Simultaneously non-dense orbits under different expanding maps

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. It is well-known that in many cases such sets have full Hausdorff dimension. We prove that such sets have a large intersection property, i.e. countable intersections of such sets also have full Hausdorff dimension. This result applies to a class of maps including multiplication by integers modulo 1 and x -> 1/x modulo 1. We prove that the same properties hold for multiplication modulo 1 by a dense set of non-integer numbers between 1 and 2.