  We develop the theory of multiresolutions in the context of Hausdorff measure
of fractional dimension between 0 and 1. While our fractal wavelet theory has
points of similarity that it shares with the standard case of Lebesgue measure
on the line, there are also sharp contrasts. These are stated in our main
result, a dichotomy theorem. The first section is the case of the middle-third
Cantor set. This is followed by a review of the essentials on Hausdorff
measure. The remaining sections of the paper cover multiresolutions in the
general context of affine iterated function systems.
