Large subspaces of compositionally universal functions with maximal cluster sets

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