We present a technique to lift some tilings of the discrete hyperbolic plane –tilings defined by a 1D substitution– into a zero entropy subshift of finite type (SFT) on non-abelian amenable groups for . For well chosen hyperbolic tilings, this SFT is also aperiodic and minimal. As an application we construct a strongly aperiodic SFT on with a hierarchical structure, which is an analogue of Robinson’s construction on or Goodman–Strauss’s on .
Nous présentons une technique pour relever certains pavages du plan hyperbolique discret, ceux définis par une substitution 1D, au sein d’un sous-décalage de type fini (SFT) d’entropie nulle sur les groupes de Baumslag–Solitar moyennables et non abéliens avec . Lorsque ces pavages hyperboliques sont bien choisis, on montre que ce SFT est également apériodique et minimal. En guise d’application nous construisons un SFT fortement apériodique sur avec une structure hiérarchique, qui est l’analogue de la construction de Robinson sur ou de celle de Goodman–Strauss sur .
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Nathalie Aubrun 1; Michael Schraudner 2
CC-BY 4.0
@article{CRMATH_2024__362_G5_553_0,
author = {Nathalie Aubrun and Michael Schraudner},
title = {Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on {Baumslag{\textendash}Solitar} groups $\mathit{BS}(1,n)$},
journal = {Comptes Rendus. Math\'ematique},
pages = {553--580},
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volume = {362},
year = {2024},
doi = {10.5802/crmath.571},
language = {en},
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Nathalie Aubrun; Michael Schraudner. Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on Baumslag–Solitar groups $\mathit{BS}(1,n)$. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 553-580. doi : 10.5802/crmath.571. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.571/
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