Indistinguishable Asymptotic Pairs and Multidimensional Sturmian Configurations

Two asymptotic configurations on a full Zd-shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation Zd of. We prove that indistinguishable asymptotic pairs satisfying a flip condition are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to Zd of the characterization of Sturmian sequences by their factor complexity n + 1. Many open questions are raised by the current work and are listed in the introduction. © The Author(s), 2024.