37 research outputs found
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
, where
.
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 $\{\boldsymbol{M}\in\mathbb{N}^{3\times 3} |
\boldsymbol{M}\boldsymbol{E}\boldsymbol{M}^T = \pm\boldsymbol{E},\
\det\boldsymbol{M}=\pm 1\}\boldsymbol{E} =
\Big(\begin{smallmatrix}0&1&1 -1&0&1 -1&-1&0\end{smallmatrix}\Big)$.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 (``-integers'') are those numbers which are the
counterparts of integers when real numbers are expressed in irrational basis
. In quasicrystalline studies -integers supersede the
``crystallographic'' ordinary integers. When the number is a Parry
number, the corresponding -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 -integers.Comment: 17 page
Construction Of A Rich Word Containing Given Two Factors
A finite word with contains at most distinct
palindromic factors. If the bound is attained, the word is called
\emph{rich}. Let \Factor(w) be the set of factors of the word . 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 if there is a rich word such that \{u,v\}\subseteq
\Factor(w). We present a response to this open question:\\ If are
rich words, , and
\{w_1,w_2\}\subseteq \Factor(w) then there exists also a rich word
such that \{w_1,w_2\}\subseteq \Factor(\bar w) and , where and is the size
of the alphabet. Hence it is enough to check all rich words of length equal or
lower to in order to decide if there is a rich word containing
factors
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