**Graduate Studies in Mathematics**

Volume: 198;
2019;
463 pp;
Hardcover

MSC: Primary 37;

**Print ISBN: 978-1-4704-4688-8
Product Code: GSM/198**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-5106-6
Product Code: GSM/198.E**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

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#### Supplemental Materials

# Dynamics in One Non-Archimedean Variable

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*Robert L. Benedetto*

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.

#### Readership

Graduate students and researchers interested in arithmetic and non-archimedean dynamics.

#### Reviews & Endorsements

This book would be good for a topics course in Berkovich space for a graduate student familiar with real and complex analysis. [Bennedeto] makes Berkovich space accessible to the new researcher.

-- Bianca Thompson, MAA Reviews

#### Table of Contents

# Table of Contents

## Dynamics in One Non-Archimedean Variable

- Cover Cover11
- Title page iii4
- List of Notation xiii14
- Preface xvii18
- Introduction 120
- Part 1 . Background 928
- Part 2 . Elementary Non-Archimedean Dyanmics 6988
- Part 3 . The Berkovich Line 119138
- Chapter 6. The Berkovich Projective Line 121140
- 6.1. Seminorms as Berkovich points 122141
- 6.2. Disks in the Berkovich affine line 125144
- 6.3. Berkovich’s classification 128147
- 6.4. The Berkovich projective line 129148
- 6.5. Disks and affinoids in \PBerk 133152
- 6.6. Paths and path-connectedness 138157
- 6.7. Directions at Berkovich points 143162
- 6.8. The hyperbolic metric 144163
- Exercises for Chapter 6 146165

- Chapter 7. Rational Functions and Berkovich Space 153172

- Part 4 . Dynamics on the Berkovich Line 181200
- Chapter 8. Introduction to Dynamics on Berkovich Space 183202
- Chapter 9. Classifying Berkovich Fatou Components 203222
- Chapter 10. Further Results on Periodic Components 231250
- Chapter 11. Wandering Domains 253272
- Chapter 12. Repelling Points in Berkovich Space 291310
- Chapter 13. The Equilibrium Measure 323342
- 13.1. Some measure theory 323342
- 13.2. Weak convergence 327346
- 13.3. Potential functions 328347
- 13.4. The Laplacian operator 331350
- 13.5. Construction of the equilibrium measure 336355
- 13.6. Local heights 342361
- 13.7. Equidistribution of points of small canonical height 347366
- Exercises for Chapter 13 352371

- Part 5 . Proofs from Non-Archimedean Analysis 359378
- Chapter 14. Proofs of Results from Non-Archimedean Analysis 361380
- 14.1. Basic power series proofs 361380
- 14.2. The Weierstrass preparation theorem 364383
- 14.3. Proofs related to Newton polygons 368387
- 14.4. Proofs related to mapping properties of disks 370389
- 14.5. Weierstrass preparation theorem for open annuli 374393
- 14.6. Proofs about Laurent series on annuli 379398
- 14.7. An overview of rigid analysis (optional) 380399
- Exercises for Chapter 14 382401

- Chapter 15. Proofs of Berkovich Space Results 385404
- Chapter 16. Proofs of Results on Berkovich Maps 411430

- Appendices 427446
- Back Cover Back Cover1486