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Rauzy substitutions and multi-dimensional Sturmian words

Robert Tijdeman
2005 Theoretical Computer Science  
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

V BERTHE
2004 Theoretical Computer Science  
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

Valérie Berthé, Robert Tijdeman
2004 Theoretical Computer Science  
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]

Timo Jolivet, Anne Siegel
2015 Lecture Notes in Computer Science  
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]

Vincent Delecroix, Tomáš Hejda, Wolfgang Steiner
2013 Lecture Notes in Computer Science  
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

Amy Glen, Jacques Justin
2009 RAIRO - Theoretical Informatics and Applications  
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

Pierre Arnoux, Valerie Berthe, Hiromi Ei, Shunji Ito
2001 Discrete Mathematics & Theoretical Computer Science  
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]

Valérie Berthé, Sébastien Labbé
2011 Lecture Notes in Computer Science  
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]

Valérie Berthé, Dirk Frettlöh, Victor Sirvent
2012 arXiv   pre-print
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

Boris Adamczewski
2003 Theoretical Computer Science  
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

Valérie Berthé, Milton Minervino, Wolfgang Steiner, Jörg Thuswaldner
2016 Topology and its Applications  
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]

N. Pytheas Fogg, C. Noûs
2020 arXiv   pre-print
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]

Valérie Berthé, Milton Minervino, Wolfgang Steiner, Jörg Thuswaldner
2015 arXiv   pre-print
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

Anne Siegel, Jörg M. Thuswaldner
2009 Mémoires de la Société mathématique de France  
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

David Damanik
2005 arXiv   pre-print
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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