  Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing
probability measure which is positive on all cylinders in $X$. We show that
there exist arbitrarily small finite overlapping union of shifted cylinders
which intersect every orbit under the shift map.
  We also show that for any proper subshift $Y$ of $X$ there exists a finite
overlapping unions of shifted cylinders such that its survivor set contains $Y$
(in particular, it can have entropy arbitrarily close to the entropy of $X$).
Both results may be seen as somewhat counter-intuitive.
  Finally, we apply these results to a certain class of hyperbolic algebraic
automorphisms of a torus.
