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E-raamat: Newton's Method Applied to Two Quadratic Equations in $\mathbb {C}^2$ Viewed as a Global Dynamical System

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Newton's Method Applied to Two Quadratic Equations in $\mathbb {C}^2$ Viewed as a Global Dynamical SystemSuurem pilt
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This article examines the Newton map associated to two equations in two unknowns as a dynamical system. The authors analyze the Russakovskii-Shiffman measure for intersections of graphs, define the Farey blow-up, compute the cohomology of the smooth part of the Farey blow-up, and construct a compact space X on which the Newton map operates. Color illustrations are provided, but no index. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)
Introduction
1(10)
Introduction
1(1)
Outline of paper
2(2)
Acknowledgements
4(1)
A computer tour of Newton's method
4(5)
Some open questions
9(2)
Fundamental properties of Newton maps
11(24)
Generalities about Newton's method
11(2)
The intersection of graphs
13(5)
The Russakovskii-Shiffman measure
18(4)
Invariant currents
22(1)
The intersection of conics
23(6)
Degenerate cases
29(3)
The one-variable rational functions associated to the roots
32(3)
Invariant 3-manifolds associated to invariant circles
35(26)
The circles in the invariant lines
35(3)
Periodic cycles on invariant circles
38(4)
Unstable manifolds at infinity
42(4)
The invariant manifolds of circles
46(8)
The extension of Φ and the origin of ``bubbles''
54(7)
The behavior at infinity when a=b=0.
61(7)
The primitive space
61(2)
Newton's method and the primitive space
63(5)
The Farey blow-up
68(23)
Definition of the Farey blow-up
68(3)
Naturality of the Farey blow-up
71(1)
The real oriented blow-up of the Farey blow-up
72(4)
Naturality and real oriented blow-ups
76(1)
Inner products on spaces of homogeneous functions
77(5)
Homology of the Farey blow-up
82(4)
The action of mappings F(k/l) on homology
86(5)
The compactification when a=b=0
91(32)
The tower of blow-ups when a=b=0
91(4)
Sequence spaces
95(2)
The real oriented blow-up of X∞
97(3)
The homology of X1
100(4)
The action of Np on homology and cohomology
104(10)
The (co)homology H2 (X*∞)
114(4)
The action of N on homology
118(5)
The case where a and b are arbitrary
123(5)
A curve of order two
123(2)
The primitive space for arbitrary a and b
125(1)
Building the space X∞
126(1)
The basins of the roots
126(1)
Real oriented blow-ups and homology
127(1)
Bibliography 128
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