The Arithmetic of Polynomial Dynamical Pairs
(AMS-214)
1st Edition
Published on 1. June 2022
252 pages
978-0-691-23548-6 (ISBN)
Description
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.
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.
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Charles Favre | Thomas Gauthier
The Arithmetic of Polynomial Dynamical Pairs
Book
06/2022
Princeton University Press
€84.00
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Charles Favre | Thomas Gauthier
The Arithmetic of Polynomial Dynamical Pairs
Book
06/2022
Princeton University Press
€197.50
Shipment within 10-20 days
Persons
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.
Content
- Cover
- Contents
- List of figures
- Preface
- List of abbreviations
- Introduction
- 1. Geometric background
- 1.1 Analytic geometry
- 1.1.1 Analytic varieties
- 1.1.2 The non-Archimedean affine and projective lines
- 1.1.3 Non-Archimedean Berkovich curves
- 1.2 Potential theory
- 1.2.1 Pluripotential theory on complex manifolds
- 1.2.2 Potential theory on Berkovich analytic curves
- 1.2.3 Subharmonic functions on singular curves
- 1.3 Line bundles on curves
- 1.3.1 Metrizations of line bundles
- 1.3.2 Positive line bundles
- 1.4 Adelic metrics, Arakelov heights, and equidistribution
- 1.4.1 Number fields
- 1.4.2 Adelic metrics
- 1.4.3 Heights
- 1.4.4 Equidistribution
- 1.5 Adelic series and Xie's algebraization theorem
- 2. Polynomial dynamics
- 2.1 The parameter space of polynomials
- 2.2 Fatou-Julia theory
- 2.3 Green functions and equilibrium measure
- 2.3.1 Basic definitions
- 2.3.2 Estimates on the Green function
- 2.4 Examples
- 2.4.1 Integrable polynomials
- 2.4.2 Potential good reduction
- 2.4.3 PCF maps
- 2.5 Böttcher coordinates and Green functions
- 2.5.1 Expansion of the Böttcher coordinate
- 2.5.2 Böttcher coordinate and Green function
- 2.6 Polynomial dynamics over a global field
- 2.7 Bifurcations in holomorphic dynamics
- 2.8 Components of preperiodic points
- 3. Dynamical symmetries
- 3.1 The group of dynamical symmetries of a polynomial
- 3.2 Symmetry groups in family
- 3.3 Algebraic characterization of dynamical symmetries
- 3.4 Primitive families of polynomials
- 3.5 Ritt's theory of composite polynomials
- 3.5.1 Decomposability
- 3.5.2 Intertwined polynomials
- 3.5.3 Uniform bounds and invariant subvarieties
- 3.5.4 Intertwining classes
- 3.5.5 Intertwining classes of a generic polynomial
- 3.6 Stratification of the parameter space in low degree
- 3.7 Open problems
- 4. Polynomial dynamical pairs
- 4.1 Holomorphic dynamical pairs and proof of Theorem 4.10
- 4.1.1 Basics on holomorphic dynamical pairs
- 4.1.2 Density of transversely prerepelling parameters
- 4.1.3 Rigidity of the bifurcation locus
- 4.1.4 A renormalization procedure
- 4.1.5 Bifurcation locus of a dynamical pair and J-stability
- 4.1.6 Proof of Theorem 4.10
- 4.2 Algebraic dynamical pairs
- 4.2.1 Algebraic dynamical pairs
- 4.2.2 The divisor of a dynamical pair
- 4.2.3 Meromorphic dynamical pairs parametrized by the punctured disk
- 4.2.4 Metrizations and dynamical pairs
- 4.2.5 Characterization of passivity
- 4.3 Family of polynomials and Green functions
- 4.4 Arithmetic polynomial dynamical pairs
- 5. Entanglement of dynamical pairs
- 5.1 Dynamical entanglement
- 5.1.1 Definition
- 5.1.2 Characterization of entanglement
- 5.1.3 Overview of the proof of Theorem B
- 5.2 Dynamical pairs with identical measures
- 5.2.1 Equality at an Archimedean place
- 5.2.2 The implication (1) ? (2) of Theorem B
- 5.3 Multiplicative dependence of the degrees
- 5.4 Proof of the implication (2) ? (3) of Theorem B
- 5.4.1 More precise forms of Theorem B
- 5.5 Proof of Theorem C
- 5.6 Further results and open problems
- 5.6.1 Effective versions of the theorem
- 5.6.2 The integrable case
- 5.6.3 Algorithm
- 5.6.4 Application to Manin-Mumford's problem
- 6. Entanglement of marked points
- 6.1 Proof of Theorem D
- 6.2 Proof of Theorem E
- 7. The unicritical family
- 7.1 General facts
- 7.2 Unlikely intersection in the unicritical family
- 7.3 Archimedean rigidity
- 7.4 Connectedness of the bifurcation locus
- 7.5 Some experiments
- 8. Special curves
- 8.1 Special curves in the moduli space of polynomials
- 8.2 Marked dynamical graphs
- 8.2.1 Definition
- 8.2.2 Critically marked dynamical graphs
- 8.2.3 The critical graph of a polynomial
- 8.2.4 The critical graph of an irreducible subvariety in the moduli space of polynomials
- 8.3 Dynamical graphs attached to special curves
- 8.4 Realization theorem
- 8.4.1 Asymmetric special graphs
- 8.4.2 Truncated marked dynamical graphs
- 8.4.3 Polynomials with a fixed portrait
- 8.4.4 Construction of a suitable sequence of Riemann surfaces
- 8.4.5 End of the proof of Theorem 8.15
- 8.5 Special curves and dynamical graphs
- 8.5.1 Wringing deformations and marked dynamical graphs
- 8.5.2 Proof of Theorem 8.30
- 8.6 Realizability of PCF maps
- 8.6.1 Proof of Proposition 8.34
- 8.6.2 Combinatorics of strictly PCF polynomials
- 8.6.3 Proof of Proposition 8.35
- 8.7 Special curves in low degrees
- 8.8 Open questions on the geometry of special curves
- Notes
- Bibliography
- Index
