  The goal of this article is two fold. First, using transcendental shift-like
automorphisms of C^k, k > 2 we construct two examples of non-degenerate entire
mappings with prescribed ranges. The first example exhibits an entire mapping
of C^k, k > 2 whose range avoids a given polydisc but contains the complement
of a slightly larger concentric polydisc. This generalizes a result of
Dixon-Esterle in C^2. The second example shows the existence of a
Fatou--Bieberbach domain in C^k,k > 2 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.
