Indistinguishable asymptotic pairs and multidimensional Sturmian configurations

TL;DR

This paper characterizes indistinguishable asymptotic pairs in multidimensional shifts, linking their pattern complexity to Sturmian configurations and extending classical one-dimensional results to higher dimensions.

Contribution

It introduces a characterization of indistinguishable asymptotic pairs satisfying a flip condition via pattern complexity and connects uniformly recurrent pairs to multidimensional Sturmian configurations.

Findings

Indistinguishable pairs are characterized by pattern complexity on finite supports.

Uniformly recurrent pairs correspond to multidimensional Sturmian configurations.

The results generalize the one-dimensional Sturmian sequence characterization to higher dimensions.

Abstract

Two asymptotic configurations on a full $\mathbb{Z}^d$-shift are indistinguishable if for every finite pattern the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb{Z}^d$. 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 $\mathbb{Z}^d$ 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.