### Complete characterization of substitution invariant Sturmian sequences.

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We study the palindromic complexity of infinite words ${u}_{\beta}$, the fixed points of the substitution over a binary alphabet, $\varphi \left(0\right)={0}^{a}1$, $\varphi \left(1\right)={0}^{b}1$, with $a-1\ge b\ge 1$, which are canonically associated with quadratic non-simple Parry numbers $\beta $.

We study the palindromic complexity of infinite words , the fixed points of the substitution over a binary alphabet, , , with , which are canonically associated with quadratic non-simple Parry numbers .

We consider positional numeration system with negative base $-\beta $, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta $ is a quadratic Pisot number. We study a class of roots $\beta >1$ of polynomials ${x}^{2}-mx-n$, $m\ge n\ge 1$, and show that in this case the set $\mathrm{Fin}(-\beta )$ of finite $(-\beta )$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta =\tau =\frac{1}{2}(1+\sqrt{5})$, the golden ratio. For such $\beta $, we determine the exact bound on the number of fractional digits...

We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.

We study the properties of the function which determines the number of representations of an integer as a sum of distinct Fibonacci numbers . We determine the maximum and mean values of for .

We study the complexity of the infinite word ${u}_{\beta}$ associated with the Rényi expansion of $1$ in an irrational base $\beta \>1$. When $\beta $ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $\u2102\left(n\right)=n+1$. For $\beta $ such that ${d}_{\beta}\left(1\right)={t}_{1}{t}_{2}\cdots {t}_{m}$ is finite we provide a simple description of the structure of special factors of the word ${u}_{\beta}$. When ${t}_{m}=1$ we show that $\u2102\left(n\right)=(m-1)n+1$. In the cases when ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}$ or ${t}_{1}\>max\{{t}_{2},\cdots ,{t}_{m-1}\}$ we show that the first difference of the complexity function $\u2102(n+1)-\u2102\left(n\right)$ takes value in $\{m-1,m\}$ for every $n$, and consequently we determine...

We study the properties of the function $R\left(n\right)$ which determines the number of representations of an integer $n$ as a sum of distinct Fibonacci numbers ${F}_{k}$. We determine the maximum and mean values of $R\left(n\right)$ for ${F}_{k}\le n\<{F}_{k+1}$.

We add a sufficient condition for validity of Propo- sition 4.10 in the paper Frougny et al. (2004). This condition is not a necessary one, it is nevertheless convenient, since anyway most of the statements in the paper Frougny et al. (2004) use it.

We add a sufficient condition for validity of Propo- sition 4.10 in the paper Frougny (2004). This condition is not a necessary one, it is nevertheless convenient, since anyway most of the statements in the paper Frougny (2004) use it.

We consider positional numeration systems with negative real base $-\beta $, where $\beta >1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $(-\beta )$-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base ${\beta}^{2}$ with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy...

We study the complexity of the infinite word associated with the Rényi expansion of in an irrational base . When is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity . For such that is finite we provide a simple description of the structure of special factors of the word . When =1 we show that . In the cases when or max} we show that the first difference of the complexity function takes value in for every , and consequently we determine the complexity...

The aim of this article is to study certain combinatorial properties of infinite binary and ternary words associated to cut-and-project sequences. We consider here the cut-and-project scheme in two dimensions with general orientation of the projecting subspaces. We prove that a cut-and-project sequence arising in such a setting has always either two or three types of distances between adjacent points. A cut-and-project sequence thus determines in a natural way a symbolic sequence (infinite word)...

A simple Parry number is a real number $\beta \>1$ such that the Rényi expansion of $1$ is finite, of the form ${d}_{\beta}\left(1\right)={t}_{1}\cdots {t}_{m}$. We study the palindromic structure of infinite aperiodic words ${u}_{\beta}$ that are the fixed point of a substitution associated with a simple Parry number $\beta $. It is shown that the word ${u}_{\beta}$ contains infinitely many palindromes if and only if ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}\ge {t}_{m}$. Numbers $\beta $ satisfying this condition are the so-called Pisot numbers. If ${t}_{m}=1$ then ${u}_{\beta}$ is an Arnoux-Rauzy word. We show that if $\beta $ is a confluent Pisot number then $\mathcal{P}(n+1)+\mathcal{P}\left(n\right)=\mathcal{C}(n+1)-\mathcal{C}\left(n\right)+2$, where...

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