Iterative blow-ups for maps with bounded $\mathcal{A}$-variation: A refinement, with application to BD and BV

We refine the iterated blow-up techniques. This technique, combined with a rigidity result and a specific choice of the kernel projection in the Poincaré inequality, might be employed to completely linearize blow-ups along at least one sequence. We show how to implement such argument by applying it to derive affine blow-up limits for $\BD$ and $\BV$ functions around Cantor points. In doing so we identify a specific subset of points - called totally singular points having one-dimensional blow-ups with completely singular gradient measure $D\psi=D^s\psi$ - at which such linearization fails.