The Arithmetic of Polynomial Dynamical Pairs: (AMS-214)
252
The Arithmetic of Polynomial Dynamical Pairs: (AMS-214)
252Hardcover
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Overview
In The Arithmetic of Polynomial Dynamical Pairs, Charles Favre and Thomas Gauthier present new mathematical research in the field of arithmetic dynamics. Specifically, the authors study one-dimensional algebraic families of pairs given by a polynomial with a marked point. Combining tools from arithmetic geometry and holomorphic dynamics, they prove an “unlikely intersection” statement for such pairs, thereby demonstrating strong rigidity features for them. They further describe one-dimensional families in the moduli space of polynomials containing infinitely many postcritically finite parameters, proving the dynamical André-Oort conjecture for curves in this context, originally stated by Baker and DeMarco.
This is a reader-friendly invitation to a new and exciting research area that brings together sophisticated tools from many branches of mathematics.
Product Details
| ISBN-13: | 9780691235462 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 06/14/2022 |
| Series: | Annals of Mathematics Studies , #214 |
| Pages: | 252 |
| Product dimensions: | 6.12(w) x 9.25(h) x (d) |
About the Author
Table of Contents
List of figures xi
Preface xiii
List of abbreviations xv
Introduction 1
1 Geometric background 15
1.1 Analytic geometry 15
1.1.1 Analytic varieties 15
1.1.2 The non-Archimedean affine and projective lines 16
1.1.3 Non-Archimedean Berkovich curves 17
1.2 Potential theory 20
1.2.1 Pluripotential theory on complex manifolds 20
1.2.2 Potential theory on Berkovich analytic curves 22
1.2.3 Subharmonic functions on singular curves 26
1.3 Line bundles on curves 27
1.3.1 Metrizations of line bundles 27
1.3.2 Positive line bundles 28
1.4 Adelic metrics, Arakelov heights, and equidistribution 29
1.4.1 Number fields 29
1.4.2 Adelic metrics 30
1.4.3 Heights 30
1.4.4 Equidistribution 31
1.5 Adelic series and Xie's algebraization theorem 32
2 Polynomial dynamics 37
2.1 The parameter space of polynomials 37
2.2 Fatou-Julia theory 39
2.3 Green functions and equilibrium measure 43
2.3.1 Basic definitions 43
2.3.2 Estimates on the Green function 44
2.4 Examples 47
2.4.1 Integrable polynomials 47
2.4.2 Potential good reduction 48
2.4.3 PCF maps 49
2.5 Böttcher coordinates and Green functions 49
2.5.1 Expansion of the Böttcher coordinate 49
2.5.2 Böttcher coordinate and Green function 53
2.6 Polynomial dynamics over a global field 56
2.7 Bifurcations in holomorphic dynamics 58
2.8 Components of preperiodic points 60
3 Dynamical symmetries 64
3.1 The group of dynamical symmetries of a polynomial 64
3.2 Symmetry groups in family 68
3.3 Algebraic characterization of dynamical symmetries 69
3.4 Primitive families of polynomials 72
3.5 Ritt's theory of composite polynomials 76
3.5.1 Decomposability 77
3.5.2 Intertwined polynomials 78
3.5.3 Uniform bounds and invariant, subvarieties 81
3.5.4 Intertwining classes 81
3.5.5 Intertwining classes of a generic polynomial 83
3.6 Stratification of the parameter space in low degree 85
3.7 Open problems 87
4 Polynomial dynamical pairs 89
4.1 Holomorphic dynamical pairs and proof of Theorem 4.10 89
4.1.1 Basics on holomorphic dynamical pairs 90
4.1.2 Density of transversely prerepelling parameters 91
4.1.3 Rigidity of the bifurcation locus 94
4.1.4 A renormalization procedure 96
4.1.5 Bifurcation locus of a dynamical pair and J-stability 98
4.1.6 Proof of Theorem 4.10 98
4.2 Algebraic dynamical pairs 102
4.2.1 Algebraic dynamical pairs 102
4.2.2 The divisor of a dynamical pair 102
4.2.3 Meromorphic dynamical pairs parametrized by the punctured disk 103
4.2.4 Metrizations and dynamical pairs 109
4.2.5 Characterization of passivity 111
4.3 Family of polynomials and green functions 113
4.4 Arithmetic polynomial dynamical pairs 114
5 Entanglement of dynamical pairs 119
5.1 Dynamical entanglement 119
5.1.1 Definition 119
5.1.2 Characterization of entanglement 120
5.1.3 Overview of the proof of Theorem B 121
5.2 Dynamical pairs with identical measures 122
5.2.1 Equality at an Archimedean place 122
5.2.2 The implication (1) → (2) of Theorem B 126
5.3 Multiplicative dependence of the degrees 127
5.4 Proof of the implication (2) → (3) of Theorem B 131
5.4.1 More precise forms of Theorem B 139
5.5 Proof of Theorem C 140
5.6 Further results and open problems 145
5.6.1 Effective versions of the theorem 145
5.6.2 The integrable case 146
5.6.3 Algorithm 146
5.6.4 Application to Manin-Mumford's problem 147
6 Entanglement of marked points 150
6.1 Proof of Theorem D 150
6.2 Proof of Theorem E 152
7 The unicritical family 157
7.1 General facts 157
7.2 Unlikely intersection in the unicritical family 160
7.3 Archimedean rigidity 161
7.4 Connectedness of the bifurcation locus 162
7.5 Some experiments 163
8 Special Curves 169
8.1 Special curves in the moduli space of polynomials 170
8.2 Marked dynamical graphs 171
8.2.1 Definition 171
8.2.2 Critically marked dynamical graphs 173
8.2.3 The critical graph of a polynomial 174
8.2.4 The critical graph of an irreducible subvariety in the moduli space of polynomials 175
8.3 Dynamical graphs attached to special curves 176
5.3 Realization theorem 177
8.4.1 Asymmetric special graphs 179
8.4.2 Truncated marked dynamical graphs 180
8.4.3 Polynomials with a fixed portrait 181
8.4.4 Construction of a suitable sequence of Riemann surfaces 185
8.4.5 End of the proof of Theorem 8.15 197
8.5 Special curves and dynamical graphs 198
8.5.1 Wringing deformations and marked dynamical graphs 199
8.5.2 Proof of theorem 8.30 201
8.6 Realizability of PCF maps 202
8.6.1 Proof of Proposition 8.34 203
8.6.2 Combinatorics of strictly PCF polynomials 205
8.6.3 Proof of Proposition 8.35 208
8.7 Special curves in low degrees 210
8.8 Open questions on the geometry of special curves 212
Notes 215
Bibliography 217
Index 231