An interpolation theorem for slice-regular functions with application to very tame sets and slice Fatou–Bieberbach domains in H 2

We prove an interpolation theorem for slice-regular quaternionic functions. We define very tame sets in H 2 to be the sets which can be mapped by compositions of automorphisms with volume 1 to the set T = {(2n − 1, 0), n ∈ N} ∪ {(2n + S, 0), n ∈ N}. We then show that any zero set of a slice-regular function of one variable embedded in H × {0} ⊂ H 2 is very tame in H 2 . A notion of slice Fatou–Bieberbach domain in H 2 is introduced and, finally, a slice Fatou–Bieberbach domain in H 2 avoiding T is constructed in the last section.