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Imbalances in Arnoux-Rauzy sequences

Julien Cassaigne, Sébastien Ferenczi, Luca Q. Zamboni (2000)

Annales de l'institut Fourier

In a 1982 paper Rauzy showed that the subshift ( X , T ) generated by the morphism 1 ↦ 12 , 2 ↦ 13 and 3 ↦ 1 is a natural coding of a rotation on the two-dimensional torus 𝕋 2 , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in ℝ 2 , each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity 2 n + 1 satisfying a combinatorial criterion known as the ☆ condition of Arnoux and Rauzy codes the orbit of a point...

Infinite periodic points of endomorphisms over special confluent rewriting systems

Julien Cassaigne, Pedro V. Silva (2009)

Annales de l’institut Fourier

We consider endomorphisms of a monoid defined by a special confluent rewriting system that admit a continuous extension to the completion given by reduced infinite words, and study from a dynamical viewpoint the nature of their infinite periodic points. For prefix-convergent endomorphisms and expanding endomorphisms, we determine the structure of the set of all infinite periodic points in terms of adherence values, bound the periods and show that all regular periodic points are attractors.

Infinite words containing squares at every position

James Currie, Narad Rampersad (2010)

RAIRO - Theoretical Informatics and Applications

Richomme asked the following question: what is the infimum of the real numbers α > 2 such that there exists an infinite word that avoids α-powers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is α = 7/3.

Integers in number systems with positive and negative quadratic Pisot base

Z. Masáková, T. Vávra (2014)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We consider numeration systems with base β and − β, for quadratic Pisot numbers β and focus on comparing the combinatorial structure of the sets Zβ and Z− β of numbers with integer expansion in base β, resp. − β. Our main result is the comparison of languages of infinite words uβ and u− β coding the ordering of distances between consecutive β- and (− β)-integers. It turns out that for a class of roots β of x2 − mx − m, the languages coincide, while for other quadratic Pisot numbers the language...

Inverse problems of symbolic dynamics

Alexei Ya. Belov, Grigorii V. Kondakov, Ivan V. Mitrofanov (2011)

Banach Center Publications

This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by a substitutional system, and dynamical properties are considered...

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