### Bounded Displacement Non-Equivalence In Substitution Tilings

Frettlöh D, Smilansky Y, Solomon Y (2019)
arXiv:1907.01597.

Preprint | Englisch

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Autor*in
Frettlöh, DirkUniBi; Smilansky, Yotam; Solomon, Yaar
Einrichtung
Abstract / Bemerkung
In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets associated with tilings of Euclidean space. First we consider substitution tilings, and exhibit a substitution matrix associated with two distinct substitution rules. The first rule generates only periodic tilings, while the second generates tilings for which any associated Delone set is non-equivalent to any lattice in space. As an extension of this result, we introduce arbitrarily many distinct substitution rules associated with a single matrix, with the property that Delone sets generated by distinct rules are non-equivalent. We then turn to the study of mixed substitution tilings, and present a mixed substitution system that generates representatives of continuously many distinct BD equivalence classes.
Erscheinungsjahr
2019
Zeitschriftentitel
arXiv:1907.01597
Page URI
https://pub.uni-bielefeld.de/record/2939735

### Zitieren

Frettlöh D, Smilansky Y, Solomon Y. Bounded Displacement Non-Equivalence In Substitution Tilings. arXiv:1907.01597. 2019.
Frettlöh, D., Smilansky, Y., & Solomon, Y. (2019). Bounded Displacement Non-Equivalence In Substitution Tilings. arXiv:1907.01597
Frettlöh, D., Smilansky, Y., and Solomon, Y. (2019). Bounded Displacement Non-Equivalence In Substitution Tilings. arXiv:1907.01597.
Frettlöh, D., Smilansky, Y., & Solomon, Y., 2019. Bounded Displacement Non-Equivalence In Substitution Tilings. arXiv:1907.01597.
D. Frettlöh, Y. Smilansky, and Y. Solomon, “Bounded Displacement Non-Equivalence In Substitution Tilings”, arXiv:1907.01597, 2019.
Frettlöh, D., Smilansky, Y., Solomon, Y.: Bounded Displacement Non-Equivalence In Substitution Tilings. arXiv:1907.01597. (2019).
Frettlöh, Dirk, Smilansky, Yotam, and Solomon, Yaar. “Bounded Displacement Non-Equivalence In Substitution Tilings”. arXiv:1907.01597 (2019).

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### Quellen

arXiv: 1907.01597

Inspire: 1743116