On self-similarities of cut-and-project sets

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F17%3A00316073" target="_blank" >RIV/68407700:21340/17:00316073 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.14311/AP.2017.57.0430" target="_blank" >http://dx.doi.org/10.14311/AP.2017.57.0430</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.14311/AP.2017.57.0430" target="_blank" >10.14311/AP.2017.57.0430</a>

Result language
angličtina
Original language name
On self-similarities of cut-and-project sets
Original language description
Among the commonly used mathematical models of quasicrystals are Delone sets constructed using a cut-and-project scheme, the so-called cut-and-project sets. A cut-and-project scheme (L; pi_1; pi_2) is given by a lattice L in R^s and projections pi_1, pi_2 to suitable subspaces V1, V2. In this paper we derive several statements describing the connection between self-similarity transformations of the lattice L and transformations of its projections pi_1(L), pi_2(L). For a self-similarity of a set Sigma we take any linear mapping A such that A(Sigma)subsetSigma, which generalizes the notion of self-similarity usually restricted to scaled rotations. We describe a method of construction of cut-and-project schemes with required self-similarities and apply it to produce a cut-and-project scheme such that pi_1(L) subset R2 is invariant under an isometry of order 5. We describe all linear self-similarities of this scheme and show that they form an 8-dimensional associative algebra over the ring Z. We perform an example of a cut-and-project set with linear self-similarity which is not a scaled rotation.
Czech name
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Czech description
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Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA13-03538S" target="_blank" >GA13-03538S: Algorithms, Dynamics and Geometry of Numeration systems</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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