Abstract
In his Ph.D. thesis, Michal Szabados conjectured that for a not fully periodic configuration with a minimal periodic decomposition the nonexpansive lines are exactly the lines that contain a period for some periodic configuration in such decomposition. In this paper, we study Szabados’s conjecture. First, we show that we may consider a minimal periodic decomposition where each periodic configuration is defined on a finite alphabet. Then we prove that Szabados’s conjecture holds for a wide class of configurations, which includes all not fully periodic configurations with low convex pattern complexity.
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Notes
We use \(\mathbb {N} = \{1,2,\ldots \}\) and \(\mathbb {Z}_+ = \mathbb {N} \cup \{0\}\).
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Acknowledgements
The author is grateful to Etienne Moutot and Pierre Guillon for remarks about the redaction and to the Referee for suggestions that certainly have contributed to a meaningful improvement of the paper.
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Colle, C.F. On periodic decompositions and nonexpansive lines. Math. Z. 303, 80 (2023). https://doi.org/10.1007/s00209-023-03239-0
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DOI: https://doi.org/10.1007/s00209-023-03239-0