In this paper we consider the times-$q$ map on the unit interval as a subshift of finite type by identifying each number with its base $q$ expansion, and we study certain non-dense orbits of this system where no element of the orbit is smaller than some fixed parameter $c$.

The Hausdorff dimension of these orbits can be calculated using the spectral radius of the transition matrix of the corresponding subshift, and using simple methods based on Euclidean division in the integers, we completely characterize the characteristic polynomials of these matrices as well as give the value of the spectral radius for certain values of $c$. It is known through work of Urbanski and Nilsson that the Hausdorff dimension of the orbits mentioned above as a map of $c$ is continuous and constant almost everywhere, and as a new result we give some asymptotic results on how this map behaves as $q \to \infty$.