Dynamics in One Non-Archimedean Variable

Description

The theory of complex dynamics in one variable, initiated by Fatou and Julia in the early twentieth century, concerns the iteration of a rational function acting on the Riemann sphere. Building on foundational investigations of $p$-adic dynamics in the late twentieth century, dynamics in one non-archimedean variable is the analogous theory over non-archimedean fields rather than over the complex numbers. It is also an essential component of the number-theoretic study of arithmetic dynamics. This textbook presents the fundamentals of non-archimedean dynamics, including a unified exposition of Rivera-Letelier's classification theorem, as well as results on wandering domains, repelling periodic points, and equilibrium measures. The Berkovich projective line, which is the appropriate setting for the associated Fatou and Julia sets, is developed from the ground up, as are relevant results in non-archimedean analysis. The presentation is accessible to graduate students with only first-year courses in algebra and analysis under their belts, although some previous exposure to non-archimedean fields, such as the $p$-adic numbers, is recommended. The book should also be a useful reference for more advanced students and researchers in arithmetic and non-archimedean dynamics. Cover Title page List of Notation Preface Introduction A brief history What’s in this book Some things not in this book Some possible paths through this book Part 1 . Background Chapter 1. Basic Dynamics on \PP¹(𝐾) 1.1. Elementary discrete dynamics 1.2. Morphisms and coordinate changes 1.3. Degrees, multiplicities, and multipliers 1.4. Critical points and exceptional sets 1.5. Dynamics in degree less than 2 1.6. An overview of complex dynamics Exercises for Chapter 1 Chapter 2. Some Background on Non-Archimedean Fields 2.1. Absolute values 2.2. Disks Exercises for Chapter 2 Chapter 3. Power Series and Laurent Series 3.1. Convergence of power series 3.2. Power series rings 3.3. Newton polygons 3.4. Images of disks under power series 3.5. \PCv-disks and affinoids 3.6. Laurent series on open annuli 3.7. Images of annuli Exercises for Chapter 3 Part 2 . Elementary Non-Archimedean Dyanmics Chapter 4. Fundamentals of Non-Archimedean Dynamics 4.1. Classifying periodic points 4.2. Local conjugacies at fixed points (optional) 4.3. Good reduction 4.4. Lattès maps 4.5. Dynamics on disks Exercises for Chapter 4 Chapter 5. Fatou and Julia Sets 5.1. The spherical metric 5.2. Fatou and Julia sets 5.3. Further properties of Fatou and Julia sets 5.4. Examples of non-archimedean Fatou and Julia sets Exercises for Chapter 5 Part 3 . The Berkovich Line Chapter 6. The Berkovich Projective Line 6.1. Seminorms as Berkovich points 6.2. Disks in the Berkovich affine line 6.3. Berkovich’s classification 6.4. The Berkovich projective line 6.5. Disks and affinoids in \PBerk 6.6. Paths and path-connectedness 6.7. Directions at Berkovich points 6.8. The hyperbolic metric Exercises for Chapter 6 Chapter 7. Rational Functions and Berkovich Space 7.1. The action of rational functions 7.2. Images of points of Types II and III 7.3. Local degrees in directions 7.4. Local degrees at Berkovich points 7.5. Computing local degrees 7.6. The injectivity and ramification loci Exercises for Chapter 7 Part 4 . Dynamics on the Berkovich Line Chapter 8. Introduction to Dynamics on Berkovich Space 8.1. Berkovich Fatou and Julia sets 8.2. Classifying Berkovich periodic points 8.3. Good reduction in Berkovich space 8.4. More basic properties of Berkovich Julia sets 8.5. Examples Exercises for Chapter 8 Chapter 9. Classifying Berkovich Fatou Components 9.1. Berkovich Fatou components 9.2. Attracting components 9.3. Indifferent components 9.4. Rivera-Letelier’s classification Exercises for Chapter 9 Chapter 10. Further Results on Periodic Components 10.1. Periodic points in the indifference domain 10.2. Indifferent components in mixed characteristic 10.3. Counting cycles of Fatou components 10.4. Infinitely many periodic components Exercises for Chapter 10 Chapter 11. Wandering Domains 11.1. Wandering domains are eventually disks 11.2. An expansion theorem 11.3. No wandering domains for 𝑝-adic fields: A theorem 11.4. Wandering domains for tame maps 11.5. Wandering domains for tame polynomials Exercises for Chapter 11 Chapter 12. Repelling Points in Berkovich Space 12.1. Preliminary repelling fixed point lemmas 12.2. Repelling fixed points exist 12.3. Repelling density 12.4. Type I repelling density: Hsia’s theorem 12.5. Type I repelling density: Bézivin’s theorem Exercises for Chapter 12 Chapter 13. The Equilibrium Measure 13.1. Some measure theory 13.2. Weak convergence 13.3. Potential functions 13.4. The Laplacian operator 13.5. Construction of the equilibrium measure 13.6. Local heights 13.7. Equidistribution of points of small canonical height Exercises for Chapter 13 Part 5 . Proofs from Non-Archimedean Analysis Chapter 14. Proofs of Results from Non-Archimedean Analysis 14.1. Basic power series proofs 14.2. The Weierstrass preparation theorem 14.3. Proofs related to Newton polygons 14.4. Proofs related to mapping properties of disks 14.5. Weierstrass preparation theorem for open annuli 14.6. Proofs about Laurent series on annuli 14.7. An overview of rigid analysis (optional) Exercises for Chapter 14 Chapter 15. Proofs of Berkovich Space Results 15.1. Basic results on seminorms 15.2. Proofs on Berkovich points 15.3. Basic proofs on the Berkovich projective line 15.4. Proving compactness 15.5. Proving path-connectedness 15.6. Other Berkovich space proofs Exercises for Chapter 15 Chapter 16. Proofs of Results on Berkovich Maps 16.1. Basic results on Berkovich maps 16.2. Proofs on local degrees 16.3. Proving Rivera-Letelier’s reduction theorem Exercises for Chapter 16 Appendices Appendix A. Fatou Components without Berkovich Space A.1. Analytic components and 𝐷-components A.2. Examples of non-archimedean Fatou components A.3. Open analytic components Exercises for Appendix A Appendix B. Other Constructions of Berkovich Spaces B.1. Berkovich disks via power series B.2. Seminorms in homogeneous coordinates B.3. Berkovich morphisms via homogeneous coordinates B.4. Rivera-Letelier’s construction of \PP¹_{𝑎𝑛} Exercises for Appendix B Bibliography Index Back Cover