The arithmetic of dynamical systems:
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 poin...
Gespeichert in:
1. Verfasser: | |
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Format: | Buch |
Sprache: | English |
Veröffentlicht: |
New York, NY
Springer
2007
|
Schriftenreihe: | Graduate texts in mathematics
241 |
Schlagworte: | |
Online-Zugang: | Inhaltsverzeichnis |
Zusammenfassung: | 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. |
Beschreibung: | IX, 511 S. graph. Darst. |
ISBN: | 0387699031 9780387699035 |
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100 | 1 | |a Silverman, Joseph H. |d 1955- |e Verfasser |0 (DE-588)118906933 |4 aut | |
245 | 1 | 0 | |a The arithmetic of dynamical systems |c Joseph H. Silverman |
264 | 1 | |a New York, NY |b Springer |c 2007 | |
300 | |a IX, 511 S. |b graph. Darst. | ||
336 | |b txt |2 rdacontent | ||
337 | |b n |2 rdamedia | ||
338 | |b nc |2 rdacarrier | ||
490 | 1 | |a Graduate texts in mathematics |v 241 | |
520 | 3 | |a 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. | |
650 | 4 | |a Dynamique - Manuels d'enseignement supérieur | |
650 | 7 | |a Dynamische systemen |2 gtt | |
650 | 7 | |a Getaltheorie |2 gtt | |
650 | 4 | |a Théorie des nombres - Manuels d'enseignement supérieur | |
650 | 4 | |a Dynamics | |
650 | 4 | |a Number theory | |
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Datensatz im Suchindex
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---|---|
adam_text | Contents
Preface
v
Introduction
1
Exercises
............................... 7
1
An Introduction to Classical Dynamics
9
1.1
Rational Maps and the
Projective
Line
................ 9
1.2
Critical Points and the Riemann-Hurwitz Formula
.......... 12
1.3
Periodic Points and Multipliers
.................... 18
1.4
The Julia Set and the Fatou Set
.................... 22
1.5
Properties of Periodic Points
..................... 27
1.6
Dynamical Systems Associated to Algebraic Groups
........ 28
Exercises
............................... 35
2
Dynamics over Local Fields: Good Reduction
43
2.1
The Nonarchimedean Chordal Metric
................ 43
2.2
Periodic Points and Their Properties
................. 47
2.3
Reduction of Points and Maps Modulo
ρ
............... 48
2.4
The Resultant of a Rational Map
................... 53
2.5
Rational Maps with Good Reduction
................. 58
2.6
Periodic Points and Good Reduction
................. 62
2.7
Periodic Points and Dynamical Units
................. 69
Exercises
............................... 74
3
Dynamics over Global Fields
81
3.1
Height Functions
........................... 81
3.2
Height Functions and Geometry
................... 89
3.3
The Uniform Boundedness Conjecture
................ 95
3.4
Canonical Heights and Dynamical Systems
............. 97
3.5
Local Canonical Heights
....................... 102
3.6
Diophantine Approximation
..................... 104
3.7
Integral Points in Orbits
........................ 108
3.8
Integrality Estimates for Points in Orbits
............... 112
3.9
Periodic Points and Galois Groups
.................. 122
vii
viii Contents
3.10
Equidistribution and Preperiodic Points
............... 126
3.11
Ramification and Units in Dynatomic Fields
............. 129
Exercises
............................... 135
4
Families of Dynamical Systems
147
4.1
Dynatomic Polynomials
........................ 148
4.2
Quadratic Polynomials and Dynatomic Modular Curves
....... 155
4.3
The Space Ratd of Rational Functions
................ 168
4.4
The Moduli Space Ma of Dynamical Systems
............ 174
4.5
Periodic Points, Multipliers, and Multiplier Spectra
......... 179
4.6
The Moduli Space Mi of Dynamical Systems of Degree
2..... 188
4.7
Automorphisms and Twists
...................... 195
4.8
General Theory of Twists
....................... 199
4.9
Twists of Rational Maps
....................... 203
4.10
Fields of Definition and the Field of Moduli
............. 206
4.11
Minimal Resultants and Minimal Models
.............. 218
Exercises
............................... 224
5
Dynamics over Local Fields: Bad Reduction
239
5.1
Absolute Values and Completions
.................. 240
5.2
A Primer on Nonarchimedean Analysis
............... 242
5.3
Newton Polygons and the Maximum Modulus Principle
....... 248
5.4
The Nonarchimedean Julia and Fatou Sets
.............. 254
5.5
The Dynamics of (z2
-
z)/p
..................... 257
5.6
A Nonarchimedean
Montei
Theorem
................. 263
5.7
Periodic Points and the Julia Set
................... 268
5.8
Nonarchimedean Wandering Domains
................ 276
5.9
Green Functions and Local Heights
................. 287
5.10
Dynamics on Berkovich Space
.................... 294
Exercises
............................... 312
6
Dynamics Associated to Algebraic Groups
325
6.1
Power Maps and the Multiplicative Group
.............. 325
6.2
Chebyshev Polynomials
........................ 328
6.3
A Primer on Elliptic Curves
..................... 336
6.4
General Properties of
Lattes
Maps
.................. 350
6.5
Flexible
Lattes
Maps
......................... 355
6.6
Rigid
Lattes
Maps
........................... 364
6.7
Uniform Bounds for
Lattes
Maps
................... 368
6.8
Affine Morphisms
and Commuting Families
............. 375
Exercises
............................... 380
Contents ix
7 Dynamics in Dimension
Greater Than One
387
7.1 Dynamics
of
Rational
Maps on
Projective
Space........... 388
7.2 Primer
on Algebraic Geometry
.................... 402
7.3
The Weil Height Machine
....................... 407
7.4
Dynamics on Surfaces with Noncommuting Involutions
....... 410
Exercises
............................... 427
Notes on Exercises
441
List of Notation
445
References
451
Index
473
|
adam_txt |
Contents
Preface
v
Introduction
1
Exercises
. 7
1
An Introduction to Classical Dynamics
9
1.1
Rational Maps and the
Projective
Line
. 9
1.2
Critical Points and the Riemann-Hurwitz Formula
. 12
1.3
Periodic Points and Multipliers
. 18
1.4
The Julia Set and the Fatou Set
. 22
1.5
Properties of Periodic Points
. 27
1.6
Dynamical Systems Associated to Algebraic Groups
. 28
Exercises
. 35
2
Dynamics over Local Fields: Good Reduction
43
2.1
The Nonarchimedean Chordal Metric
. 43
2.2
Periodic Points and Their Properties
. 47
2.3
Reduction of Points and Maps Modulo
ρ
. 48
2.4
The Resultant of a Rational Map
. 53
2.5
Rational Maps with Good Reduction
. 58
2.6
Periodic Points and Good Reduction
. 62
2.7
Periodic Points and Dynamical Units
. 69
Exercises
. 74
3
Dynamics over Global Fields
81
3.1
Height Functions
. 81
3.2
Height Functions and Geometry
. 89
3.3
The Uniform Boundedness Conjecture
. 95
3.4
Canonical Heights and Dynamical Systems
. 97
3.5
Local Canonical Heights
. 102
3.6
Diophantine Approximation
. 104
3.7
Integral Points in Orbits
. 108
3.8
Integrality Estimates for Points in Orbits
. 112
3.9
Periodic Points and Galois Groups
. 122
vii
viii Contents
3.10
Equidistribution and Preperiodic Points
. 126
3.11
Ramification and Units in Dynatomic Fields
. 129
Exercises
. 135
4
Families of Dynamical Systems
147
4.1
Dynatomic Polynomials
. 148
4.2
Quadratic Polynomials and Dynatomic Modular Curves
. 155
4.3
The Space Ratd of Rational Functions
. 168
4.4
The Moduli Space Ma of Dynamical Systems
. 174
4.5
Periodic Points, Multipliers, and Multiplier Spectra
. 179
4.6
The Moduli Space Mi of Dynamical Systems of Degree
2. 188
4.7
Automorphisms and Twists
. 195
4.8
General Theory of Twists
. 199
4.9
Twists of Rational Maps
. 203
4.10
Fields of Definition and the Field of Moduli
. 206
4.11
Minimal Resultants and Minimal Models
. 218
Exercises
. 224
5
Dynamics over Local Fields: Bad Reduction
239
5.1
Absolute Values and Completions
. 240
5.2
A Primer on Nonarchimedean Analysis
. 242
5.3
Newton Polygons and the Maximum Modulus Principle
. 248
5.4
The Nonarchimedean Julia and Fatou Sets
. 254
5.5
The Dynamics of (z2
-
z)/p
. 257
5.6
A Nonarchimedean
Montei
Theorem
. 263
5.7
Periodic Points and the Julia Set
. 268
5.8
Nonarchimedean Wandering Domains
. 276
5.9
Green Functions and Local Heights
. 287
5.10
Dynamics on Berkovich Space
. 294
Exercises
. 312
6
Dynamics Associated to Algebraic Groups
325
6.1
Power Maps and the Multiplicative Group
. 325
6.2
Chebyshev Polynomials
. 328
6.3
A Primer on Elliptic Curves
. 336
6.4
General Properties of
Lattes
Maps
. 350
6.5
Flexible
Lattes
Maps
. 355
6.6
Rigid
Lattes
Maps
. 364
6.7
Uniform Bounds for
Lattes
Maps
. 368
6.8
Affine Morphisms
and Commuting Families
. 375
Exercises
. 380
Contents ix
7 Dynamics in Dimension
Greater Than One
387
7.1 Dynamics
of
Rational
Maps on
Projective
Space. 388
7.2 Primer
on Algebraic Geometry
. 402
7.3
The Weil Height Machine
. 407
7.4
Dynamics on Surfaces with Noncommuting Involutions
. 410
Exercises
. 427
Notes on Exercises
441
List of Notation
445
References
451
Index
473 |
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author | Silverman, Joseph H. 1955- |
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dewey-hundreds | 500 - Natural sciences and mathematics |
dewey-ones | 513 - Arithmetic |
dewey-raw | 513/.39 |
dewey-search | 513/.39 |
dewey-sort | 3513 239 |
dewey-tens | 510 - Mathematics |
discipline | Mathematik |
discipline_str_mv | Mathematik |
format | Book |
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genre_facet | Lehrbuch |
id | DE-604.BV022430805 |
illustrated | Illustrated |
index_date | 2024-07-02T17:29:16Z |
indexdate | 2024-07-09T20:57:26Z |
institution | BVB |
isbn | 0387699031 9780387699035 |
language | English |
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series | Graduate texts in mathematics |
series2 | Graduate texts in mathematics |
spelling | Silverman, Joseph H. 1955- Verfasser (DE-588)118906933 aut The arithmetic of dynamical systems Joseph H. Silverman New York, NY Springer 2007 IX, 511 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 241 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. Dynamique - Manuels d'enseignement supérieur Dynamische systemen gtt Getaltheorie gtt Théorie des nombres - Manuels d'enseignement supérieur Dynamics Number theory Dynamisches System (DE-588)4013396-5 gnd rswk-swf Arithmetik (DE-588)4002919-0 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Dynamisches System (DE-588)4013396-5 s Arithmetik (DE-588)4002919-0 s DE-604 Erscheint auch als Online-Ausgabe 978-0-387-69904-2 Graduate texts in mathematics 241 (DE-604)BV000000067 241 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015638997&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
spellingShingle | Silverman, Joseph H. 1955- The arithmetic of dynamical systems Graduate texts in mathematics Dynamique - Manuels d'enseignement supérieur Dynamische systemen gtt Getaltheorie gtt Théorie des nombres - Manuels d'enseignement supérieur Dynamics Number theory Dynamisches System (DE-588)4013396-5 gnd Arithmetik (DE-588)4002919-0 gnd |
subject_GND | (DE-588)4013396-5 (DE-588)4002919-0 (DE-588)4123623-3 |
title | The arithmetic of dynamical systems |
title_auth | The arithmetic of dynamical systems |
title_exact_search | The arithmetic of dynamical systems |
title_exact_search_txtP | The arithmetic of dynamical systems |
title_full | The arithmetic of dynamical systems Joseph H. Silverman |
title_fullStr | The arithmetic of dynamical systems Joseph H. Silverman |
title_full_unstemmed | The arithmetic of dynamical systems Joseph H. Silverman |
title_short | The arithmetic of dynamical systems |
title_sort | the arithmetic of dynamical systems |
topic | Dynamique - Manuels d'enseignement supérieur Dynamische systemen gtt Getaltheorie gtt Théorie des nombres - Manuels d'enseignement supérieur Dynamics Number theory Dynamisches System (DE-588)4013396-5 gnd Arithmetik (DE-588)4002919-0 gnd |
topic_facet | Dynamique - Manuels d'enseignement supérieur Dynamische systemen Getaltheorie Théorie des nombres - Manuels d'enseignement supérieur Dynamics Number theory Dynamisches System Arithmetik Lehrbuch |
url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015638997&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
volume_link | (DE-604)BV000000067 |
work_keys_str_mv | AT silvermanjosephh thearithmeticofdynamicalsystems |