A classification of the point spectrum of constant length substitution tiling spaces and general fixed point theorems for tilings

We examine the point spectrum of the various tiling spaces that result from different choices of tile lengths for substitution of constant length on a two or a three letter alphabet. In some cases we establish insensitivity to changes in length. In a wide range of cases, we establish that the typical choice of length leads to trivial point spectrum. We also consider a problem related to tilings of the integers and their connection to fixed point theorems. Using this connection, we prove a generalization of the Banach Contraction Principle.