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Newton’s Method Applied to Two Quadratic Equations in $\mathbb{C}^2$ Viewed as a Global Dynamical System
 
John H. Hubbard Cornell University, Ithaca, NY and Université de Provence, Marseille, France
Peter Papadopol Grand Canyon University, Phoenix, AZ
Newton's Method Applied to Two Quadratic Equations in C^2 Viewed as a Global Dynamical System
eBook ISBN:  978-1-4704-0497-0
Product Code:  MEMO/191/891.E
List Price: $73.00
MAA Member Price: $65.70
AMS Member Price: $43.80
Newton's Method Applied to Two Quadratic Equations in 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 System
John H. Hubbard Cornell University, Ithaca, NY and Université de Provence, Marseille, France
Peter Papadopol Grand Canyon University, Phoenix, AZ
eBook ISBN:  978-1-4704-0497-0
Product Code:  MEMO/191/891.E
List Price: $73.00
MAA Member Price: $65.70
AMS Member Price: $43.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1912008; 146 pp
    MSC: 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 non-trivial case: two simultaneous quadratics, to intersect two conics. In the first two chapters, the authors prove among other things:

    The Russakovksi-Shiffman 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 blow-ups. The simultaneous presence of points of indeterminancy and of critical curves forces the authors to define a new kind of blow-up: the Farey blow-up.

    This construction is studied in its own right in chapter 4, where they show among others that the real oriented blow-up of the Farey blow-up 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 square-integrable 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 3-manifolds associated to invariant circles
    • 3. The behavior at infinity when $a$ = $b$ = 0
    • 4. The Farey blow-up
    • 5. The compactification when $a$ = $b$ = 0
    • 6. The case where $a$ and $b$ are arbitrary
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1912008; 146 pp
MSC: 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 non-trivial case: two simultaneous quadratics, to intersect two conics. In the first two chapters, the authors prove among other things:

The Russakovksi-Shiffman 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 blow-ups. The simultaneous presence of points of indeterminancy and of critical curves forces the authors to define a new kind of blow-up: the Farey blow-up.

This construction is studied in its own right in chapter 4, where they show among others that the real oriented blow-up of the Farey blow-up 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 square-integrable 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 3-manifolds associated to invariant circles
  • 3. The behavior at infinity when $a$ = $b$ = 0
  • 4. The Farey blow-up
  • 5. The compactification when $a$ = $b$ = 0
  • 6. The case where $a$ and $b$ are arbitrary
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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