  In this paper we prove two theorems. The first one is a structure result that
describes the extrinsic geometry of an embedded surface with constant mean
curvature (possibly zero) in a homogeneously regular Riemannian three-manifold,
in any small neighborhood of a point of large almost-maximal curvature. We next
apply this theorem and the Quadratic Curvature Decay Theorem (previously proven
by the same authors in [14]) to deduce compactness, descriptive and
dynamics-type results concerning the space $D(M)$ of non-flat limits under
dilations of any given properly embedded minimal surface $M$ in $\mathbb{R}^3$.
