Newton's method applied to two quadratic equations in C[superscript 2] viewed as a global dynamical system / John H. Hubbard, Peter Papadopol.

Format:

Book

Author/Creator:

Hubbard, John H. (John Hamal), 1945 or 1946- author.

Papadopol, Peter, 1931- author.

Series:

Memoirs of the American Mathematical Society ; no. 891.

Memoirs of the American Mathematical Society, 0065-9266 ; number 891

Language:

English

Subjects (All):

Newton-Raphson method.

Equations, Quadratic.

Differentiable dynamical systems.

Physical Description:

1 online resource (160 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

Studies the Newton map $N: \mathbb{C} DEGREES2\rightarrow\mathbb{C} DEGREES2$ associated to two equations in two unknowns, as a dynamical system. This title focuses on the first non-trivial case: two simultaneous quadratics, to intersect two conics. It proves among other things: the Russakovksi-Shiffman measure does not change the points of

Contents:

""Table of Contents""; ""Chapter 0 Introduction""; ""1. Introduction""; ""2. Outline of paper""; ""3. Acknowledgements""; ""4. A computer tour of Newton's method""; ""5. Some open questions""; ""Chapter 1 Fundamental properties of Newton maps""; ""1.1. Generalities about Newton's method""; ""1.2. The intersection of graphs""; ""1.3. The Russakovskii-Shiffman measure""; ""1.4. Invariant currents""; ""1.5. The intersection of conies""; ""1.6. Degenerate cases""; ""1.7. The one-variable rational functions associated to the roots""

""Chapter 2 Invariant 3-manifolds associated to invariant circles""""2.1. The circles in the invariant lines""; ""2.2. Periodic cycles on invariant circles""; ""2.3. Unstable manifolds at infinity""; ""2.4. The invariant manifolds of circles""; ""2.5. The extension of Î? and the origin of ""bubbles""""; ""Chapter 3 The behavior at infinity when a = b = 0""; ""3.1. The primitive space""; ""3.2. Newton's method and the primitive space""; ""Chapter 4 The Farey blow-up""; ""4.1. Definition of the Farey blow-up""; ""4.2. Naturality of the Farey blow-up""

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0497-4