  For a map $S:X\to X$ and an open connected set ($=$ a hole) $H\subset X$ we
define $\mathcal J_H(S)$ to be the set of points in $X$ whose $S$-orbit avoids
$H$. We say that a hole $H_0$ is supercritical if (i) for any hole $H$ such
that $\bar{H_0}\subset H$ the set $\mathcal J_H(S)$ is either empty or contains
only fixed points of $S$; (ii) for any hole $H$ such that $\barH\subset H_0$
the Hausdorff dimension of $\mathcal J_H(S)$ is positive.
  The purpose of this note to completely characterize all supercritical holes
for the doubling map $Tx=2x\bmod1$.
