Large subspaces of compositionally universal functions with maximal cluster sets

Abstract

Let (φn) be a sequence of holomorphic self-maps of a Jordan domain G in the complex plane. Under appropriate conditions on (φn), we construct an H(G)-dense linear manifold –as well as a closed infinite-dimensional linear manifold– all of whose non-zero functions have H(G)-dense orbits under the action of the sequence of composition operators associated to (φn). Simultaneously, these functions also present maximal cluster sets along each member of a large class of curves in G tending to the boundary.