Introduction This book has several goals. The first goal is to develop the foundations of potential theory on P1 Berk , including the definition of a measure-valued Laplacian operator, capacity theory, and a theory of harmonic and subhar- monic functions. A second goal is to give applications of potential theory on P1 Berk , especially to the dynamics of rational maps defined over an arbitrary complete and algebraically closed non-Archimedean field K. A third goal is to provide the reader with a concrete introduction to Berkovich’s theory of analytic spaces by focusing on the special case of the Berkovich projective line. We now outline the contents of the book. The Berkovich affine and projective lines. Let K be an alge- braically closed field which is complete with respect to a nontrivial non- Archimedean absolute value. The topology on K induced by the given absolute value is Hausdorff, but it is also totally disconnected and not lo- cally compact. This makes it difficult to define a good notion of an analytic function on K. Tate dealt with this problem by developing the subject now known as rigid analysis, in which one works with a certain Grothendieck topology on K. This leads to a satisfactory theory of analytic functions, but since the underlying topological space is unchanged, difficulties remain for other applications. For example, using only the topology on K, there is no evident way to define a Laplacian operator analogous to the classical Laplacian on C or to work sensibly with probability measures on K. However, these difficulties, and many more, can be resolved in a very satisfactory way using Berkovich’s theory. The Berkovich affine line A1 Berk over K is a locally compact, Hausdorff, and path-connected topological space which contains K (with the topology induced by the given absolute value) as a dense subspace. One obtains the Berkovich projective line P1 Berk by adjoining to A1 Berk in a suitable manner a point at infinity the resulting space P1 Berk is a compact, Hausdorff, path-connected topological space which contains P1(K) (with its natural topology) as a dense subspace. In fact, A1 Berk and P1 Berk are more than just path-connected: they are uniquely path- connected, in the sense that any two distinct points can be joined by a unique arc. The unique path-connectedness is closely related to the fact that A1 Berk and P1 Berk are endowed with a natural tree structure. (More specifically, they are R-trees, as defined in §1.4.) The tree structure on A1 Berk (resp. P1 Berk ) can xv
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