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Indistinguishable asymptotic pairs and multidimensional Sturmian configurations
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 45 / Issue 2 / February 2025
- Published online by Cambridge University Press:
- 31 May 2024, pp. 337-395
- Print publication:
- February 2025
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- Article
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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.
