Morphisms preserving the set of words coding three interval exchange

Any amicable pair \phi, \psi{} of Sturmian morphisms enables a construction of a ternary morphism \eta{} 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 \eta.Comment: 16 page

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

On Words with the Zero Palindromic Defect

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word

On morphisms preserving palindromic richness

It is known that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric $d$-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow to construct new rich words from already known rich words. We focus on morphisms in Class $P_{ret}$. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism $\varphi$ in the class there exists a palindrome $w$ such that $\varphi(a)w$ is a first complete return word to $w$ for each letter $a$. We characterize $P_{ret}$ morphisms which preserve richness over a binary alphabet. We also study marked $P_{ret}$ morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class $P_{ret}$ and that it preserves richness

Rauzy induction of polygon partitions and toral Z2\mathbb{Z}^2-rotations

We extend the notion of Rauzy induction of interval exchange transformations to the case of toral $\mathbb{Z}^2$-rotation, i.e., $\mathbb{Z}^2$-action defined by rotations on a 2-torus. If $\mathcal{X}_{\mathcal{P},R}$ denotes the symbolic dynamical system corresponding to a partition $\mathcal{P}$ and $\mathbb{Z}^2$-action $R$ such that $R$ is Cartesian on a sub-domain $W$, we express the 2-dimensional configurations in $\mathcal{X}_{\mathcal{P},R}$ as the image under a $2$-dimensional morphism (up to a shift) of a configuration in $\mathcal{X}_{\widehat{\mathcal{P}}|_W,\widehat{R}|_W}$ where $\widehat{\mathcal{P}}|_W$ is the induced partition and $\widehat{R}|_W$ is the induced $\mathbb{Z}^2$-action on $W$. We focus on one example $\mathcal{X}_{\mathcal{P}_0,R_0}$ for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift $X_0$ of the Jeandel-Rao Wang shift computed in an earlier work by the author. As a consequence, $\mathcal{P}_0$ is a Markov partition for the associated toral $\mathbb{Z}^2$-rotation $R_0$. It also implies that the subshift $X_0$ is uniquely ergodic and is isomorphic to the toral $\mathbb{Z}^2$-rotation $R_0$ which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.Comment: v1:36 p, 11 fig; v2:40 p, 12 fig, rewritten before submission; v3:after reviews; v4:typos and updated references; v5:typos and abstract; v6: added a paragraph commenting that Algo 1 may not halt. Jupyter notebook available at https://nbviewer.jupyter.org/url/www.slabbe.org/Publications/arXiv_1906_01104.ipyn

Inverse problems of symbolic dynamics

This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by substitutional system, and dynamical properties are considered as criteria of superword being generated by interval exchange transformation. As a consequence, one can get a morphic word appearing in interval exchange transformation such that frequencies of letters are algebraic numbers of an arbitrary degree. Concerning multydimensional systems, our main result is the following. Let P(n) be a polynomial, having an irrational coefficient of the highest degree. A word $w$ (w=(w_n), n\in \nit) consists of a sequence of first binary numbers of $\{P(n)\}$ i.e. $w_n=[2\{P(n)\}]$. Denote the number of different subwords of $w$ of length $k$ by $T(k)$ . \medskip {\bf Theorem.} {\it There exists a polynomial $Q(k)$, depending only on the power of the polynomial $P$, such that $T(k)=Q(k)$ for sufficiently great $k$.

S-adic characterization of minimal ternary dendric shifts

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive S-adic representation where the morphisms in S are positive tame automorphisms of the free group generated by the alphabet. In this paper we investigate those S-adic representations, heading towards an S-adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with 2 vertices

Extremal properties of (epi)Sturmian sequences and distribution modulo 1

Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in distribution of real numbers modulo 1 via combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some of which have been rediscovered several times