Multidimensional constant-length substitution sequences

AbstractWe consider multidimensional substitutions of constant length in a primarily expository setting, explaining how results from both symbolic dynamics and tiling dynamical systems can be applied. We focus in particular on ergodic and spectral theoretic concepts in an analysis that includes results and proofs extending what is known to our case. Tools such as the frequency measure, spectral measures, and the multidimensional odometer are used. We investigate several examples, among them the class of bijective substitutions. Bijective substitutions are of particular interest due to their mixed dynamical spectrum and because they are skew products over multidimensional odometers. For these, a condition is given allowing a full decomposition of the spectral measures