Lubomira Dvorakova, Tereza Velka
We introduce a class of fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words. We conjecture that this class contains all fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words that are not standard Sturmian words.
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Generalized pseudostandard words were introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. In this paper we provide a necessary and sufficient condition for their periodicity over binary and ternary alphabet. More precisely, we describe how the directive bi-sequence of a generalized pseudostandard word has to look like in order to correspond to a periodic word. We state moreover a conjecture concerning a necessary and su…
Generalized pseudostandard words were introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. We present a counterexample to Conjecture 43 from a paper by Blondin Massé et al. that estimated the complexity of binary generalized pseudostandard words as ${\mathcal C}(n) \leq 4n$ for all sufficiently large $n$. We conjecture that ${\mathcal C}(n)<6n$ for all $n \in \mathbb N$.
This paper focuses on generalized pseudostandard words, defined by de Luca and De Luca in 2006. In every step of the construction, the involutory antimorphism to be applied for the pseudopalindromic closure changes and is given by a so called directive bi-sequence. The concept of a normalized form of directive bi-sequences was introduced by Blondin-Massé et al. in 2013 and an algorithm for finding the normalized directive bi-sequence over a binary alphabet was provided. In this paper, we present an algorithm to find the normalized form of any directive bi-sequence over a ternary alphabet. More…
Generalized pseudostandard words have been introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. We present two new results concerning these words. The first one is a necessary and sufficient condition for their periodicity. The second result is a counterexample to Conjecture 43 from the paper: A. B. Masse, G.Paquin, H. Tremblay, and L. Vuillon, On Generalized Pseudostandard Words over Binary Alphabet (Journal of Int. Seq…
Generalized pseudostandard word $\bf u$, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet ${\cal A}$ and by a directive sequence of involutory antimorphisms acting on ${\cal A}^*$. Prefixes of $\bf u$ with increasing length are constructed using pseudopalindromic closure operator. We show that generalized Thue--Morse words ${\bf t}_{b,m}$, with $b, m \in \N$ and $b, m \geq 2$, are generalized pseudostandard words if and only if ${\bf t}_{b,m}$ is a periodic word or $b \leq m$. This extends the result of de Luca and De Luca obtained for …
First introduced in the study of the Sturmian words, the iterated palindromic closure was recently generalized to pseudopalindromes. This operator allows one to construct words with an infinity of pseudopalindromic prefixes, called pseudostandard words. We provide here several combinatorial properties of the fixed points under the iterated pseudopalindromic closure.
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.
Any infinite uniformly recurrent word ${\bf u}$ can be written as concatenation of a finite number of return words to a chosen prefix $w$ of ${\bf u}$. Ordering of the return words to $w$ in this concatenation is coded by derivated word $d_{\bf u}(w)$. In 1998, Durand proved that a fixed point ${\bf u}$ of a primitive morphism has only finitely many derivated words $d_{\bf u}(w)$ and each derivated word $d_{\bf u}(w)$ is fixed by a primitive morphism as well. In our article we focus on Sturmian words fixed by a primitive morphism. We provide an algorithm which to a given Sturmian morphism $ψ$ …
An infinite word, which is aperiodic and codes the orbit of a transformation of the exchange of three intervals is called 3iet word. Such a word is thus a natural generalization of a sturmian word to a word over 3-letter alphabet. A morphism is said to be 3iet preserving if it maps any 3iet word to another 3iet word. It is known that the monoid of morphisms preserving sturmian words is finitely generated. On the contrary, in this note we prove that the monoid of 3iet preserving morphisms is not finitely generated, that is, there are infinitely many 3iet preserving morphisms, which cannot be wr…
We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism $\mathrm{E}:\{0,1\}^*\to\{0,1\}^*$, defined by $\mathrm{E}(w_1\cdots w_n)=(1-w_{n})\cdots(1-w_1)$. We conjecture that these two classes contain all morphisms (up to conjugation) which generate infinite words with infinitely many antipalindromes. This is an analogue to the famous HKS conjecture concerning infinite words containing infinitely many palindromes. We prove our conjectu…
Any amicable pair ϕ, ψ of Sturmian morphisms enables a construction of a ternary morphism η which preserves the set of infinite words coding 3-interval exchange. We determine the number of amicable pairs with the same incidence matrix in $SL^\pm(2,N)$ and we study incidence matrices associated with the corresponding ternary morphisms η.
We study the palindromic length of factors of infinite words fixed by morphisms of the so-called class $\mathcal{P}$ introduced by Hof, Knill and Simon. We show that it grows at most logarithmically with the length of the factor. For the Fibonacci word and the Thue-Morse word we provide estimates on the constants of the growth. We also construct an infinite word rich in palindromes for which the palindromic length grows as $\sqrt{n}$.
In 1982, Seebold showed that the only overlap-free binary words that are the fixed points of non-identity morphisms are the Thue-Morse word and its complement. We strengthen Seebold's result by showing that the same result holds if the term 'overlap-free' is replaced with '7/3-power-free'. Furthermore, the number 7/3 is best possible.
We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a m…
A graph $G=(V,E)$ is word-representable if and only if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x\neq y$, alternate in $w$ if and only if $xy\in E$. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. There is a long line of research on word-representable graphs in the literature, and recently, word-representability of split graphs has attracted interest. In this paper, we first give a characterization of word-representable split graphs in terms of permutations of columns of the adjacency matrices. Then, w…
We study morphisms from certain classes and their action on episturmian words. The first class is $P_{ret}$. In general, a morphism of class $P_{ret}$ can map an infinite word having zero palindromic defect to a word having infinite palindromic defect. We show that the image of an episturmian word, which has zero palindromic defect, under a morphism of class $P_{ret}$ has always its palindromic defect finite. We also focus on letter-to-letter morphisms to binary alphabet: we show that images of ternary episturmian words under such morphisms have zero palindromic defect. These results contribut…
In this paper, we provide a new characterization of uniformly recurrent words with finite defect based on a relation between the palindromic and factor complexity. Furthermore, we introduce a class of morphisms P_ret closed under composition and we show that a uniformly recurrent word with finite defect is an image of a rich (also called full) word under a morphism of class P_ret. This class is closely related to the well-known class P defined by Hof, Knill, and Simon; every morphism from P_ret is conjugate to a morphism of class P.
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 interv…
We observe that word reading is a crystal morphism. This leads us to prove that, in the case of the complex special linear group, the map from all galleries to MV cycles is a morphism of crystals.
Weakly and strongly quasiperiodic morphisms are tools introduced to study quasiperiodic words. Formally they map respectively at least one or any non-quasiperiodic word to a quasiperiodic word. Considering them both on finite and infinite words, we get four families of morphisms between which we study relations. We provide algorithms to decide whether a morphism is strongly quasiperiodic on finite words or on infinite words.
We consider an infinite word $\boldsymbol{u}$ fixed by a primitive morphism. We show a necessary condition under which $\boldsymbol{u}$ has a non-trivial geometric representation which is bounded distance equivalent to a lattice.
Morphic words are letter-to-letter images of fixed points $x$ of morphisms on finite alphabets. There are situations where these letter-to-letter maps do not occur naturally, but have to be replaced by a morphism. We call this a decoration of $x$. Theoretically, decorations of morphic words are again morphic words, but in several problems the idea of decorating the fixed point of a morphism is useful. We present two of such problems. The first considers the so called $AA$ sequences, where $α$ is a quadratic irrational, $A$ is the Beatty sequence defined by $A(n)=\lfloor αn\rfloor$, and $AA$ is…
We say $x \in \{0,1,2 \}^{\NN}$ is a word with Sturmian erasures if for any $a\in \{0,1,2 \}$ the word obtained erasing all $a$ in $x$ is a Sturmian word. A large family of such words is given coding trajectories of balls in the game of billiards in the cube. We prove that the monoid of morphisms mapping all words with Sturmian erasures to words with Sturmian erasures is not finitely generated.
Mapping words to powers by morphisms
本文刻画了能被单射态射映射至任意高次幂的字，证明了其余字的最高次幂线性上界，并证明判定字能否经非周期态射映射为 $n$ 次幂或非本原字是 NP-hard 且属于 PSPACE 的。
Permutation complexity of images of Sturmian words by marked morphisms
本文证明了 Sturmian 词经二元标记态射作用后的像，其置换复杂度在足够长时呈线性形式 n+k。
Pairs of square-free arithmetic progressions in infinite words
研究互质整数p,q≥3时，是否存在无限三元无平方单词，其模p和模q的子序列也是无平方的，证明了满足条件的(p,q)对只有有限个，并引入了无平方转换器的概念。
A word is cubefree if it contains no non-empty subword of the form xxx. A morphism h : Sigma^* -> Sigma^* is k-uniform if h(a) has length k for all a in Sigma. A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all k >= 0 there exists a k-uniform cubefree binary morphism.
The analogue of overlap-freeness for the Fibonacci morphism
本文定义了不含$4^-$-幂和因子11的faux-bonacci词，证明其与Fibonacci态射的关系等同于overlap-free词与Thue-Morse态射的关系，并给出了Fife定理的类似物及字典序极值无穷词的刻画。
This article is concerned with characteristic Sturmian words of slope $α$ and $1-α$ (denoted by $c_α$ and $c_{1-α}$ respectively), where $α\in (0,1)$ is an irrational number such that $α= [0;1+d_1,\bar{d_2,...,d_n}]$ with $d_n \geq d_1 \geq 1$. It is known that both $c_α$ and $c_{1-α}$ are fixed points of non-trivial (standard) morphisms $σ$ and $\hatσ$, respectively, if and only if $α$ has a continued fraction expansion as above. Accordingly, such words $c_α$ and $c_{1-α}$ are generated by the respective morphisms $σ$ and $\hatσ$. For the particular case when $α= [0;2,\bar{r}]$ ($r\geq1$), we…
We study infinite words fixed by a morphism and their derived words. A derived word is a coding of return words to a factor. We exhibit two examples of sets of morphisms which are closed under derivation --- any derived word with respect to any factor of the fixed point is again fixed by a morphism from this set. The first example involves standard episturmian morphisms, and the second concerns the period doubling morphism.
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。
WELLDOC property for words generated by morphisms
本文研究了由形变生成的无限词的WELLDOC性质，提出了基于Parikh向量的判据，用于判断无限词是否满足特定的分布特性。
In this work we consider morphisms that preserve well-known non-repeating properties: squarefreeness, cubefreeness, overlap-freeness and weak squarefreeness. Up to the present moment only the morphisms preserving three out of four non-repeating properties have been known. The problem of the existence of weakly squarefree morphisms was open. The essential result of this work is the positive solution to this problem. An example of the morphism preserving all four properties is provided. Also, it is proved that there are no morphisms with the same properties and a lower rank.
The Narayana Morphism and Related Words
本文利用自动机理论研究 Narayana 态射的不动点词及相关词的性质。
On Occurrence-Preserving Morphisms
本文研究了保持出现的映射，引入了无干扰映射的概念，并通过该概念对保持出现的映射进行了精确刻画，应用于Fibonacci和Thue-Morse词以识别其最小唯一子字符串，并简化了现有关于净出现的证明。
We investigate the computational power of periodically iterated morphisms, also known as D0L systems with periodic control, PD0L systems for short. These systems give rise to a class of one-sided infinite sequences, called PD0L words. We construct a PD0L word with exponential subword complexity, thereby answering a question raised by Lepisto (1993) on the existence of such words. We solve another open problem concerning the decidability of the first-order theories of PD0L words; we show it is already undecidable whether a certain letter occurs in a PD0L word. This stands in sharp contrast to t…
Fixed points ${\bf u}=φ({\bf u})$ of marked and primitive morphisms $φ$ over arbitrary alphabet are considered. We show that if ${\bf u}$ is palindromic, i.e., its language contains infinitely many palindromes, then some power of $φ$ has a conjugate in class ${\mathcal P}$. This class was introduced by Hof, Knill, Simon (1995) in order to study palindromic morphic words. Our definitions of marked and well-marked morphisms are more general than the ones previously used by Frid (1999) or Tan (2007). As any morphism with aperiodic fixed point over binary alphabet is marked, our result generalizes…
Mapped Exponent and Asymptotic Critical Exponent of Words
本文研究单射态射对单词重复性的影响，刻画了映射后分数指数可任意高的有限单词，并证明无限单词的渐近临界指数至多增长一个与字母表大小有关的常数因子。
On the Asymptotic Palindrome Density of Fibonacci Infinite Words
本文研究Fibonacci型形态生成的无限词的回文密度，推导出其密度公式和界限，并证明了回文前缀密度的上限为黄金比例的倒数。
Two words are $k$-binomially equivalent if each subword of length at most $k$ occurs the same number of times in both words. The $k$-binomial complexity of an infinite word is a counting function that maps $n$ to the number of $k$-binomial equivalence classes represented by its factors of length $n$. Cassaigne et al. [Int. J. Found. Comput. S., 22(4) (2011)] characterized a family of morphisms, which we call Parikh-collinear, as those morphisms that map all words to words with bounded $1$-binomial complexity. Firstly, we extend this characterization: they map words with bounded $k$-binomial co…
We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F_2) of automorphisms of the rank two free group F_2 and show that it can be realized as a monoid in the group B_4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F_2 lifting any given basis of the free abelian group Z^2. We further give an algorithm allowing to decide whether two elements of F_2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palind…
The Fibonacci word $W$ on an infinite alphabet was introduced in [Zhang et al., Electronic J. Combinatorics 2017 24(2), 2-52] as a fixed point of the morphism $2i\rightarrow (2i)(2i+1)$, $(2i+1) \rightarrow (2i+2)$, $i\geq 0$. Here, for any integer $k>2$, we define the infinite $k$-bonacci word $W^{(k)}$ on the infinite alphabet as the fixed point of the morphism $φ_k$ on the alphabet $\mathbb{N}$ defined for any $i\geq 0$ and any $0\leq j\leq k-1$, as \begin{equation*} φ_k(ki+j) = \left\{ \begin{array}{ll} (ki)(ki+j+1) & \text{if } j = 0,\cdots ,k-2,\\ (ki+j+1)& \text{otherwise}. \end{array} …
Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing a common episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.
Power-free Complementary Binary Morphisms
本文研究二元互补态射的无幂性质，证明Thue-Morse词前缀生成3^+-无态射，并刻画无立方情形的长度存在性及其α-无边界。
Complementary symmetric Rote sequences are binary sequences which have factor complexity $\mathcal{C}(n) = 2n$ for all integers $n \geq 1$ and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derivated sequences to the prefixes of any complementary symmetric Rote sequence $\mathbf{v}$ which is associated with a standard Sturmian sequence $\mathbf{u}$. We show that any non-empty prefix of $\mathbf{v}$ has three return words. We prove that any derivated sequence of …
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.
G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism $τ: 0\mapsto 01, 1\mapsto 02 and 2\mapsto 0$ is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $R^2$, each domain being translated by the same vector modulo a lattice. In this paper we study the Abelian complexity AC(n) of the Tribonacci word $t$ which is the unique fixed point of $τ$. We show that $AC(n)\in {3,4,5,6,7}$ for each $n\geq 1$, and that each of these five values is assumed. Our proof relies on the fact that the Tribonacci word is 2-balanced, i.e., for all fac…
We describe factor frequencies of the generalized Thue-Morse word t_{b,m} defined for integers b greater than 1, m greater than 0 as the fixed point starting in 0 of the morphism ϕ_{b,m} given by ϕ_{b,m}(k)=k(k+1)...(k+b-1), where k = 0,1,..., m-1 and where the letters are expressed modulo m. We use the result of A. Frid, On the frequency of factors in a D0L word, Journal of Automata, Languages and Combinatorics 3 (1998), 29-41 and the study of generalized Thue-Morse words by S. Starosta, Generalized Thue-Morse words and palindromic richness, arXiv:1104.2476v2 [math.CO].
In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of so-called "rich words" is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with X. Droubay and G. Pirillo in 2001. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of w…
In a recent work, A. Berger and C. Defant showed that if $x$ is a fixed point of a binary uniform and primitive morphism, then there exists a constant $C$ such that for all positive integers $i,k,$ beginning in position $n$ in $x$ is a $k$-anti-power with block length at most $Ck$. They ask whether this result extends to a broader class of morphic words. In this note we extend their results to fixed points of uniform primitive morphisms on arbitrary finite alphabets. Our methods make use of the recognisability of uniform primitive morphisms. This result was proved independantly by S. Garg, usi…
Motivated by applications in the theory of numeration systems and recognizable sets of integers, this paper deals with morphic words when erasing morphisms are taken into account. Cobham showed that if an infinite word $w =g(f^ω(a))$ is the image of a fixed point of a morphism $f$ under another morphism $g$, then there exist a non-erasing morphism $σ$ and a coding $τ$ such that $w =τ(σ^ω(b))$. Based on the Perron theorem about asymptotic properties of powers of non-negative matrices, our main contribution is an in-depth study of the growth type of iterated morphisms when one replaces erasing m…
We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite words are letter-to-letter projections of fixed points of these morphisms. We characterize the first differences of both functions a) with generalized Beatty sequences, or unions of generalized Beatty sequences, and b) with morphic sequences.
Let $A$ be a finite or countable alphabet and let $θ$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $θ$ ($θ$-invariant for short).We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin $θ$-invariant codes, maximality and completeness are two equivalent notions. We prove that a similar property holds in the framework of some special families of $θ$-invariant codes such as prefix (bifix) codes, codes with a finite dec…
Avoiding fractional powers over the natural numbers
本研究通过自动化方法确定自然数上字典序最小分数幂自由词的结构，涵盖多个有理数族及零星情形，并揭示其与有限字母表及自动序列的联系。
Fici, Restivo, Silva, and Zamboni define a $\textit{$k$-anti-power}$ to be a concatenation of $k$ consecutive words that are pairwise distinct and have the same length. They ask for the maximum $k$ such that every aperiodic recurrent word must contain a $k$-anti-power, and they prove that this maximum must be 3, 4, or 5. We resolve this question by demonstrating that the maximum is 5. We also conjecture that if $W$ is a reasonably nice aperiodic morphic word, then there is some constant $C = C(W)$ such that for all $i,k\geq 1$, $W$ contains a $k$-anti-power with blocks of length at most $Ck$ b…
We prove that the generalized Thue-Morse word $\mathbf{t}_{b,m}$ defined for $b \geq 2$ and $m \geq 1$ as $\mathbf{t}_{b,m} = (s_b(n) \mod m)_{n=0}^{+\infty}$, where $s_b(n)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering simultaneously antimorphisms $Θ\in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $Θ$-palindromes up to the highest possible level. Using the terminology generalizing the notion o…
We introduce and study natural derivatives for Christoffel and finite standard words, as well as for characteristic Sturmian words. These derivatives, which are realized as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map $a$ to $a^{k+1}b$ (resp.,~$ab^{k}$) and $b$ to $a^{k}b$ (resp.,~$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are longer than one letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or standard word as the smallest order for w…
We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Fibonacci-automatic". This class includes, for example, the famous Fibonacci word f = 01001010..., the fixed point of the morphism 0 -> 01 and 1 -> 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth. As an application of our method we prove a new result: there exists an aperiodic infinite binary word avoiding the …
We show that the equality language of two non-periodic binary morphisms is generated by at most two words. If its rank is two, then the generators start (and end) with different letters. This in particular implies that any binary language has a test set of cardinality at most two.
In a 1995 paper, Hof, Knill and Simon obtain a sufficient combinatorial criterion on the hull $Ω$ of the potential of a discrete Schrödinger operator which guarantees purely singular continuous spectrum on a generic subset of $Ω.$ In part, this condition requires the existence of infinitely many palindromic factors. In this same paper, they introduce the class P of morphisms $f:A^*\rightarrow B^*$ of the form $a\mapsto pq_a$ and ask whether every palindromic subshift generated by a primitive substitution arises from morphisms of class P or by morphisms of the form $a\mapsto q_ap$ where again $…
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