  Polymer adsorption on fractally rough walls of varying dimensionality is
studied by renormalization group methods on hierarchical lattices. Exact
results are obtained for deterministic walls. The adsorption transition is
found continuous for low dimension $d_w$ of the adsorbing wall and the
corresponding crossover exponent $\phi$ monotonically increases with $d_w$,
eventually overcoming previously conjectured bounds. For $d_w$ exceeding a
threshold value $d_w^*$, $\phi$ becomes 1 and the transition turns
first--order. $d_w^*>d_{saw}$, the fractal dimension of the polymer in the
bulk. An accurate numerical approach to the same problem with random walls
gives evidence of the same scenario.
