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Abstract Numeration Systems and Tilings

Valerie Berthe 1 Michel Rigo 2 
1 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : An abstract numeration system is a triple S = (L,Σ,<) where (Σ,<) is a totally ordered alphabet and L a regular language over Σ; the associated numeration is defined as follows: by enumerating the words of the regular language L over Σ with respect to the induced genealogical ordering, one obtains a one-to-one correspondence between ℕ and L. Furthermore, when the language L is assumed to be exponential, real numbers can also be expanded. The aim of the present paper is to associate with S a self-replicating multiple tiling of ăthe space, under the following assumption: the adjacency matrix of the trimmed minimal automaton recognizing L is primitive with a dominant eigenvalue being a Pisot unit. This construction generalizes the classical constructions performed for Rauzy fractals associated with Pisot substitutions [16], and for central tiles associated with a Pisot beta-numeration [23].
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-00106058
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Submitted on : Thursday, January 20, 2022 - 11:09:38 AM
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Valerie Berthe, Michel Rigo. Abstract Numeration Systems and Tilings. MFCS 2005 - 30th International Symposium on Mathematical Foundations of Computer Science, Aug 2005, Gdansk, Poland. pp.131-143, ⟨10.1007/11549345_13⟩. ⟨lirmm-00106058⟩

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