We will use transcendental shift-like automorphisms of C^k ,k>2 to construct two examples of non-degenerate entire mappings with prescribed ranges. The first example generalizes a result of Dixon-Esterle in C^2, i.e., we construct an entire mapping of C^k, k>2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. 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.

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Last updated: 18 Sep 2024