Preface This book is a revised and expanded version of the authors’ manuscript “Analysis and Dynamics on the Berkovich Projective Line” ([91], July 2004). Its purpose is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line. The theory developed here has applications in arithmetic geometry, arithmetic intersection theory, and arithmetic dynamics. In an effort to create a reference which is as useful as possible, we work over an arbitrary complete and algebraically closed non-Archimedean field. We also state our global applications over an arbitrary product formula field whenever pos- sible. Recent work has shown that such generality is essential, even when addressing classical problems over C. As examples, we note the first au- thor’s proof of a Northcott-type finiteness theorem for the dynamical height attached to a nonisotrivial rational function of degree at least 2 over a func- tion field [5] and his joint work with Laura DeMarco [6] on finiteness results for preperiodic points of complex dynamical systems. We first give a detailed description of the topological structure of the Berkovich projective line. We then introduce the Hsia kernel, the fundamen- tal kernel for potential theory (closely related to the Gromov kernel of [47]). Next we define a Laplacian operator on P1 Berk and construct theories of capac- ities, harmonic functions, and subharmonic functions, all strikingly similar to their classical counterparts over C. We develop a theory of multiplici- ties for rational maps and give applications to non-Archimedean dynamics, including the construction of a canonical invariant probability measure on P1 Berk analogous to the well-known measure on P1(C) constructed by Lyu- bich and by Freire, Lopes, and Ma˜e. Finally, we investigate Berkovich space analogues of the classical Fatou-Julia theory for rational iteration over C. In §7.8, we give an updated treatment (in the special case of P1) of the Fekete and Fekete-Szeg¨ o theorems from [88], replacing the somewhat esoteric notion of “algebraic capacitability” with the simple notion of com- pactness. In §7.9, working over an arbitrary product formula field, we prove a generalization of Bilu’s equidistribution theorem [24] for algebraic points which are ‘small’ with respect to the height function attached to a compact Berkovich adelic set. In §10.3, again working over a product formula field, we prove an adelic equidistribution theorem for algebraic points which are ‘small’ with respect to the dynamical height attached to a rational function of degree at least 2, extending results in [9], [35], and [47]. ix
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