It is known that a polynomial on $\mathbb{C}$ is holomorphically conjugate to its term of highest degree near infinity. By assigning suitable weights, we generalize this fact to polynomial skew products on $\mathbb{C}^{2}$.
Abstract. Let f : (C 2 , 0) → (C 2 , 0) be a germ of holomorphic skew product with a superattracting fixed point at the origin. If it has a suitable weight, then we can construct a Böttcher coordinate which conjugates f to the associated monomial map. This Böttcher coordinate is defined on an invariant open set whose interior or boundary contains the origin.
Let f (z, w) = (p(z), q(z, w)) be a holomorphic skew product with a superattracting fixed point at the origin. Under one or two assumptions, we prove that f is conjugate to a monomial map on an invariant open set whose closure contains the origin. The monomial map and the open set are determined by the degree of p and the Newton polygon of q.for some rational numbers 0 ≤ l 1 < ∞ and 0 < l 2 ≤ ∞, which are also determined by the degree of p and the Newton polygon of q.Let f 0 (z, w) = (a δ z δ , b γd z γ w d ) and ||(z, w)|| = max{|z|, |w|}.Lemma 1.1. If d ≥ 2, then (1) for any small ε > 0 there is r > 0 such that ||f − f 0 || < ε||f 0 || on U r , and(2) f (U r ) ⊂ U r for small r > 0.In particular, f is rigid on U r . As in the one dimensional case, this lemma induces a conjugacy on U r from f to the monomial map f 0 .Theorem 1.2. If d ≥ 2, then there is a biholomorphic map φ defined on U r that conjugates f to f 0 for small r > 0. Moreover, for any small ε > 0, there is r > 0 such that ||φ − id|| < ε||id|| on U r .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.