Dimensions associated with surjective local homeomorphisms and subshifts with low complexity

Abstract

We prove that the Cuntz-Pimsner algebra associated to any surjective aperiodic one-sided subshift with finitely many left special elements has finite nuclear dimension, which is especially the case for every surjective aperiodic subshift with nonsuperlinear-growth complexity. As a generalization, we define the notions of left speical set, the topological Rokhlin dimension, the tower dimension and the amenability dimension for every local homeomorphism. Then we turn to prove that, for every surjective local homeomorphism with a finite left special set consisting of isolated points, these dimensions along with the dynamic asymptotic dimension are all finite.