Hardcover ISBN:  9780821849248 
Product Code:  SURV/159 
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Electronic ISBN:  9781470413866 
Product Code:  SURV/159.E 
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Book DetailsMathematical Surveys and MonographsVolume: 159; 2010; 428 ppMSC: 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 nonArchimedean 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 selfcontained 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 nonArchimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical FatouJulia theory of rational iteration. They illustrate the theory with concrete examples and exposit RiveraLetelier'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 FeketeSzegö theorem and Bilu's equidistribution theorem.ReadershipGraduate students and research mathematicians interested in number theory, algebraic geometry, and nonArchimedean dynamics.

Table of Contents

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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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 nonArchimedean 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 selfcontained 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 nonArchimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical FatouJulia theory of rational iteration. They illustrate the theory with concrete examples and exposit RiveraLetelier'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 FeketeSzegö theorem and Bilu's equidistribution theorem.
Graduate students and research mathematicians interested in number theory, algebraic geometry, and nonArchimedean 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