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Potential Theory and Dynamics on the Berkovich Projective Line

Matthew Baker Georgia Institute of Technology, Atlanta, GA
Robert Rumely University of Georgia, Athens, GA
Available Formats:
Hardcover ISBN: 978-0-8218-4924-8
Product Code: SURV/159
428 pp
List Price: $117.00 MAA Member Price:$105.30
AMS Member Price: $93.60 Electronic ISBN: 978-1-4704-1386-6 Product Code: SURV/159.E 428 pp List Price:$110.00
MAA Member Price: $99.00 AMS Member Price:$88.00
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List Price: $175.50 MAA Member Price:$157.95
AMS Member Price: $140.40 Click above image for expanded view Potential Theory and Dynamics on the Berkovich Projective Line Matthew Baker Georgia Institute of Technology, Atlanta, GA Robert Rumely University of Georgia, Athens, GA Available Formats:  Hardcover ISBN: 978-0-8218-4924-8 Product Code: SURV/159 428 pp  List Price:$117.00 MAA Member Price: $105.30 AMS Member Price:$93.60
 Electronic ISBN: 978-1-4704-1386-6 Product Code: SURV/159.E 428 pp
 List Price: $110.00 MAA Member Price:$99.00 AMS Member Price: $88.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$175.50
MAA Member Price: $157.95 AMS Member Price:$140.40
• Book Details

Mathematical Surveys and Monographs
Volume: 1592010
MSC: Primary 14; Secondary 37; 31;

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.

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

• Chapters
• 1. The Berkovich unit disc
• 2. The Berkovich projective line
• 3. Metrized graphs
• 4. The Hsia kernel
• 5. The Laplacian on the Berkovich projective line
• 6. Capacity theory
• 7. Harmonic functions
• 8. Subharmonic functions
• 9. Multiplicities
• 10. Applications to the dynamics of rational maps
• 11. Some results from analysis and topology
• 12. $\mathbb {R}$-trees and Gromov hyperbolicity
• 13. Brief overview of Berkovich’s theory

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Volume: 1592010
MSC: Primary 14; Secondary 37; 31;

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.

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

• Chapters
• 1. The Berkovich unit disc
• 2. The Berkovich projective line
• 3. Metrized graphs
• 4. The Hsia kernel
• 5. The Laplacian on the Berkovich projective line
• 6. Capacity theory
• 7. Harmonic functions
• 8. Subharmonic functions
• 9. Multiplicities
• 10. Applications to the dynamics of rational maps
• 11. Some results from analysis and topology
• 12. $\mathbb {R}$-trees and Gromov hyperbolicity
• 13. Brief overview of Berkovich’s theory
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