A degenerate Newton’s map in two complex variables: Linking with currents

Abstract

Little is known about the global structure of the basins of attraction of Newton's method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of P (x, y) = x(1 − x) and Q(x, y) = y 2 + Bxy − y. There are invariant circles S0 and S1 within the lines x = 0 and x = 1 which are superattracting in the x-direction and hyperbolically repelling within the vertical line. We show that S0 and S1 have local super-stable manifolds, which when pulled back under iterates of N form global super-stable spaces W0 and W1. By blowing-up the points of indeterminacy p and q of N and all of their inverse images under N we prove that W0 and W1 are real-analytic varieties.We define linking between closed 1-cycles in Wi (i = 0, 1) and an appropriate closed 2 current providing a homomorphism lk : H1(Wi, Z) → Q. If Wi intersects the critical value locus of N , this homomorphism has dense image, proving that H1(Wi, Z) is infinitely generated. Using the Mayer-Vietoris exact sequence and an algebraic trick, we show that the same is true for the closures of the basins of the roots W (ri). Newton's method is one of the fundamental algorithms of mathematics, so it is evidently important to understand its dynamics, particularly the structure of the basins of attraction of the roots. Even in one dimension, the topology of these basins can be complicated and there has been a good deal of research on this subject. In higher dimensions, next to nothing is known about the topology of the basins. In this paper we make the first steps at understanding their topology in two complex variables.We focus on a specific system: the Newton's Method used to solve for the common roots of P (x, y) = x(1 − x) and Q(x, y) = y 2 + Bxy − y. While this is one specific and relatively simple system, we believe that some of the techniques developed in this paper can be used to study more general systems. Dynamical systems g : C n → C n are often classified in terms of: (1) The number of inverse images of a generic point by g, which is called the topological degree d t (g), and (2) Whether g has points of indeterminacy.Mappings g : Meanwhile, birational maps g : P n P n (rational maps with rational inverse) are examples of systems with points of indeterminacy, but with d t (g) = 1. The famous Henon mappings H : P 2 P 2 fall into this class. Such systems have been studied extensively by Bedford and Smillie [3,4,5,6,7,9,8]

Superstable manifolds of invariant circles and codimension-one Böttcher functions

“…Therefore, it follows immediately from Theorem A that the local stable manifolds W s loc (S 0 ) and W s loc (S 1 ) are real analytic. This was proven previously in [31] using more specific details of the mapping.…”

“…Existence of such stable laminations has also been proved in the holomorphic context by Ushiki [35]. It can be proved in the following simple way as well, which is a direct generalization of what was done in [31,Proposition 4.2] and [6,Proposition 9.2].…”

“…This is described more precisely at the beginning of §2. Although this is a rather special situation, it has appeared in examples from [31,16,6,5]. For such maps, the Julia set of f |L is an invariant circle S, which is a hyperbolic set for f .…”

“…The proof of Theorem A' is followed by §3, where we provide applications to certain specific examples, including those from [16, §6.2] and [31].…”

Topology of Fatou components for endomorphisms of CPk: linking with the Green's current

“…Given any Γ and Γ both having boundary γ we have Γ − Γ , S − Ω = 0, since S and Ω are cohomologous. (In the language of [Ro,p. 132], we say that S − Ω is in the "linking kernel of P k ".)…”