
The Arithmetic of Polynomial Dynamical Pairs
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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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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
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