The Markoff injectivity conjecture states that $wmapstomu(w)_{12}$ is injective on the set of Christoffel words where $mu:{mathtt{0},mathtt{1}}^*tomathrm{SL}_2(mathbb{Z})$ 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 $mu_q$ of $mu$ such that $mu_{qto1}(w)=mu(w)$ is the Markoff number associated to the Christoffel word $w$. We show that for every $q>0$, the map ${mathtt{0},mathtt{1}}^*tomathbb{Z}[q]$ defined by $wmapstomu_q(w)_{12}$ is injective over the language $mathcal{L}(s)$ of a balanced sequence $sin{mathtt{0},mathtt{1}}^mathbb{Z}$. The proof is based on new equivalent definitions of balanced sequences.
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 det
erministic finite automaton taking as input the representation of a position $(m,n)inmathbb{Z}^2$ and outputting a Wang tile.
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 $mu$-almost every $mathcal{C}$-adic sequence is balanced, where $mu$ 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 $mu$ is negative.
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 $mathcal{A}
={0,dots,18}$ for which we give three characterizations: as the subshift $mathcal{X}_phi$ generated by a 2-dimensional morphism $phi$ defined on $mathcal{A}$; as the Wang shift $Omega_mathcal{U}$ defined by a set $mathcal{U}$ of 19 Wang tiles; as the symbolic dynamical system $mathcal{X}_{mathcal{P}_mathcal{U},R_mathcal{U}}$ representing the orbits under some $mathbb{Z}^2$-action $R_mathcal{U}$ defined by rotations on $mathbb{T}^2$ and coded by some topological partition $mathcal{P}_mathcal{U}$ of $mathbb{T}^2$ into 19 polygonal atoms. We prove their equality $Omega_mathcal{U} =mathcal{X}_phi=mathcal{X}_{mathcal{P}_mathcal{U},R_mathcal{U}}$ by showing they are self-similar with respect to the substitution $phi$. 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.
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 co
incide up to a finitely supported permutation. This characterization can be seen as an extension to biinfinite sequences of Pirillos 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.
We propose a method for proving that a toral partition into polygons is a Markov partition for a given toral $mathbb{Z}^2$-rotation, i.e., $mathbb{Z}^2$-action defined by rotations on a torus. If $mathcal{X}_{mathcal{P},R}$ denotes the symbolic dynam
ical 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 the sub-domain $W$. The induced $mathbb{Z}^2$-action extends the notion of Rauzy induction of IETs to the case of $mathbb{Z}^2$-actions where subactions are polytope exchange transformations. This allows to describe $mathcal{X}_{mathcal{P},R}$ by a $S$-adic sequence of 2-dimensional morphisms. We apply the method on one example and we obtain a sequence of 2-dimensional morphisms which is eventually periodic leading to a self-induced partition. We prove that its substitutive structure 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, we deduce the equality of the two subshifts and it implies that the partition is a Markov partition for the associated toral $mathbb{Z}^2$-rotation since $X_0$ is a shift of finite type. It also implies that $X_0$ 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 the code to reproduce the proofs are provided.
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 def
ined 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.
Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a
nontrivial translation. We study herein the Wang shift $Omega_0$ made of all valid tilings using the set $mathcal{T}_0$ of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift $X_0$ of $Omega_0$ such that every tiling in $X_0$ can be decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112 that are equivalent to a set of 19 self-similar and aperiodic Wang tiles. We suggest that this provides an almost complete description of the substitutive structure of Jeandel-Rao tilings, as we believe that $Omega_0setminus X_0$ is a null set for any shift-invariant probability measure on $Omega_0$. The proof is based on 12 elementary steps, 10 of which involve the same procedure allowing one to desubstitute Wang tilings from the existence of a subset of marker tiles. The 2 other steps involve the addition of decorations to deal with fault lines and changing the base of the $mathbb{Z}^2$-action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions, and shear conjugacy from a set of Wang tiles.
We define a Wang tile set $mathcal{U}$ of cardinality 19 and show that the set $Omega_mathcal{U}$ of all valid Wang tilings $mathbb{Z}^2tomathcal{U}$ is self-similar, aperiodic and is a minimal subshift of $mathcal{U}^{mathbb{Z}^2}$. Thus $mathcal{U}
$ is the second smallest self-similar aperiodic Wang tile set known after Ammanns set of 16 Wang tiles. The proof is based on the unique composition property. We prove the existence of an expansive, primitive and recognizable $2$-dimensional morphism $omega:Omega_mathcal{U}toOmega_mathcal{U}$ that is onto up to a shift. The proof of recognizability is done in two steps using at each step the same criteria (the existence of marker tiles) for proving the existence of a recognizable one-dimensional substitution that sends each tile either on a single tile or on a domino of two tiles.
For any primitive matrix $Minmathbb{R}^{ntimes n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $mathbf{x}=(x_1,dots,x_n)^t$ such that $Mmathbf{x}=(frac{1}{x_1},dots,frac{1}{x_n})^t$. The contribution of
this note is to provide an alternative proof of a result of Brualdi et al. (1966) on the diagonal equivalence of a nonnegative matrix to a stochastic matrix.
