Holomorphic automorphisms of complex Banach spaces

We consider holomorphic automorphisms of infinite dimensional complex Banach spaces. First we look at automorphisms with an attracting fixed point to construct Fatou–Bieberbach domains in Banach spaces. Second, we look tame sets in Banach spaces. Recall that a discrete set in X is tame if it can be mapped onto an arithmetic progression via an automorphism of X. We show that bounded discrete sets of Banach spaces allowing a Schauder basis are tame. In contrast, \(l_\infty \) has several bounded discrete sets which are not tame.