Studying of complexity of infinite aperiodic words, i.e., the number of different factors of the infinite word of a fixed length, is an interesting combinatorial problem. Moreover, investigation of infinite words associated with $\beta$-integers can be interpreted as investigation of one-dimensional quasicrystals. In such a way of interpretation, complexity corresponds to the number of local configurations of atoms.
"Complexity for Infinite Words Associated with Quadratic Non-Simple Parry Numbers." J. Geom. Symmetry Phys. 7 1 - 11, 2006. https://doi.org/10.7546/jgsp-7-2006-1-11