Introduction

This book has several goals. The first goal is to develop the foundations

of potential theory on PBerk,

1

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

PBerk,

1

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

ABerk1

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 PBerk

1

by

adjoining to ABerk

1

in a suitable manner a point at infinity; the resulting

space PBerk

1

is a compact, Hausdorff, path-connected topological space which

contains

P1(K)

(with its natural topology) as a dense subspace. In fact,

ABerk

1

and PBerk

1

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

ABerk1

and PBerk

1

are endowed with a natural tree structure. (More specifically, they

are R-trees, as defined in §1.4.) The tree structure on ABerk

1

(resp. PBerk)

1

can

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