On non-repetitive complexity of Arnoux-Rauzy words

A Rauzy fractal unbounded in all directions of the plane

We construct an Arnoux-Rauzy word for which the set of all differences of two abelianized factors is equal to $\mathbb{Z}^3$. In particular, the imbalance of this word is infinite - and its Rauzy fractal is unbounded in all directions of the plane.

（最终）dendric词的态射像的一些性质

Some properties of morphic images of (eventually) dendric words

本文研究将非擦除态射应用于最终dendric词时因子复杂性的演化，并证明保持所有dendric词dendric性的态射恰好由Arnoux-Rauzy态射生成。

A generalized palindromization map in free monoids

The palindromization map $ψ$ in a free monoid $A^*$ was introduced in 1997 by the first author in the case of a binary alphabet $A$, and later extended by other authors to arbitrary alphabets. Acting on infinite words, $ψ$ generates the class of standard episturmian words, including standard Arnoux-Rauzy words. In this paper we generalize the palindromization map, starting with a given code $X$ over $A$. The new map $ψ_X$ maps $X^*$ to the set $PAL$ of palindromes of $A^*$. In this way some properties of $ψ$ are lost and some are saved in a weak form. When $X$ has a finite deciphering delay on…

Infinite words rich and almost rich in generalized palindromes

We focus on $Θ$-rich and almost $Θ$-rich words over a finite alphabet $\mathcal{A}$, where $Θ$ is an involutive antimorphism over $\mathcal{A}^*$. We show that any recurrent almost $Θ$-rich word $\uu$ is an image of a recurrent $Θ'$-rich word under a suitable morphism, where $Θ'$ is again an involutive antimorphism. Moreover, if the word $\uu$ is uniformly recurrent, we show that $Θ'$ can be set to the reversal mapping. We also treat one special case of almost $Θ$-rich words. We show that every $Θ$-standard words with seed is an image of an Arnoux-Rauzy word.

因子平衡性、线性递归与因子复杂度

Factor-balancedness, linear recurrence, and factor complexity

本文研究了无限字的因子平衡性与线性递归关系，提出通用条件并构造了具有指数复杂度的因子平衡字示例。

Palindromic complexity of infinite words associated with simple Parry numbers

A simple Parry number is a real number β>1 such that the Rényi expansion of 1 is finite, of the form d_β(1)=t_1...t_m. We study the palindromic structure of infinite aperiodic words u_βthat are the fixed point of a substitution associated with a simple Parry number β. It is shown that the word u_βcontains infinitely many palindromes if and only if t_1=t_2= ... =t_{m-1} \geq t_m. Numbers βsatisfying this condition are the so-called confluent Pisot numbers. If t_m=1 then u_βis an Arnoux-Rauzy word. We show that if βis a confluent Pisot number then P(n+1)+ P(n) = C(n+1) - C(n)+ 2, where P(n) is t…

Central sets defined by words of low factor complexity

A subset $A$ of $\mathbb{N}$ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $(x_n)_{n\in \mathbb{N}} $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing additional nice combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. In this paper we show how certain families of aperiodic words of low factor complexity may be used to …