eBook ISBN:  9781470404970 
Product Code:  MEMO/191/891.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 
eBook ISBN:  9781470404970 
Product Code:  MEMO/191/891.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 191; 2008; 146 ppMSC: Primary 32; 37; Secondary 65
The authors study the Newton map \(N:\mathbb{C}^2\rightarrow\mathbb{C}^2\) associated to two equations in two unknowns, as a dynamical system. They focus on the first nontrivial case: two simultaneous quadratics, to intersect two conics. In the first two chapters, the authors prove among other things:
The RussakovksiShiffman measure does not change the points of indeterminancy.
The lines joining pairs of roots are invariant, and the Julia set of the restriction of \(N\) to such a line has under appropriate circumstances an invariant manifold, which shares features of a stable manifold and a center manifold.
The main part of the article concerns the behavior of \(N\) at infinity. To compactify \(\mathbb{C}^2\) in such a way that \(N\) extends to the compactification, the authors must take the projective limit of an infinite sequence of blowups. The simultaneous presence of points of indeterminancy and of critical curves forces the authors to define a new kind of blowup: the Farey blowup.
This construction is studied in its own right in chapter 4, where they show among others that the real oriented blowup of the Farey blowup has a topological structure reminiscent of the invariant tori of the KAM theorem. They also show that the cohomology, completed under the intersection inner product, is naturally isomorphic to the classical Sobolev space of functions with squareintegrable derivatives.
In chapter 5 the authors apply these results to the mapping \(N\) in a particular case, which they generalize in chapter 6 to the intersection of any two conics.

Table of Contents

Chapters

0. Introduction

1. Fundamental properties of Newton maps

2. Invariant 3manifolds associated to invariant circles

3. The behavior at infinity when $a$ = $b$ = 0

4. The Farey blowup

5. The compactification when $a$ = $b$ = 0

6. The case where $a$ and $b$ are arbitrary


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The authors study the Newton map \(N:\mathbb{C}^2\rightarrow\mathbb{C}^2\) associated to two equations in two unknowns, as a dynamical system. They focus on the first nontrivial case: two simultaneous quadratics, to intersect two conics. In the first two chapters, the authors prove among other things:
The RussakovksiShiffman measure does not change the points of indeterminancy.
The lines joining pairs of roots are invariant, and the Julia set of the restriction of \(N\) to such a line has under appropriate circumstances an invariant manifold, which shares features of a stable manifold and a center manifold.
The main part of the article concerns the behavior of \(N\) at infinity. To compactify \(\mathbb{C}^2\) in such a way that \(N\) extends to the compactification, the authors must take the projective limit of an infinite sequence of blowups. The simultaneous presence of points of indeterminancy and of critical curves forces the authors to define a new kind of blowup: the Farey blowup.
This construction is studied in its own right in chapter 4, where they show among others that the real oriented blowup of the Farey blowup has a topological structure reminiscent of the invariant tori of the KAM theorem. They also show that the cohomology, completed under the intersection inner product, is naturally isomorphic to the classical Sobolev space of functions with squareintegrable derivatives.
In chapter 5 the authors apply these results to the mapping \(N\) in a particular case, which they generalize in chapter 6 to the intersection of any two conics.

Chapters

0. Introduction

1. Fundamental properties of Newton maps

2. Invariant 3manifolds associated to invariant circles

3. The behavior at infinity when $a$ = $b$ = 0

4. The Farey blowup

5. The compactification when $a$ = $b$ = 0

6. The case where $a$ and $b$ are arbitrary