Associated to every shift space, the Cuntz-Krieger algebra (C-K algebra for abbreviation) is an invariant of conjugacy defined and developed by K. Matsumoto, S. Eilers, T. Carlsen, and many of their collaborators in the last decade. In particular, Carlsen defined the C-K algebra to be the full groupoid C*-algebra of the “cover”, which is a topological system consisting of a surjective local homeomorphism on a zero-dimensional space induced by the shift space.
In 2022, K. Brix proved that the C-K algebra of the Sturmian shift has finite nuclear dimension, where the Sturmian shift is the (unique) minimal shift space with the smallest complexity function: p_X(n)=n+1. In recent results (joint with Z. He), we show that for any minimal shift space with finitely many left special elements, its C-K algebra always have finite nuclear dimension. In fact, this can be further applied to the class of aperiodic shift spaces with non-superlinear growth complexity.