It is known that for 0 < p < ∞
the Hardy space Hp contains a residual set of functions, each of which has range
equal to the whole plane at every boundary point of the unit disk. With
quite new general techniques, we are able to show that this result holds for
numerous other spaces. The space BMOA of analytic functions of bounded
mean oscillation, the Bloch spaces, the Nevanlinna space and the Dirichlet
spaces Da for 0 ≤ a ≤ 1∕2 are examples. Our methods involve hyperbolic
geometry, cluster set analysis and the “depth” function which we have used
previously for determining geometric properties of the image surfaces of
functions.
