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El. knyga: Arithmetic of Polynomial Dynamical Pairs

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  • Formatas: 252 pages
  • Serija: Annals of Mathematics Studies
  • Išleidimo metai: 14-Jun-2022
  • Leidėjas: Princeton University Press
  • Kalba: eng
  • ISBN-13: 9780691235486
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Arithmetic of Polynomial Dynamical PairsDidesnis paveikslėlis
  • Formatas: 252 pages
  • Serija: Annals of Mathematics Studies
  • Išleidimo metai: 14-Jun-2022
  • Leidėjas: Princeton University Press
  • Kalba: eng
  • ISBN-13: 9780691235486
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New mathematical research in arithmetic dynamics

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.
List of figures
xi
Preface xiii
List of abbreviations
xv
Introduction 1(14)
1 Geometric background
15(22)
1.1 Analytic geometry
15(5)
1.1.1 Analytic varieties
15(1)
1.1.2 The non-Archimedean affine and projective lines
16(1)
1.1.3 Non-Archimedean Berkovich curves
17(3)
1.2 Potential theory
20(7)
1.2.1 Pluripotential theory on complex manifolds
20(2)
1.2.2 Potential theory on Berkovich analytic curves
22(4)
1.2.3 Subharmonic functions on singular curves
26(1)
1.3 Line bundles on curves
27(2)
1.3.1 Metrizations of line bundles
27(1)
1.3.2 Positive line bundles
28(1)
1.4 Adelic metrics, Arakelov heights, and equidistribution
29(3)
1.4.1 Number fields
29(1)
1.4.2 Adelic metrics
30(1)
1.4.3 Heights
30(1)
1.4.4 Equidistribution
31(1)
1.5 Adelic series and Xie's algebraization theorem
32(5)
2 Polynomial dynamics
37(27)
2.1 The parameter space of polynomials
37(2)
2.2 Fatou-Julia theory
39(4)
2.3 Green functions and equilibrium measure
43(4)
2.3.1 Basic definitions
43(1)
2.3.2 Estimates on the Green function
44(3)
2.4 Examples
47(2)
2.4.1 Integrable polynomials
47(1)
2.4.2 Potential good reduction
48(1)
2.4.3 PCF maps
49(1)
2.5 Bottcher coordinates and Green functions
49(7)
2.5.1 Expansion of the Bottcher coordinate
49(4)
2.5.2 Bottcher coordinate and Green function
53(3)
2.6 Polynomial dynamics over a global field
56(2)
2.7 Bifurcations in holornorphic dynamics
58(2)
2.8 Components of preperiodic points
60(4)
3 Dynamical symmetries
64(25)
3.1 The group of dynamical symmetries of a polynomial
64(4)
3.2 Symmetry groups in family
68(1)
3.3 Algebraic characterization of dynamical symmetries
69(3)
3.4 Primitive families of polynomials
72(4)
3.5 Ritt's theory of composite polynomials
76(9)
3.5.1 Decomposability
77(1)
3.5.2 Intertwined polynomials
78(3)
3.5.3 Uniform bounds and invariant subvarieties
81(1)
3.5.4 Intertwining classes
81(2)
3.5.5 Intertwining classes of a generic polynomial
83(2)
3.6 Stratification of the parameter space in low degree
85(2)
3.7 Open problems
87(2)
4 Polynomial dynamical pairs
89(30)
4.1 Holornorphic dynamical pairs and proof of Theorem 4.10
89(13)
4.1.1 Basics on holornorphic dynamical pairs
90(1)
4.1.2 Density of transversely prerepelling parameters
91(3)
4.1.3 Rigidity of the bifurcation locus
94(2)
4.1.4 A renormalization procedure
96(2)
4.1.5 Bifurcation locus of a dynamical pair and J-stability
98(1)
4.1.6 Proof of Theorem 4.10
98(4)
4.2 Algebraic dynamical pairs
102(11)
4.2.1 Algebraic dynamical pairs
102(1)
4.2.2 The divisor of a dynamical pair
102(1)
4.2.3 Meromorphic dynamical pairs parametrized by the punctured disk
103(6)
4.2.4 Metrizations and dynamical pairs
109(2)
4.2.5 Characterization of passivity
111(2)
4.3 Family of polynomials and Green functions
113(1)
4.4 Arithmetic polynomial dynamical pairs
114(5)
5 Entanglement of dynamical pairs
119(31)
5.1 Dynamical entanglement
119(3)
5.1.1 Definition
119(1)
5.1.2 Characterization of entanglement
120(1)
5.1.3 Overview of the proof of Theorem B
121(1)
5.2 Dynamical pairs with identical measures
122(5)
5.2.1 Equality at an Archimedean place
122(4)
5.2.2 The implication (1) ⇒ (2) of Theorem B
126(1)
5.3 Multiplicative dependence of the degrees
127(4)
5.4 Proof of the implication (2) ⇒ (3) of Theorem B
131(9)
5.4.1 More precise forms of Theorem B
139(1)
5.5 Proof of Theorem C
140(5)
5.6 Further results and open problems
145(5)
5.6.1 Effective versions of the theorem
145(1)
5.6.2 The integrable case
146(1)
5.6.3 Algorithm
146(1)
5.6.4 Application to Manin-Mumford's problem
147(3)
6 Entanglement of marked points
150(7)
6.1 Proof of Theorem D
150(2)
6.2 Proof of Theorem E
152(5)
7 The unicritical family
157(12)
7.1 General facts
157(3)
7.2 Unlikely intersection in the unicritical family
160(1)
7.3 Archimedean rigidity
161(1)
7.4 Connectedness of the bifurcation locus
162(1)
7.5 Some experiments
163(6)
8 Special curves
169(46)
8.1 Special curves in the moduli space of polynomials
170(1)
8.2 Marked dynamical graphs
171(5)
8.2.1 Definition
171(2)
8.2.2 Critically marked dynamical graphs
173(1)
8.2.3 The critical graph of a polynomial
174(1)
8.2.4 The critical graph of an irreducible subvariety in the moduli space of polynomials
175(1)
8.3 Dynamical graphs attached to special curves
176(1)
8.4 Realization theorem
177(21)
8.4.1 Asymmetric special graphs
179(1)
8.4.2 Truncated marked dynamical graphs
180(1)
8.4.3 Polynomials with a fixed portrait
181(4)
8.4.4 Construction of a suitable sequence of Riemann surfaces
185(12)
8.4.5 End of the proof of Theorem 8.15
197(1)
8.5 Special curves and dynamical graphs
198(4)
8.5.1 Wringing deformations and marked dynamical graphs
199(2)
8.5.2 Proof of Theorem 8.30
201(1)
8.6 Realizability of PCF maps
202(8)
8.6.1 Proof of Proposition 8.34
203(2)
8.6.2 Combinatorics of strictly PCF polynomials
205(3)
8.6.3 Proof of Proposition 8.35
208(2)
8.7 Special curves in low degrees
210(2)
8.8 Open questions on the geometry of special curves
212(3)
Notes 215(2)
Bibliography 217(14)
Index 231
Charles Favre is a CNRS senior researcher based at the École Polytechnique in Paris. He is the coauthor of The Valuative Tree and the coeditor of Berkovich Spaces and Applications. Thomas Gauthier is professor of mathematics at the Université Paris-Saclay.
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