Balance properties of the fixed point  of the substitution
associated to quadratic simple Pisot numbers

Ondřej Turek

Displaying similar documents to “Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers”

Palindromes in infinite ternary words

L&amp;#039;ubomíra Balková, Edita Pelantová, Štěpán Starosta (2009)

RAIRO - Theoretical Informatics and Applications

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We study infinite words over an alphabet $𝒜$ satisfying the property $𝒫 : 𝒫 (n) + 𝒫 (n + 1) = 1 + # 𝒜 for any n \in ℕ$ , where $𝒫 (n)$ denotes the number of palindromic factors of length occurring in the language of . We study also infinite words satisfying a stronger property $𝒫ℰ$ : every palindrome of has exactly one palindromic extension in . For binary words, the properties $𝒫$ and $𝒫ℰ$ coincide and these properties characterize Sturmian words, i.e., words with the complexity + 1 for any $n \in ℕ$ . In this paper, we focus on ternary infinite...

Two-variable word equations

Lucian Ilie, Wojciech Plandowski (2010)

RAIRO - Theoretical Informatics and Applications

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We consider languages expressed by word equations in two variables and give a complete characterization for their complexity functions, that is, the functions that give the number of words of the same length. Specifically, we prove that there are only five types of complexities: constant, linear, exponential, and two in between constant and linear. For the latter two, we give precise characterizations in terms of the number of solutions of Diophantine equations of certain types. In...

Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words

Ondřej Turek (2010)

RAIRO - Theoretical Informatics and Applications

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A word defined over an alphabet $𝒜$ is -balanced ( $ℕ$ ) if for all pairs of factors , of of the same length and for all letters $𝒜$ , the difference between the number of letters in and is less or equal to . In this paper we consider a ternary alphabet $𝒜$ = {, , } and a class of substitutions $ϕ_{p}$ defined by $ϕ_{p}$ () = , $ϕ_{p}$ () = , $ϕ_{p}$ () = where 1. We prove that the fixed point of $ϕ_{p}$ , formally written as $ϕ_{p}^{\infty}$ (), is 3-balanced...

On the distribution of free path lengths for the periodic Lorentz gas II

François Golse, Bernt Wennberg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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Consider the domain $Z_{ϵ} = {x \in ℝ^{n}; d i s t (x, ϵ ℤ^{n}) > ϵ^{γ}}$ and let the free path length be defined as $τ_{ϵ} (x, v) = inf {t > 0; x - t v \in Z_{ϵ}} .$ In the Boltzmann-Grad scaling corresponding to $γ = \frac{n}{n - 1}$ , it is shown that the limiting distribution $φ_{ϵ}$ of $τ_{ϵ}$ is bounded from below by an expression of the form , for some . A numerical study seems to indicate that asymptotically for large , $φ_{ϵ} \sim C / t$ . This is an extension of a previous work [J. Bourgain , (1998) 491-508]. As a consequence, it is proved that the linear Boltzmann type transport equation is inappropriate...

The Helping Hierarchy

Patrizio Cintioli, Riccardo Silvestri (2010)

RAIRO - Theoretical Informatics and Applications

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Schöning [14] introduced a notion of helping and suggested the study of the class $P_{help} (𝒞)$ of the languages that can be helped by oracles in a given class $𝒞$ . Later, Ko [12], in order to study the connections between helping and "witness searching" , introduced the notion of self-helping for languages. We extend this notion to classes of languages and show that there exists a self-helping class that we call SH which contains all the self-helping classes. We introduce the Helping hierarchy whose levels...