of Points and Maps Modulo p | p. 48 |
2.4 The Resultant of a Rational Map | p. 53 |
2.5 Rational Maps with Good Reduction | p. 58 |
2.6 Periodic Points and Good Reduction | p. 62 |
2.7 Periodic Points and Dynamical Units | p. 69 |
Exercises | p. 74 |
3 Dynamics over Global Fields | p. 81 |
3.1 Height Functions | p. 81 |
3.2 Height Functions and Geometry | p. 89 |
3.3 The Uniform Boundedness Conjecture | p. 95 |
3.4 Canonical Heights and Dynamical Systems | p. 97 |
3.5 Local Canonical Heights | p. 102 |
3.6 Diophantine Approximation | p. 104 |
3.7 Integral Points in Orbits | p. 108 |
3.8 Integrality Estimates for Points in Orbits | p. 112 |
3.9 Periodic Points and Galois Groups | p. 122 |
3.10 Equidistribution and Preperiodic Points | p. 126 |
3.11 Ramification and Units in Dynatomic Fields | p. 129 |
Exercises | p. 135 |
4 Families of Dynamical Systems | p. 147 |
4.1 Dynatomic Polynomials | p. 148 |
4.2 Quadratic Polynomials and Dynatomic Modular Curves | p. 155 |
4.3 The Space Rat[superscript d] of Rational Functions | p. 168 |
4.4 The Moduli Space M[subscript d] of Dynamical Systems | p. 174 |
4.5 Periodic Points, Multipliers, and Multiplier Spectra | p. 179 |
4.6 The Moduli Space M[subscript 2] of Dynamical Systems of Degree 2 | p. 188 |
4.7 Automorphisms and Twists | p. 195 |
4.8 General Theory of Twists | p. 199 |
4.9 Twists of Rational Maps | p. 203 |
4.10 Fields of Definition and the Field of Moduli | p. 206 |
4.11 Minimal Resultants and Minimal Models | p. 218 |
Exercises | p. 224 |
5 Dynamics over Local Fields: Bad Reduction | p. 239 |
5.1 Absolute Values and Completions | p. 240 |
5.2 A Primer on Nonarchimedean Analysis | p. 242 |
5.3 Newton Polygons and the Maximum Modulus Principle | p. 248 |
5.4 The Nonarchimedean Julia and Fatou Sets | p. 254 |
5.5 The Dynamics of (z[superscript 2] - z)/p | p. 257 |
5.6 A Nonarchimedean Montel Theorem | p. 263 |
5.7 Periodic Points and the Julia Set | p. 268 |
5.8 Nonarchimedean Wandering Domains | p. 276 |
5.9 Green Functions and Local Heights | p. 287 |
5.10 Dynamics on Berkovich Space | p. 294 |
Exercises | p. 312 |
6 Dynamics Associated to Algebraic Groups | p. 325 |
6.1 Power Maps and the Multiplicative Group | p. 325 |
6.2 Chebyshev Polynomials | p. 328 |
6.3 A Primer on Elliptic Curves | p. 336 |
6.4 General Properties of Lattes Maps | p. 350 |
6.5 Flexible Lattes Maps | p. 355 |
6.6 Rigid Lattes Maps | p. 364 |
6.7 Uniform Bounds for Lattes Maps | p. 368 |
6.8 Affine Morphisms and Commuting Families | p. 375 |
Exercises | p. 380 |
7 Dynamics in Dimension Greater Than One | p. 387 |
7.1 Dynamics of Rational Maps on Projective Space | p. 388 |
7.2 Primer or Algebraic Geometry | p. 402 |
7.3 The Weil Height Machine | p. 407 |
7.4 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 |