  We give a fractal-geometric condition for a measure on [0,1] to be supported
on points x that are normal in base n, i.e. such that the sequence x,nx,n^2
x,... equidistributes modulo 1. This condition is robust under C^1 coordinate
changes, and it applies also when n is a Pisot number and equidistribution is
understood with respect to the beta-map and Parry measure. As applications we
obtain new results (and strengthen old ones) about the prevalence of normal
numbers in fractal sets, and new results on measure rigidity, specifically
completing Host's theorem to multiplicatively independent integers and proving
a Rudolph-Johnson-type theorem for certain pairs of beta transformations.
