arXiv subjects

Sébastien Labbé

Publications and source records attributed to Sébastien Labbé.

At least 19 recordsLinked to original sources

arXiv2026

An aperiodic set of Wang tiles for every quadratic irrational

We propose a sufficient condition for the non-periodicity of a set of Wang tiles. It applies to sets of Wang tiles whose tiles have vertical or horizontal stripes. The proof is based on a geometric argument involving a quadrilateral circumscribed to a parabola from which we conclude the irrationality of the densities of the vertical and horizontal stripes. We apply the sufficient condition to propose new proofs of non-periodicity of known sets of Wang tiles, including an encoding of Penrose tilings into 24 Wang tiles and the family of metallic mean Wang tiles. Conversely, for every pair $(α,β)\in[0,1]^2$ of irrational numbers in the same quadratic number field, we construct a finite aperiodic set of Wang tiles with stripes that admits a valid tiling whose density of vertical stripes is $α$ and density of horizontal stripes is $β$.

math.NT↗

arXiv2026

Positionality of Dumont--Thomas numeration systems for integers

Introduced in 2001 by Lecomte and Rigo, abstract numeration systems provide a way of expressing natural numbers with words from a language $L$ accepted by a finite automaton. As it turns out, these numeration systems are not necessarily positional, i.e., we cannot always find a sequence $U=(U_i)_{i\ge 0}$ of integers such that the value of every word in the language $L$ is determined by the position of its letters and the first few values of $U$. Finding the conditions under which an abstract numeration system is positional seems difficult in general. In this paper, we thus consider this question for a particular sub-family of abstract numeration systems called Dumont--Thomas numeration systems. They are derived from substitutions and were introduced in 1989 by Dumont and Thomas. We exhibit conditions on the underlying substitution so that the corresponding Dumont--Thomas numeration is positional. We first work in the most general setting, then particularize our results to some practical cases. Finally, we link our numeration systems to existing literature, notably properties studied by Rényi in 1957, Parry in 1960, Bertrand-Mathis in 1989, and Fabre in 1995

math.CO↗

arXiv2026

A construction of the hat tilings by a Markov partition

We present a simple construction of hat tilings. The construction can be carried out by superimposing a triangular grid on a specially colored image and reading off the orientation of the tiles. We show that our construction produces valid hat tilings, and conversely, in an appropriate sense that is made precise in the paper, that every valid hat tiling can be obtained in this way.

math.CO↗

arXiv2025

$q$-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings

We consider the $q$-deformation of rational numbers introduced recently by Morier-Genoud and Ovsienko. We propose three enumerative interpretations of these $q$-rationals: in terms of a new version of Ostrowski's numeration system for integers, in terms of order ideals of fence posets and in terms of perfect matchings of snake graphs. Contrary to previous results which are restricted to rational numbers greater than one, our interpretations work for all positive rational numbers and are based on a single combinatorial object for defining both the numerator and denominator. The proofs rest on order-preserving bijections between posets over these objects. We recover a formula for a $q$-analog of Markoff numbers. We also deduce a fourth interpretation given in terms of the integer points inside a polytope in $\mathbb{R}^k$ on both sides of a hyperplane where $k$ is the length of the continued fraction expansion.

math.CO↗

arXiv2025

Metallic mean Wang tiles II: the dynamics of an aperiodic computer chip

We consider a new family $(\mathcal{T}_n)_{n\geq1}$ of aperiodic sets of Wang tiles and we describe the dynamical properties of the set $Ω_n$ of valid configurations $\mathbb{Z}^2\to\mathcal{T}_n$. The tiles can be defined as the different instances of a square-shaped computer chip whose inputs and outputs are 3-dimensional integer vectors. The family include the Ammann aperiodic set of 16 Wang tiles and gathers the hallmarks of other small aperiodic sets of Wang tiles. Notably, the tiles satisfy additive versions of equations verified by the Kari--Culik aperiodic sets of 14 and 13 Wang tiles. Also configurations in $Ω_n$ are the codings of a $\mathbb{Z}^2$-action on a 2-dimensional torus like the Jeandel--Rao aperiodic set of 11 Wang tiles. The family broadens the relation between quadratic integers and aperiodic tilings beyond the omnipresent golden ratio as the dynamics of $Ω_n$ involves the positive root $β$ of the polynomial $x^2-nx-1$, also known as the $n$-th metallic mean. We show the existence of an almost one-to-one factor map $Ω_n\to\mathbb{T}^2$ which commutes with the shift action on $Ω_n$ with horizontal and vertical translations by $β$ on $\mathbb{T}^2$. The factor map can be explicitly defined by the average of the top labels from the same row of tiles as in Kari and Culik examples. The proofs are based on the minimality of $Ω_n$ (proved in a previous article) and a polygonal partition of $\mathbb{T}^2$ which we show is a Markov partition for the toral $\mathbb{Z}^2$-action. The partition and the sets of Wang tiles are symmetric which makes them, like Penrose tilings, worthy of investigation.

math.DS↗

arXiv2025

Three characterizations of a self-similar aperiodic 2-dimensional subshift

The goal of this chapter is to illustrate a generalization of the Fibonacci word to the case of 2-dimensional configurations on $\mathbb{Z}^2$. More precisely, we consider a particular subshift of $\mathcal{A}^{\mathbb{Z}^2}$ on the alphabet $\mathcal{A}=\{0,\dots,15\}$ for which we give three characterizations: as the subshift $\mathcal{X}_Φ$ generated by a 2-dimensional morphism $Φ$ defined on $\mathcal{A}$; as the Wang shift $Ω_\mathcal{Z}$ defined by a set $\mathcal{Z}$ of 16 Wang tiles; as the symbolic dynamical system $\mathcal{X}_{\mathcal{P}_\mathcal{Z},R_\mathcal{Z}}$ representing the orbits under some $\mathbb{Z}^2$-action $R_\mathcal{Z}$ defined by rotations on $\mathbb{T}^2$ and coded by some topological partition $\mathcal{P}_\mathcal{Z}$ of $\mathbb{T}^2$ into 16 polygonal atoms. We prove their equality $Ω_\mathcal{Z} =\mathcal{X}_Φ=\mathcal{X}_{\mathcal{P}_\mathcal{Z},R_\mathcal{Z}}$ by showing that they are self-similar with respect to the substitution $Φ$. This chapter provides a transversal reading of results divided into four different articles obtained through the study of the Jeandel-Rao Wang shift. It gathers in one place the methods introduced to desubstitute Wang shifts and to desubstitute codings of $\mathbb{Z}^2$-actions by focussing on a simple 2-dimensional self-similar subshift. SageMath code to find marker tiles and compute the Rauzy induction of $\mathbb{Z}^2$-rotations is provided allowing to reproduce the computations. The chapter contains many exercises whose solutions are provided at the end.

math.DS↗

arXiv2025

Metallic mean Wang tiles I: self-similarity, aperiodicity and minimality

For every positive integer $n$, we introduce a set $\mathcal{T}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb{Z}^2\to\mathcal{T}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n\geq1$, we show that the Wang shift $Ω_n$, defined as the set of valid configurations over the tiles $\mathcal{T}_n$, is self-similar, aperiodic and minimal for the shift action. We say that $\{Ω_n\}_{n\geq1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of $Ω_n$ is the positive root of the polynomial $x^2-nx-1$. This root is sometimes called the $n$-th metallic mean, and in particular, the golden mean when $n=1$, and the silver mean when $n=2$. When $n=1$, the set of Wang tiles $\mathcal{T}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles.

math.DS↗

arXiv2024

Dumont-Thomas complement numeration systems for $\mathbb{Z}$

We extend the well-known Dumont--Thomas numeration systems to $\mathbb{Z}$ using an approach inspired by the two's complement numeration system. Integers in $\mathbb{Z}$ are canonically represented by a finite word (starting with $\mathtt{0}$ when nonnegative and with $\mathtt{1}$ when negative). The systems are based on two-sided periodic points of substitutions as opposed to the right-sided fixed points. For every periodic point of a substitution, we construct an automaton which returns the letter at position $n\in\mathbb{Z}$ of the periodic point when fed with the representation of $n$ in the corresponding numeration system. The numeration system naturally extends to $\mathbb{Z}^d$. We give an equivalent characterization of the numeration system in terms of a total order on a regular language. Lastly, using particular periodic points, we recover the well-known two's complement numeration system and the Fibonacci analogue of the two's complement numeration system.

math.CO↗

arXiv2024

Indistinguishable asymptotic pairs and multidimensional Sturmian configurations

Two asymptotic configurations on a full $\mathbb{Z}^d$-shift are indistinguishable if for every finite pattern the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb{Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a "flip condition" are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together the two results provide a generalization to $\mathbb{Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction.

math.DS↗

arXiv2024

Skyline Operators for Document Spanners

When extracting a relation of spans (intervals) from a text document, a common practice is to filter out tuples of the relation that are deemed dominated by others. The domination rule is defined as a partial order that varies along different systems and tasks. For example, we may state that a tuple is dominated by tuples which extend it by assigning additional attributes, or assigning larger intervals. The result of filtering the relation would then be the skyline according to this partial order. As this filtering may remove most of the extracted tuples, we study whether we can improve the performance of the extraction by compiling the domination rule into the extractor. To this aim, we introduce the skyline operator for declarative information extraction tasks expressed as document spanners. We show that this operator can be expressed via regular operations when the domination partial order can itself be expressed as a regular spanner, which covers several natural domination rules. Yet, we show that the skyline operator incurs a computational cost (under combined complexity). First, there are cases where the operator requires an exponential blowup on the number of states needed to represent the spanner as a sequential variable-set automaton. Second, the evaluation may become computationally hard. Our analysis more precisely identifies classes of domination rules for which the combined complexity is tractable or intractable.

cs.DB↗

arXiv2023

A Fibonacci analogue of the two's complement numeration system

Using the classic two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. We introduce a Fibonacci-equivalent of the two's complement notation and we show that addition in this numeration system can be performed by a deterministic finite-state transducer. The result is based on the Berstel adder, which performs addition of the usual Fibonacci representations of nonnegative integers and for which we provide a new constructive proof. Moreover, we characterize the Fibonacci-equivalent of the two's complement notation as an increasing bijection between $\mathbb{Z}$ and a particular language.

cs.FL↗

arXiv2023

Nonexpansive directions in the Jeandel-Rao Wang shift

We show that $\{0,φ+3,-3φ+2,-φ+\frac{5}{2}\}$ is the set of slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao Wang shift, where $φ=(1+\sqrt{5})/2$ is the golden mean. This set is a topological invariant allowing to distinguish the Jeandel-Rao Wang shift from other subshifts. Moreover, we describe the combinatorial structure of the two resolutions of the Conway worms along the nonexpansive directions in terms of irrational rotations of the unit interval. The introduction finishes with pictures of nonperiodic Wang tilings corresponding to what Conway called the cartwheel tiling in the context of Penrose tilings. The article concludes with open questions regarding the description of octopods and essential holes in the Jeandel-Rao Wang shift.

math.DS↗

arXiv2022

Almost everywhere balanced sequences of complexity $2n+1$

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $μ$-almost every $\mathcal{C}$-adic sequence is balanced, where $μ$ is any shift-invariant ergodic Borel probability measure on $\{1,2\}^\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $μ$ is negative.

math.DS↗

arXiv2021

The $q$-analog of the Markoff injectivity conjecture over the language of a balanced sequence

The Markoff injectivity conjecture states that $w\mapstoμ(w)_{12}$ is injective on the set of Christoffel words where $μ:\{\mathtt{0},\mathtt{1}\}^*\to\mathrm{SL}_2(\mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2\times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $μ_q$ of $μ$ such that $μ_{q}(w)_{12}|_{q=1}=μ(w)_{12}$ is the Markoff number associated to the Christoffel word $w$ when evaluated at $q=1$. We show that there exists an order $<_{radix}$ on $\{\mathtt{0},\mathtt{1}\}^*$ such that for every balanced sequence $s \in \{\mathtt{0},\mathtt{1}\}^\mathbb{Z}$ and for all factors $u, v$ in the language of $s$ with $u <_{radix} v$, the difference $μ_q(v)_{12} - μ_q(u)_{12}$ is a nonzero polynomial of indeterminate $q$ with nonnegative integer coefficients. Therefore, the map $u\mapstoμ_q(u)_{12}$ is injective over the language of a balanced sequence. The proof uses an equivalence between balanced sequences satisfying some Markoff property and indistinguishable asymptotic pairs.

math.CO↗

arXiv2021

Rauzy induction of polygon partitions and toral $\mathbb{Z}^2$-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.

math.DS↗

arXiv2021

A Numeration System for Fibonacci-like Wang Shifts

Motivated by the study of Fibonacci-like Wang shifts, we define a numeration system for $\mathbb{Z}$ and $\mathbb{Z}^2$ based on the binary alphabet $\{0,1\}$. We introduce a set of 16 Wang tiles that admits a valid tiling of the plane described by a deterministic finite automaton taking as input the representation of a position $(m,n)\in\mathbb{Z}^2$ and outputting a Wang tile.

math.CO↗

arXiv2021

A characterization of Sturmian sequences by indistinguishable asymptotic pairs

We give a new characterization of biinfinite Sturmian sequences in terms of indistinguishable asymptotic pairs. Two asymptotic sequences on a full $\mathbb{Z}$-shift are indistinguishable if the sets of occurrences of every pattern in each sequence coincide up to a finitely supported permutation. This characterization can be seen as an extension to biinfinite sequences of Pirillo's theorem which characterizes Christoffel words. Furthermore, we provide a full characterization of indistinguishable asymptotic pairs on arbitrary alphabets using substitutions and biinfinite characteristic Sturmian sequences. The proof is based on the well-known notion of derived sequences.

math.CO↗

arXiv2020

Markov partitions for toral $\mathbb{Z}^2$-rotations featuring Jeandel-Rao Wang shift and model sets

We define a partition $\mathcal{P}_0$ and a $\mathbb{Z}^2$-rotation ($\mathbb{Z}^2$-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition $\mathcal{P}_\mathcal{U}$ and a $\mathbb{Z}^2$-rotation on $\mathbb{T}^2$ whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that $\mathcal{P}_\mathcal{U}$ is a Markov partition for the $\mathbb{Z}^2$-rotation on $\mathbb{T}^2$. We prove in both cases that the toral $\mathbb{Z}^2$-rotation is the maximal equicontinuous factor of the minimal subshifts and that the set of fiber cardinalities of the factor map is $\{1,2,8\}$. The two minimal subshifts are uniquely ergodic and are isomorphic as measure-preserving dynamical systems to the toral $\mathbb{Z}^2$-rotations. It provides a construction of these Wang shifts as model sets of 4-to-2 cut and project schemes. A do-it-yourself puzzle is available in the appendix to illustrate the results.

math.DS↗