  It is well known that a purely unrectifiable set cannot support a harmonic
measure which is absolutely continuous with respect to the Hausdorff measure of
this set. We show that nonetheless there exist elliptic operators on (purely
unrectifiable) Cantor sets in ${\mathbb{R}}^2$ whose elliptic measure is
absolutely continuous, and in fact, essentially proportional to the Hausdorff
measure.
