Indistinguishable asymptotic pairs and multidimensional Sturmian configurations

TL;DR

This work extends the one-dimensional Sturmian characterization to multidimensional shifts by introducing indistinguishable asymptotic pairs and a flip condition with difference set $F=\{0,-e_1,\dots,-e_d\}$. The authors connect pattern complexity to indistinguishability, proving that $\#\mathcal{L}_S(x)=\#(F-S)$ characterizes such pairs, and show that uniformly recurrent pairs under the flip condition are encoded by codimension-one cut-and-project schemes, i.e., multidimensional Sturmian configurations. They establish that characteristic $d$-dimensional Sturmian configurations $c_{\alpha},c'_{\alpha}$ (for totally irrational $\alpha$) are indistinguishable and satisfy the flip condition, and prove a full characterization: every uniformly recurrent indistinguishable pair with the flip condition arises from a Sturmian pair up to affine equivalence. The results generalize Morse-Hedlund-type complexity characterizations to higher dimensions and link symbolic dynamics with geometric tiling via cut-and-project theory, raising open questions about étale limits and Nivat-type questions in 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.

Figures (13)

• Figure 1: The indistinguishable asymptotic configurations $x,y\in\{\mathtt{0},\mathtt{1},\mathtt{2}\}^{\mathbb{Z}^2}$ are shown on the support $\llbracket -7,7\rrbracket \times \llbracket -7,7\rrbracket$. The two configurations are equal except on their difference set $F=\{0,-e_1,-e_2\}$ shown in red.
• Figure 2: The 8 patterns of shape $\{0,e_1,2e_1,e_2\}$ appearing in the configurations $x$ and $y$. All of them appear intersecting the difference set in $x$ and $y$.
• Figure 3: The configurations $x$ and $y$ from \ref{['fig:intro-sturmian-config-pair']} are encoding a tiling of the plane MR2074952 by three types of pointed rhombus drawn using Jolivet's notation jolivet_phd_2013. The tilings shown above correspond to the projection of the surface of a discrete plane of normal vector $(1-\alpha_1,\alpha_1-\alpha_2,\alpha_2) \approx(0.293, 0.348, 0.359)$, with $\alpha=(\alpha_1,\alpha_2)=(\sqrt{2}/2,\sqrt{19}-4)$, in 3 dimensional space, and their difference can be interpreted as the flip of a unit cube shown in yellow.
• Figure 4: An indistinguishable asymptotic pair $(c,c')$ which satisfies the flip condition obtained by taking the limit of the Sturmian configurations given by $\alpha_n = (\frac{1}{n}(\sqrt{2}-1),\frac{1}{n}(\sqrt{3}-1))$.
• Figure 5: On the left, an L-shape pattern of support $\{(1,0),(2,0),(3,0),(4,0),(4,1),(4,2),(4,3),(4,4)\}$ is shown. It is bispecial at positions $a=(0,0)$ and $b=(4,5)$ because it can be extended in more than one way at these positions within the language of the configurations $x$ and $y$ shown in \ref{['fig:intro-sturmian-config-pair']}. Thus $b-a=(4,5)\in V_\alpha$ when $\alpha=(\sqrt{2}/2,\sqrt{19}-4)$.

Theorems & Definitions (113)

• Theorem A
• Corollary 1
• Theorem B
• Corollary 2
• Corollary 3
• Conjecture 1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4