Simultaneously Non-dense Orbits Under Different Expanding Maps

Abstract

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.