Fine properties of branch point singularities: stationary two-valued graphs and stable minimal hypersurfaces near points of density $< 3$

We study (higher order) asymptotic behaviour near branch points of stationary $n$-dimensional two-valued $C^{1, \mu}$ graphs in an open subset of ${\mathbb R}^{n+m}$. Specifically, if $M$ is the graph of a two-valued $C^{1, \mu}$ function $u$ on an open subset $\Omega \subset {\mathbb R}^{n}$ taking values in the space of un-ordered pairs of points in ${\mathbb R}^{m}$, and if the integral varifold $V = (M, \theta),$ where the multiplicity function $\theta \, : \, M \rightarrow \{1, 2\}$ is such that $\theta =2$ on the set where the two values of $u$ agree and $\theta =1$ otherwise, is stationary in $\Omega \times {\mathbb R}^{m}$ with respect to the mass functional, we show that at ${\mathcal H}^{n-2}$-a.e.\ point $Z$ along its branch locus $u$ decays asymptotically, modulo its single valued average, to a unique non-zero two-valued cylindrical harmonic tangent function $\varphi^{(Z)}$ which is homogeneous of some degree $\geq 3/2$. As a corollary, we obtain that the branch locus of $u$ is countably $(n-2)$-rectifiable, and near points $Z$ where the degree of homogeneity of $\varphi^{(Z)}$ is equal to $3/2$, the branch locus is an embedded real analytic submanifold of dimension $n-2$. These results, combined with the recent works \cite{M} and \cite{MW}, imply a stratification theorem for the (relatively open) set of density $< 3$ points of a stationary codimension 1 integral $n$-varifold with stable regular part and no triple junction singularities.