  Given a point and an expanding map on the unit interval, we consider the set
of points for which the forward orbit under this map is bounded away from the
given point. For maps like multiplication by an integer modulo 1, such sets
have full Hausdorff dimension. We prove that such sets have a large
intersection property, i.e. that countable intersections of such sets also have
full Hausdorff dimension. This result applies to maps like multiplication by
integers modulo 1, but also to nonlinear maps like $x \mapsto 1/x$ modulo 1. We
prove that the same thing holds for multiplication modulo 1 by a dense set of
non-integer numbers between 1 and 2.
