Matrices of 3iet preserving morphisms

We study matrices of morphisms preserving the family of words coding 3-interval exchange transformations. It is well known that matrices of morphisms preserving sturmian words (i.e. words coding 2-interval exchange transformations with the maximal possible factor complexity) form the monoid $\{\boldsymbol{M}\in\mathbb{N}^{2\times 2} | \det\boldsymbol{M}=\pm1\} = \{\boldsymbol{M}\in\mathbb{N}^{2\times 2} | \boldsymbol{M}\boldsymbol{E}\boldsymbol{M}^T = \pm\boldsymbol{E}\}$, where $\boldsymbol{E} = (\begin{smallmatrix}0&1 -1&0\end{smallmatrix})$. We prove that in case of exchange of three intervals, the matrices preserving words coding these transformations and having the maximal possible subword complexity belong to the monoid $\{\boldsymbol{M}\in\mathbb{N}^{3\times 3} | \boldsymbol{M}\boldsymbol{E}\boldsymbol{M}^T = \pm\boldsymbol{E},\ \det\boldsymbol{M}=\pm 1\}$, where$\boldsymbol{E} = \Big(\begin{smallmatrix}0&1&1 -1&0&1 -1&-1&0\end{smallmatrix}\Big)$.Comment: 26 pages, 4 figure

Diophantine equations related to quasicrystals: a note

We give the general solution of three Diophantine equations in the ring of integer of the algebraic number field {\bf Q}[{\sqr 5}]. These equations are related to the problem of determination of the minimum distance in quasicrystals with fivefold symmetry.Comment: 4 page

On a class of infinite words with affine factor complexity

In this article, we consider the factor complexity of a fixed point of a primitive substitution canonically defined by a beta-numeration system. We provide a necessary and sufficient condition on the Renyi expansion of 1 for having an affine factor complexity map C(n), that is, such that C(n)=an+b for any integer n.Comment: 14 page

Asymptotic behavior of beta-integers

Beta-integers (``$\beta$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the ``crystallographic'' ordinary integers. When the number $\beta$ is a Parry number, the corresponding $\beta$-integers realize only a finite number of distances between consecutive elements and somewhat appear like ordinary integers, mainly in an asymptotic sense. In this letter we make precise this asymptotic behavior by proving four theorems concerning Parry $\beta$-integers.Comment: 17 page

Construction Of A Rich Word Containing Given Two Factors

A finite word $w$ with $\vert w\vert=n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is attained, the word $w$ is called \emph{rich}. Let \Factor(w) be the set of factors of the word $w$. It is known that there are pairs of rich words that cannot be factors of a common rich word. However it is an open question how to decide for a given pair of rich words $u,v$ if there is a rich word $w$ such that \{u,v\}\subseteq \Factor(w). We present a response to this open question:\\ If $w_1, w_2,w$ are rich words, $m=\max{\{\vert w_1\vert,\vert w_2\vert\}}$, and \{w_1,w_2\}\subseteq \Factor(w) then there exists also a rich word $\bar w$ such that \{w_1,w_2\}\subseteq \Factor(\bar w) and $\vert \bar w\vert\leq m2^{k(m)+2}$, where $k(m)=(q+1)m^2(4q^{10}m)^{\log_2{m}}$ and $q$ is the size of the alphabet. Hence it is enough to check all rich words of length equal or lower to $m2^{k(m)+2}$ in order to decide if there is a rich word containing factors $w_1,w_2$

Ito-Sadahiro numbers vs. Parry numbers

We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of −β/(β + 1) is eventually periodic. We call such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in the same context for the Rényi positive base numeration system