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Mathematics > Dynamical Systems
Title: Indistinguishable asymptotic pairs and multidimensional Sturmian configurations
(Submitted on 13 Apr 2022 (v1), last revised 25 Apr 2024 (this version, v3))
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.
Submission history
From: Sébastien Labbé [view email][v1] Wed, 13 Apr 2022 14:16:58 GMT (93kb,D)
[v2] Tue, 31 Oct 2023 22:35:48 GMT (97kb,D)
[v3] Thu, 25 Apr 2024 13:43:37 GMT (93kb,D)
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