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TOPOLOGY FOR THE BASINS OF ATTRACTION OF NEWTON’S METHOD IN TWO COMPLEX VARIABLES

By Roland K. Roeder

Abstract

In a recent paper, John H. Hubbard and Peter Papadopol study the dynamics of the Newton map, N: C2 → C2, for finding the common zeros of two quadratic equations P(x,y) = 0 and Q(x,y) = 0. The map N has points of indeterminacy, critical curves, and invariant circles that are non-uniformly hyperbolic. Most of the work in their paper is spent resolving the points of indeterminacy of N, and creating a compactification of C2 in a way that is both compatible with the dynamics of N and that has “tame ” topology. This part of their work requires two very technical tools called Farey Blow-ups and Real-oriented blow-ups. In a different direction, Hubbard and Papadopol show that the basin of attraction for each of the four common zeros of P and Q is path connected. However, most further questions about the topology of these basins of attraction remain a mystery. The dynamics of N is much simpler if the common roots of P and Q lie on parallel lines, for instance when P(x,y) = x(x − 1) = 0 and Q(x,y) = y2 + Bxy − y = 0. The first component of N depends only on x, while the second component depends on both x and y. Many of the complexities described by Hubbard and Papadopol disappear: one must still do an infinite sequence of blow-ups in order to make N a well defined dynamical system

Year: 2005
OAI identifier: oai:CiteSeerX.psu:10.1.1.371.5240
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