# Newton’s Method Applied to Two Quadratic Equations in \(\mathbb{C}^{2}\) Viewed as a Global Dynamical System

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*John H. Hubbard; Peter Papadopol*

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

# Table of Contents

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

- Table of Contents iii4 free
- Chapter 0 Introduction 18 free
- Chapter 1 Fundamental properties of Newton maps 1118
- Chapter 2 Invariant 3-manifolds associated to invariant circles 3542
- Chapter 3 The behavior at infinity when a = b = 0 6168
- Chapter 4 The Farey blow-up 6875
- 4.1. Definition of the Farey blow-up 6875
- 4.2. Naturality of the Farey blow-up 7178
- 4.3. The real oriented blow-up of the Farey blow-up 7279
- 4.4. Naturality and real oriented blow-ups 7683
- 4.5. Inner products on spaces of homogeneous functions 7784
- 4.6. Homology of the Farey blow-up 8289
- 4.7. The action of mappings F[sub((k l))] on homology 8693

- Chapter 5 The compactification when a = b = 0 9198
- 5.1. The tower of blow-ups when a = b = 0 9198
- 5.2. Sequence spaces 95102
- 5.3. The real oriented blow-up of X 97104
- 5.4. The homology of X[sub(1)] 100107
- 5.5. The action of N[sub(p)] on homology and cohomology 104111
- 5.6. The (co) homology H[sub(2)](X*[sub(∞)] 114121
- 5.7. The action of N on homology 118125

- Chapter 6 The case where a and b are arbitrary 123130
- Bibliography 128135