**Mathematical Surveys and Monographs**

Volume: 159;
2010;
428 pp;
Hardcover

MSC: Primary 14;
Secondary 37; 31

Print ISBN: 978-0-8218-4924-8

Product Code: SURV/159

List Price: $110.00

Individual Member Price: $88.00

**Electronic ISBN: 978-1-4704-1386-6
Product Code: SURV/159.E**

List Price: $110.00

Individual Member Price: $88.00

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#### Supplemental Materials

# Potential Theory and Dynamics on the Berkovich Projective Line

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*Matthew Baker; Robert Rumely*

The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean field. In addition to providing a concrete and “elementary” introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, the book contains applications to arithmetic geometry and arithmetic dynamics. A number of results in the book are new, and most have not previously appeared in book form. Three appendices—on analysis, \(\mathbb{R}\)-trees, and Berkovich's general theory of analytic spaces—are included to make the book as self-contained as possible.

The authors first give a detailed description of the topological structure of the Berkovich projective line and then introduce the Hsia kernel, the fundamental kernel for potential theory. Using the theory of metrized graphs, they define a Laplacian operator on the Berkovich line and construct theories of capacities, harmonic and subharmonic functions, and Green's functions, all of which are strikingly similar to their classical complex counterparts. After developing a theory of multiplicities for rational functions, they give applications to non-Archimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical Fatou-Julia theory of rational iteration. They illustrate the theory with concrete examples and exposit Rivera-Letelier's results concerning rational dynamics over the field of \(p\)-adic complex numbers. They also establish Berkovich space versions of arithmetic results such as the Fekete-Szegö theorem and Bilu's equidistribution theorem.

#### Table of Contents

# Table of Contents

## Potential Theory and Dynamics on the Berkovich Projective Line

- Contents v6 free
- Preface ix10 free
- Introduction xv16 free
- Notation xxix30 free
- Chapter 1. The Berkovich unit disc 136 free
- Chapter 2. The Berkovich projective line 1954
- Chapter 3. Metrized graphs 4984
- Chapter 4. The Hsia kernel 73108
- Chapter 5. The Laplacian on the Berkovich projective line 87122
- Chapter 6. Capacity theory 121156
- Chapter 7. Harmonic functions 145180
- Chapter 8. Subharmonic functions 193228
- Chapter 9. Multiplicities 249284
- Chapter 10. Applications to the dynamics of rational maps 291326
- Appendix A. Some results from analysis and topology 377412
- Appendix B. $\bb R$-trees and Gromov hyperbolicity 393428
- Appendix C. A brief overview of Berkovich's theory 405440
- Bibliography 417452
- Index 423458 free

#### Readership

Graduate students and research mathematicians interested in number theory, algebraic geometry, and non-Archimedean dynamics.