Substitution dynamical systems on infinite alphabets

Sébastien Ferenczi

Displaying similar documents to “Substitution dynamical systems on infinite alphabets”

Palindromic complexity of infinite words associated with simple Parry numbers

Petr Ambrož, Zuzana Masáková, Edita Pelantová, Christiane Frougny (2006)

Annales de l’institut Fourier

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A simple Parry number is a real number $β > 1$ such that the Rényi expansion of $1$ is finite, of the form $d_{β} (1) = t_{1} \dots t_{m}$ . We study the palindromic structure of infinite aperiodic words $u_{β}$ that are the fixed point of a substitution associated with a simple Parry number $β$ . It is shown that the word $u_{β}$ contains infinitely many palindromes if and only if $t_{1} = t_{2} = \dots = t_{m - 1} \geq t_{m}$ . Numbers $β$ satisfying this condition are the so-called Pisot numbers. If $t_{m} = 1$ then $u_{β}$ is an Arnoux-Rauzy word. We show that if $β$ is a confluent Pisot number then...

Substitutions, abstract number systems and the space filling property

Clemens Fuchs, Robert Tijdeman (2006)

Annales de l’institut Fourier

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In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo $1$ and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.

Dynamical directions in numeration

Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner (2006)

Annales de l’institut Fourier

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This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to $β$ -numeration and its generalisations, abstract numeration...

Weak mixing and eigenvalues for Arnoux-Rauzy sequences

Julien Cassaigne, Sébastien Ferenczi, Ali Messaoudi (2008)

Annales de l’institut Fourier

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We define by simple conditions two wide subclasses of the so-called Arnoux-Rauzy systems; the elements of the first one share the property of (measure-theoretic) weak mixing, thus we generalize and improve a counter-example to the conjecture that these systems are codings of rotations; those of the second one have eigenvalues, which was known hitherto only for a very small set of examples.

Billiard complexity in the hypercube

Nicolas Bedaride, Pascal Hubert (2007)

Annales de l’institut Fourier

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We consider the billiard map in the hypercube of $ℝ^{d}$ . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that $n^{3 d - 3}$ is the order of magnitude of the complexity.