
Preface  Introduction  Preview Material  Table of Contents  Supplementary Material 
Mathematical Surveys and Monographs 2010; 428 pp; hardcover Volume: 159 ISBN10: 0821849247 ISBN13: 9780821849248 List Price: US$110 Member Price: US$88 Order Code: SURV/159 See also: \(p\)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 Arithmetic Differential Equations  Alexandru Buium  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 appendiceson analysis, \(\mathbb{R}\)trees, and Berkovich's general theory of analytic spacesare 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. Readership Graduate students and research mathematicians interested in number theory, algebraic geometry, and nonArchimedean dynamics. 


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