| When dividing a 1- or 2-dimensional region according to a specified scheme, the resulting division points can be assigned weights, which gives rise to a measure on the space. In the 1-dimensional setting, results concerning the distribution of division points are well known; they suggest how to study analogous processes in two dimensions. We explore two different division schemes on a triangular region and three different measures based on each division, finding the limiting measures in each case.;One question to ask when studying a dynamical system is, what are its endomorphisms. Expanding on the results of Ethan Coven, we find that every endomorphism of a Sturmian system is an element of the action, but for generalized Sturmian systems an additional endomorphism may exist, depending on the defining parameters. For the discrete chair substitution tiling system, every endomorphism is a power of the action. |
