  In this paper we study critical sets of solutions $u_\e$ of second-order
elliptic equations in divergence form with rapidly oscillating and periodic
coefficients. We show that the $(d-2)$-dimensional Hausdorff measures of the
critical sets are bounded uniformly with respect to the period $\e$, provided
that doubling indices for solutions are bounded. The key step is an estimate of
"turning " for the projection of a non-constant solution $u_\e$ onto the
subspace of spherical harmonics of order $\ell$, when the doubling index for
$u_\e$ on a sphere $\partial B(0, r)$ is trapped between $\ell -\delta$ and
$\ell +\delta$, for $r$ between $1$ and a minimal radius $r^*\ge C_0\e$. This
estimate is proved by using harmonic approximation successively. With a
suitable $L^2$ renormalization as well as rescaling we are able to control the
accumulated errors introduced by homogenization and projection. Our proof also
gives uniform bounds for Minkowski contents of the critical sets.
