Contents Preface ix History x Related works xi Acknowledgments xii Differences from the preliminary version xiii Introduction xv Notation xxix Chapter 1. The Berkovich unit disc 1 1.1. Definition of D(0, 1) 1 1.2. Berkovich’s classification of points in D(0, 1) 2 1.3. The topology on D(0, 1) 7 1.4. The tree structure on D(0, 1) 9 1.5. Metrizability 17 1.6. Notes and further references 18 Chapter 2. The Berkovich projective line 19 2.1. The Berkovich affine line A1 Berk 19 2.2. The Berkovich “Proj” construction 23 2.3. The action of a rational map ϕ on P1 Berk 30 2.4. Points of P1 Berk revisited 35 2.5. The tree structure on H Berk and P1 Berk 38 2.6. Discs, annuli, and simple domains 40 2.7. The strong topology 42 2.8. Notes and further references 47 Chapter 3. Metrized graphs 49 3.1. Definitions 49 3.2. The space CPA(Γ) 50 3.3. The potential kernel jz(x, y) 52 3.4. The Zhang space Zh(Γ) 54 v
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