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 ABerk
1
19
2.2. The Berkovich “Proj” construction 23
2.3. The action of a rational map ϕ on PBerk
1
30
2.4. Points of PBerk
1
revisited 35
2.5. The tree structure on HBerk and PBerk
1
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
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