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Potential Theory and Dynamics on the Berkovich Projective Line
Matthew Baker, Georgia Institute of Technology, Atlanta, GA, and Robert Rumely, University of Georgia, Athens, GA
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Mathematical Surveys and Monographs
2010; 428 pp; hardcover
Volume: 159
ISBN-10: 0-8218-4924-7
ISBN-13: 978-0-8218-4924-8
List Price: US$110 Member Price: US$88
Order Code: SURV/159

$$p$$-adic Geometry: Lectures from the 2007 Arizona Winter School - Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran S Kedlaya, Jeremy Teitelbaum, edited by David Savitt and Dinesh S Thakur

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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 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.