Some aspects of shift-like automorphisms of $$\pmb {\mathbb {C}}^{\varvec{k}}$$ C k

Abstract

The goal of this article is two fold. First, using transcendental shift-like automorphisms of \(\mathbb C^k, ~k \ge 3\) we construct two examples of non-degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of \(\mathbb C^k, ~k \ge 3\) whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon–Esterle in \(\mathbb C^2.\) The second example shows the existence of a Fatou–Bieberbach domain in \(\mathbb C^k,~k \ge 3\) that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay–Rudin. In the second part we compute the order and type of entire mappings that parametrize one dimensional unstable manifolds for shift-like polynomial automorphisms and show how they can be used to prove a Yoccoz type inequality for this class of automorphisms.