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

Abstract

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