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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