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 $S_0$ and $S_1$ 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 $S_0$ and $S_1$ have local super-stable manifolds, which when pulled back under iterates of $N$ form global super-stable spaces $W_0$ and $W_1$. By blowing-up the points of indeterminacy $p$ and $q$ of $N$ and all of their inverse images under $N$ we prove that $W_0$ and $W_1$ are real-analytic varieties. We define linking between closed 1-cycles in $W_i$ ($i=0,1$) and an appropriate positive closed $(1,1)$ current providing a homomorphism $lk:H_1(W_i,\mathbb{Z}) \to \mathbb{Q}$. If $W_i$ intersects the critical value locus of $N$, this homomorphism has dense image, proving that $H_1(W_i,\mathbb{Z})$ 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 $\bar{W(r_i)}$.