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Rauzy substitutions and multi-dimensional Sturmian words
2005
Theoretical Computer Science
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Recently, Berthé and the author developed a method to construct a multi-dimensional Sturmian word as a limit of a sequence of words on lattice domains. ...
In the present paper, the reach of the approach is studied in the special case of Rauzy substitutions. ...
Acknowledgement I am grateful to Valerie Berthé for discussions and remarks on earlier versions of the paper. ...
doi:10.1016/j.tcs.2005.08.013
fatcat:sjv6lgxylffm7cxeikjwskknvm
Lattices and multi-dimensional words
2004
Theoretical Computer Science
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We show that the produced multi-dimensional words are intrinsically connected with k-dimensional Sturmian words. ...
The construction was introduced in a special case by Simpson and Tijdeman in order to derive a multi-dimensional generalization of the theorem of Fine and Wilf. ...
Acknowledgements We thank Pierre Arnoux, Anne Siegel and Laurent Vuillon for many fruitful discussions on the subject. ...
doi:10.1016/s0304-3975(04)00101-x
fatcat:nbz5rmv4cbh43fggxzk4z2b25a
Lattices and multi-dimensional words
2004
Theoretical Computer Science
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We show that the produced multi-dimensional words are intrinsically connected with k-dimensional Sturmian words. ...
The construction was introduced in a special case by Simpson and Tijdeman in order to derive a multi-dimensional generalization of the theorem of Fine and Wilf. ...
Acknowledgements We thank Pierre Arnoux, Anne Siegel and Laurent Vuillon for many fruitful discussions on the subject. ...
doi:10.1016/j.tcs.2004.02.016
fatcat:pdlw46372jasfaxt2vhvzh62qy
Decidability Problems for Self-induced Systems Generated by a Substitution
[chapter]
2015
Lecture Notes in Computer Science
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They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). ...
We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties of the substitution underlying the fractal construction make these properties ...
In Section 2 we define Rauzy fractals and explain how they are used in the study of one-dimensional substitution dynamics. ...
doi:10.1007/978-3-319-23111-2_1
fatcat:el3vc7rc5natvlx6xjwqrgjit4
Balancedness of Arnoux-Rauzy and Brun Words
[chapter]
2013
Lecture Notes in Computer Science
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We study balancedness properties of words given by the Arnoux-Rauzy and Brun multi-dimensional continued fraction algorithms. ...
We show that almost all Brun words on 3 letters and Arnoux-Rauzy words over arbitrary alphabets are finitely balanced; in particular, boundedness of the strong partial quotients implies balancedness. ...
This work was supported by the Grant Agency of the Czech Technical University in Prague grant SGS11/162/OHK4/3T/14, Czech Science Foundation grant 13-03538S, and the ANR/FWF project "FAN -Fractals and ...
doi:10.1007/978-3-642-40579-2_14
fatcat:l65oqsbgpferfaeiolhbdhwlc4
Episturmian words: a survey
2009
RAIRO - Theoretical Informatics and Applications
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We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and ...
"episkew words" that generalize the skew words of Morse and Hedlund. ...
The authors would like to thank Jean Berstel and Pierre Arnoux for their helpful comments on a preliminary version of this paper. ...
doi:10.1051/ita/2009003
fatcat:b2fwwhlldjftjmgruabyj45xr4
Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions
2001
Discrete Mathematics & Theoretical Computer Science
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tool is a generalization of the notion of substitution. ...
continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions. ...
Rauzy substitution; this explains the difference of some properties between these tilings and the classical Rauzy tilings. ...
doi:10.46298/dmtcs.2291
fatcat:mriirdo67jfdva7nso2cdonaqy
An Arithmetic and Combinatorial Approach to Three-Dimensional Discrete Lines
[chapter]
2011
Lecture Notes in Computer Science
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We associate with the steps of the algorithm substitutions, that is, rules that replace letters by words, which allow us to generate the Freeman coding of a discrete segment. ...
The aim of this paper is to discuss from an arithmetic and combinatorial viewpoint a simple algorithmic method of generation of discrete segments in the three-dimensional space. ...
Note that the use of generalized substitutions (see Section 2.2) associated with multi-dimensional continued fraction algorithms has also already proved its efficiency in discrete geometry for the generation ...
doi:10.1007/978-3-642-19867-0_4
fatcat:evtsuuzruvdbzakwoioagsgofi
Selfdual Substitutions in Dimension One
[article]
2012
arXiv
pre-print
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There are several notions of the 'dual' of a word/tile substitution. ...
Furthermore, we obtain necessary and sufficient arithmetic conditions for substitutions being selfdual in this case. ...
DF and VS would like to express their gratitude to the German Research Council (DFG) within the CRC 701. ...
arXiv:1108.5053v2
fatcat:tg6unwssunhbzppv75cqezijkq
Balances for fixed points of primitive substitutions
2003
Theoretical Computer Science
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An inÿnite word deÿned over a ÿnite alphabet A is balanced if for any pair (!; ! ) of factors of the same length and for any letter a in the alphabet !|a − |! |a| 6 1; where |! ...
|a denotes the number of occurrences of the letter a in the word !. In this paper, we generalize this notion and introduce a measure of balance for an inÿnite sequence. ...
For instance, Berthà e and Tijdeman [7] study balance properties of multi-dimensional words; in particular balanced multi-dimensional words are proved to be fully periodic. ...
doi:10.1016/s0304-3975(03)00092-6
fatcat:vf6chlqxtvgjnnelwugprrglxu
The S -adic Pisot conjecture on two letters
2016
Topology and its Applications
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The proof relies on the use of Rauzy fractals and on the fact that strong coincidences hold in this framework. ...
We prove an extension of the well-known Pisot substitution conjecture to the S-adic symbolic setting on two letters. ...
They occur e.g. under the term "mixed substitution" or "multi-substitution" [GM13, PV13] , or else as "fusion systems" [PFS14a, PFS14b] . ...
doi:10.1016/j.topol.2016.01.019
fatcat:vhqo2tf6zbgwzlkavjyazfznpq
Symbolic coding of linear complexity for generic translations of the torus, using continued fractions
[article]
2020
arXiv
pre-print
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The partitions are constructed with Rauzy fractals associated with sequences of substitutions, which are produced by a particular extended continued fraction algorithm in projective dimension 2. ...
We give some conditions which imply the existence, for almost every direction, of a translation of the torus T^d and a nice generating partition, such that the associated coding is a conjugacy with the ...
Thanks The authors would like to thank Mélodie Andrieu, Nicolas Bédaride, Jean-François Bertazzon, Julien Cassaigne, Paul Mercat, and Thierry Monteil for their help in preparing this article. ...
arXiv:2005.12229v1
fatcat:3s3dddgp4bdw7nkvvfvawf5k24
The S-adic Pisot conjecture on two letters
[article]
2015
arXiv
pre-print
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The proof relies on the use of Rauzy fractals and on the fact that strong coincidences hold in this framework. ...
We prove an extension of the well-known Pisot substitution conjecture to the S-adic symbolic setting on two letters. ...
They occur e.g. under the term 'mixed substitutions' or 'multi-substitution' [GM13, PV13] , or else as 'fusion systems' [PFS14a, PFS14b] . ...
arXiv:1501.07085v1
fatcat:wpoo3o3ehrckzjy5brp3e5tqly
Topological properties of Rauzy fractals
2009
Mémoires de la Société mathématique de France
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In the present monograph we deal with topological and geometric properties of substitutions, i.e., we study properties of the Rauzy fractals associated to substitutions. ...
Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. ...
Loridant and M. ...
doi:10.24033/msmf.430
fatcat:gyfniae3i5hpliepf5zjbqokmm
Strictly Ergodic Subshifts and Associated Operators
2005
arXiv
pre-print
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We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices. ...
There is an extension to multi-dimensional Schrödinger operators by Kiselev and Last [96] . Two remarks are in order. ...
Here, p is allowed to be the empty word and, if p is not empty, q a may be the empty word. Clearly, if a subshift is generated by a class P substitution, it contains arbitrarily long palindromes. ...
arXiv:math/0509197v1
fatcat:bzzietmthzgvfgt5x5jgo3h34a
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