Holomorphic surjectivity near the boundary: topological and algebraic genericity

Abstract

ABSTRACT Let Ω be an open set in the complex plane. Given a subdomain V that is not relatively compact in Ω, we prove the existence of infinitely generated free subalgebras as well as of dense vector subspaces and of infinite dimensional Banach spaces consisting of holomorphic functions on Ω all of whose non-zero members are surjective on V. Some of these results are extended to several collections of subdomains and to the action of differential operators on this kind of functions.