  This paper discusses the regularity of multiple-valued Dirichlet minimizing
maps into the sphere. It shows that even at branched point, as long as the
normalized energy is small enough, we have the energy decay estimate. Combined
with the previous work by Chun-Chi Lin, we get our first estimate that m-2
dimensional Hausdorff measure of singular set is zero. Furthermore, by looking
at the tangent map and using dimension reduction argument, we show that the
singular set is at least of codimension 3.
