Chapter 0 Introduction .......................................... 1
1. Introduction ....................................... 1
2. Outline of paper ................................... 2
3. Acknowledgements ................................... 4
4. A computer tour of Newton's method ................. 4
5. Some open questions ................................ 9
Chapter 1 Fundamental properties of Newton maps ................ 11
1.1. Generalities about Newton's method .............. 11
1.2. The intersection of graphs ...................... 13
1.3. The Russakovskii-Shiffman measure ............... 18
1.4. Invariant currents .............................. 22
1.5. The intersection of conies ...................... 23
1.6. Degenerate cases ................................ 29
1.7. The one-variable rational functions associated
to the roots .................................... 32
Chapter 2 Invariant 3-manifolds associated to invariant
circles .............................................. 35
2.1. The circles in the invariant lines .............. 35
2.2. Periodic cycles on invariant circles ............ 38
2.3. Unstable manifolds at infinity .................. 42
2.4. The invariant manifolds of circles .............. 46
2.5. The extension of Φ and the origin of
"bubbles" ....................................... 54
Chapter 3 The behavior at infinity when a = b = 0 .............. 61
3.1. The primitive space ............................. 61
3.2. Newton's method and the primitive space ......... 63
Chapter 4 The Farey blow-up .................................... 68
4.1. Definition of the Farey blow-up ................. 68
4.2. Naturality of the Farey blow-up ................. 71
4.3. The real oriented blow-up of the Farey
blow-up ......................................... 72
4.4. Naturality and real oriented blow-ups ........... 76
4.5. Inner products on spaces of homogeneous
functions ....................................... 77
4.6. Homology of the Farey blow-up ................... 82
4.7. The action of mappings F(kl) on homology ......... 86
Chapter 5 The compactification when a = b = 0 .................. 91
5.1. The tower of blow-ups when a = b = 0 ............ 91
5.2. Sequence spaces ................................. 95
5.3. The real oriented blow-up of Х∞ ................. 97
5.4. The homology of Xi ............................. 100
5.5. The action of Np on homology and cohomology .... 104
5.6. The (co)homology H2(X*∞) ........................ 114
5.7. The action of N on homology .................... 118
Chapter 6 The case where a and b are arbitrary ................ 123
6.1. A curve of order two ........................... 123
6.2. The primitive space for arbitrary a and b ...... 125
6.3. Building the space X∞ .......................... 126
6.4. The basins of the roots ........................ 126
6.5. Real oriented blow-ups and homology ............ 127
Bibliography .................................................. 128
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