An aperiodic computer chip

For every integer $n\geq 1$ , we define a finite subset $V_n\subset \mathbb {N}^3$ of vectors

$$\begin{align*}V_n = \{(v_0,v_1,v_2)\in\mathbb{N}^3\colon 0\leq v_0\leq v_1\leq 1 \text{ and } v_1\leq v_2\leq n+1\} \end{align*}$$

with nondecreasing entries where the middle entry is at most 1. We introduce a function

$$\begin{align*}\begin{array}{rccl} \theta_n:&V_n\times V_n & \to & \mathbb{Z}^3\\ &(u_0,u_1,u_2), (v_0,v_1,v_2) & \mapsto & (r_0,r_1,r_2), \end{array} \end{align*}$$

taking two vectors as input and returning one vector. Its image is defined by the rule

(2.1) $$ \begin{align} \left\{ \begin{array}{ll} &r_0=u_0,\\ &r_1=\begin{cases} v_2-n & \text{ if } u_0 = 0,\\ 1 & \text{ if } u_0 = 1, \end{cases}\\ &r_2=\begin{cases} v_1+u_0 & \text{ if } v_0 = 0,\\ u_2+1 & \text{ if } v_0 = 1. \end{cases} \end{array} \right. \end{align} $$

Notice that $(r_0,r_1,r_2)$ does not depend on $u_1$ . For every integer $n\geq 1$ , we construct a symmetric $\theta _n$ -chip, that is, a computer chip taking as inputs $u\in V_n$ on the left and $v\in V_n$ on the bottom and producing as outputs $\theta _n(u,v)$ on the right and $\theta _n(v,u)$ on the top (see Figure 5).

Figure 5

The $\theta _n$ -chip is a computer chip computing $\theta _n(u,v)$ and $\theta _n(v,u)$ from the left input u and bottom input v.

If $\theta _n(u,v)$ and $\theta _n(v,u)$ are in $V_n$ , then one can use multiple copies of the $\theta _n$ -chip and connect them to each other horizontally and vertically into an arbitrarily large rectangular cluster of $\theta _n$ -chips (see Figure 6).

Figure 6

A rectangular cluster of copies of the $\theta _n$ -chip.

We prove in this work the existence of arbitrarily large rectangular clusters of the $\theta _n$ -chip all of them performing correct computations. Also we show that no rectangular cluster of the $\theta _n$ -chip performs a periodic computation. Thus, we say that the $\theta _n$ -chip is an aperiodic computer chip. Perhaps we can say it is an aperiodic monochip, but we cannot say it is an aperiodic monotile as in [Reference Smith, Myers, Kaplan and Goodman-Strauss55, Reference Smith, Myers, Kaplan and Goodman-Strauss56] because the same chip with different inputs has to be considered a distinct Wang tile.

Instances of the chip are metallic mean Wang tiles

If we consider all possible values of inputs u and v in $V_n$ and if we restrict the outputs to be in the set $V_n$ , then we obtain a finite set of Wang tiles

(2.2)

which is the finite set of all possible instances of the $\theta _n$ -chip.

Something unexpected and surprising happens in the proof of Theorem A. The set $\mathcal {C}_n$ of instances of the $\theta _n$ -chip is exactly equal to the extended set $\mathcal {T}_n'$ of metallic mean Wang tiles introduced in [Reference Labbé37] in order to prove the self-similarity of $\Omega _n$ , see Proposition 5.1.

Tile labels satisfy Equations

The next result states that every tile in $\mathcal {C}_n$ satisfy a system of equations. While the equations associated with Kari’s [Reference Kari24] and Culik’s [Reference Culik11] aperiodic set of Wang tiles are multiplicative, the ones associated with $\mathcal {C}_n$ are additive.

Theorem B. Let $n\geq 1$ be an integer, $d=(0,-1,1)$ and $e=(1,0,0)$ . The set of Wang tiles defined by the $\theta _n$ -chip satisfy the following system of equations:

where $\langle \_,\_\rangle $ denotes the canonical inner product of ${\mathbb {Z}^{3}}$ .

Equivalently, if we let $\ell =(\ell _0,\ell _1,\ell _2)$ , $b=(b_0,b_1,b_2)$ , $r=(r_0,r_1,r_2)$ and $t=(t_0,t_1,t_2)$ , the equations in the theorem say that tiles in ${\mathcal {C}_{n}}$ satisfy $\ell _0=r_0$ , $b_0=t_0$ and

(2.3) $$ \begin{align} {\frac{t_2-t_1+\ell_2-\ell_1}{n}}-\ell_0 = {\frac{b_2-b_1+r_2-r_1}{n}}-b_0 \end{align} $$

which reminds of Equation (1.1).

Like Kari’s and Culik’s tiles, these equations behave well with tilings and more equations can be deduced for valid tilings of a rectangle, see Section 6. In particular, Equation (6.2) says that in a tiling of a cylinder of height k, the average of the inner product with $\textstyle \frac {1}{n}d$ of the top labels of the cylinder is obtained from the average of the inner product with $\textstyle \frac {1}{n}d$ of the bottom labels of the cylinder by k rotations on the unit circle by a fixed angle. The angle is equal to the frequency of columns in the cylinder containing junction tiles and vertical strip colored tiles, which is a rational number. Therefore, the existence of a cyclic rectangle is not directly forbidden from these equations. Note that we know from the self-similarity of $\Omega _n$ that the frequency of columns containing junction tile in every valid configuration in $\Omega _n$ is equal to $\beta ^{-1}$ , which is an irrational number [Reference Labbé37].

It remains an open problem to deduce the aperiodicity of the Wang shift $\Omega _n$ from the equations satisfied by the labels of $\theta _n$ -chip as this is nicely done for Kari and Culik sets of tiles. See Section 11 for related open questions.

Existence of valid tilings

Valid configurations in $\Omega _n$ can be constructed using the floor function on linear forms. Let $\Lambda _n:[0,1)^2\to \mathbb {Z}^3$ be defined as

$$\begin{align*}\Lambda_n(x,y) = \left( \begin{array}{r} \lfloor y-\beta^{-1}+1\rfloor\\ \lfloor \beta^{-1}x + y-\beta^{-1}+1\rfloor\\ \lfloor \beta x + y-\beta^{-1}+1\rfloor \end{array} \right). \end{align*}$$

where $\beta $ is the $n^{th}$ metallic mean, that is, the positive root of the polynomial $x^2-nx-1$ . For every $(x,y)\in \mathbb {R}^2$ , let

be a Wang tile where $\{x\}=x-\lfloor x\rfloor $ is the fractional part of a number $x\in \mathbb {R}$ .

Theorem C. For every integer $n\geq 1$ and every $(x,y)\in [0,1)^2$ , the configuration

is a valid tiling of the plane by the set of metallic mean Wang tiles $\mathcal {T}_n$ .

This construction reminds of the proof of existence of tilings with Kari and Culik tiles based on the balanced representation of real numbers and first difference of Beatty sequences [Reference Kari24, Reference Culik11], see also [Reference Eigen, Navarro and Prasad15, Reference Siefken54].

A factor map defined from averages of tile labels

In Kari–Culik tilings [Reference Kari24, Reference Culik11], there is a well-defined notion of average [Reference Durand, Gamard and Grandjean14] of the top tile labels along a bi-infinite horizontal row. The change of value from one row to the next row is described by a piecewise rationally multiplicative map. In this context, metallic mean Wang shifts also behave like Kari–Culik tilings. It involves the consideration of the average of specific inner products and irrational rotations instead of multiplications, see Figure 7 which can be compared with Figure 1.

Figure 7

A $10\times 5$ valid rectangular tiling with the set $\mathcal {T}_n$ with $n=3$ . The numbers indicated in the right margin are the average of the inner products $\langle \frac {1}{n}d,v\rangle $ over the vectors v appearing as top (or bottom) labels of a horizontal row of tiles and where $d=(0,-1,1)$ . We observe that these numbers increase by $\frac {3}{10}\ \pmod 1$ from row to row. The number $\frac {3}{10}$ is equal to the frequency of columns containing junction tiles (a junction tile is a tile whose labels all start with 0). Observe that this is a cylindrical tiling (left and right outer labels of the rectangle match) which simplifies the equations involved because the left and right carries cancel.

We show that the average of the dot products of the vector $\frac {1}{n}d=\frac {1}{n}(0,-1,1)$ with the top labels of a given row in a valid configuration $\mathbb {Z}^2\to \mathcal {T}_n$ in $\Omega _n$ is well-defined and takes a value in the interval $[0,1]$ (see Equation (8.1)). By symmetry of the set $\mathcal {T}_n$ , the same holds for the right labels of a given column. By considering the row and column going through the origin of a configuration, the two averages define a map $\Phi _n:\Omega _n\to \mathbb {T}^2$ (see Equation (8.2)). We prove that this map is a factor map from the Wang shift to the 2-torus.

Theorem D. Let $d=(0,-1,1)$ , $n\geq 1$ be an integer and $\Omega _n$ be the $n^{th}$ metallic mean Wang shift. The map

(2.4)

is a factor map, that is, it is continuous, onto and commutes the shift $\mathbb {Z}^2\overset {\sigma }{\curvearrowright }\Omega _n$ with the toral $\mathbb {Z}^2$ -rotation $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ by the equation $ \Phi _n\circ \sigma ^k = R_n^k\circ \Phi _n $ for every $k\in \mathbb {Z}^2$ where

$$\begin{align*}\begin{array}{rccl} R_n&:\mathbb{Z}^2\times\mathbb{T}^2 & \to & \mathbb{T}^2\\ &(k,x) & \mapsto & R_n^k(x):=x + \beta k \end{array} \end{align*}$$

and $\beta =\frac {n+\sqrt {n^2+4}}{2}$ is the $n^{th}$ metallic mean, that is, the positive root of the polynomial $x^2-nx-1$ .

As a consequence of Theorem D, we deduce that $\Omega _n$ is aperiodic because $\beta $ is irrational and $R_n$ is a free $\mathbb {Z}^2$ -action, see Corollary 8.3. Note that since $\beta -\beta ^{-1}=n$ , we have $\beta =\beta ^{-1}\ \pmod 1$ .

Theorem D is an analogue of a result known for Kari and Culik aperiodic Wang tilings which satisfy equations involving balanced representations of real numbers and orbits of piecewise rationally multiplicative maps, see also Theorem 16 in [Reference Eigen, Navarro and Prasad15] and Proposition 3 in [Reference Siefken54]. Here the result applies to all of the configurations in the Wang shift $\Omega _n$ .

A symbolic dynamical system and a Markov partition

The Wang shift $\Omega _n$ can be independently described as a symbolic representation of the dynamical system $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ by encoding its orbits with an appropriate topological partition of $\mathbb {T}^2$ . The partition of $\mathbb {T}^2$ naturally emerges from the set of preimages of the map and from Theorem C.

Since $\Lambda _n$ is defined as the floor of linear forms, for every tile $t\in \mathcal {T}_n$ , the set

is a polygonal open region in the unit square. It satisfies that $\mathcal {P}_n=\{P_t\mid t\in \mathcal {T}_n\}$ is a topological partition of $\mathbb {T}^2$ made of $(n+3)^2$ atoms. The polygonal partition $\mathcal {P}_n$ is the refinement of two polygonal partitions and , the second one being the image of the first under a symmetry by the positive diagonal. The partition can be constructed by drawing the following geodesics on the torus $\mathbb {T}^2$ :

• ○ two closed geodesics of slope $0$ and $\infty $ going through the origin $(0,0)$ ,

• ○ a closed geodesic of slope 0 going through the point $(0,\beta ^{-1})$ ,

• ○ a geodesic of slope $-\beta ^{-1}$ from $(0,\beta ^{-1})$ to $(1,0)$ ,

• ○ a geodesic of slope $-\beta $ from $(0,\beta ^{-1})$ to $(1,0)$ wrapping around the unit square fundamental domain n times.

See an illustration of $\mathcal {P}_n$ when $n=3$ in Figure 8. Every open region defined by the complement of the geodesics can be identified with a pair of vectors in $V_n$ and a unique tile in $\mathcal {T}_n$ with such top and right labels. As opposed to the four topological polygonal partitions associated with Jeandel-Rao tilings [Reference Labbé33], $\mathcal {P}_n$ can be computed only from and without considering the and partitions. This is because the set $\mathcal {T}_n$ of tiles is NE-deterministic, see Theorem 5.3.

Figure 8

The partition and its image under a symmetry with the positive diagonal. Their refinement is $\mathcal {P}_3$ which is a partition of the unit square into 36 polygonal atoms. Here $\beta $ is the third metallic mean, that is, the positive root of $x^2-3x-1$ .

The encoding of $\mathbb {Z}^2$ -orbits under $R_n$ by the topological partition $\mathcal {P}_n$ are 2-dimensional configurations whose topological closure is the symbolic dynamical system $\mathcal {X}_{\mathcal {P}_n,R_n}$ . We prove that $\mathcal {X}_{\mathcal {P}_n,R_n}=\Omega _n$ , and since $\Omega _n$ is a subshift of finite type by definition, we have the following theorem.

Theorem E. For every integer $n\geq 1$ , the symbolic dynamical system $\mathcal {X}_{\mathcal {P}_n,R_n}$ corresponding to $\mathcal {P}_n,R_n$ is equal to the metallic mean Wang shift $\Omega _n$ :

$$\begin{align*}\Omega_n = \mathcal{X}_{\mathcal{P}_n,R_n}. \end{align*}$$

In particular, $\mathcal {P}_n$ is a Markov partition for the dynamical system $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ .

Markov partitions were originally defined for one-dimensional dynamical systems $\mathbb {Z}\overset {T}{\curvearrowright }\mathbb {T}^2$ and were extended to $\mathbb {Z}^d$ -actions by automorphisms of compact Abelian group in [Reference Einsiedler and Schmidt16]. Following [Reference Labbé33, Reference Labbé34], we use the same terminology and extend the definition proposed in [Reference Lind and Marcus40, §6.5] for dynamical systems defined by higher-dimensional actions by rotations, see Definition 9.1.

The maximal equicontinuous factor and an isomorphism

Using Theorem E and applying the results already proved for Jeandel–Rao Wang shift [Reference Labbé33], we have the following additional topological and measurable properties for the factor map. We refer the reader to the preliminary Section 3 for the notions and vocabulary on topological and measure-preserving dynamical systems that are used in the statement. A similar result holds for Penrose tilings [Reference Robinson48].

3.1 Topological dynamical systems

Most of the notions introduced here can be found in [Reference Walters61]. A dynamical system is a triple $(X,G,T)$ , where X is a topological space, G is a topological group and T is a continuous function $G\times X\to X$ defining a left action of G on X: if $x\in X$ , e is the identity element of G and $g,h\in G$ , then using additive notation for the operation in G we have $T(e,x)=x$ and $T(g+h,x)=T(g,T(h,x))$ . In other words, if one denotes the transformation $x\mapsto T(g,x)$ by $T^g$ , then $T^{g+h}=T^g T^h$ . In this work, we consider the Abelian group $G=\mathbb {Z}\times \mathbb {Z}$ .

If $Y\subset X$ , let $\overline {Y}$ denote the topological closure of Y and let $\overline {Y}^T:=\cup _{g\in G}T^g(Y)$ denote the T-closure of Y. A subset $Y\subset X$ is T -invariant if $\overline {Y}^T=Y$ . A dynamical system $(X,G,T)$ is called minimal if X does not contain any nonempty, proper, closed T-invariant subset. The left action of G on X is free if $g=e$ whenever there exists $x\in X$ such that $T^g(x)=x$ .

Let $(X,G,T)$ and $(Y,G,S)$ be two dynamical systems with the same topological group G. A homomorphism $\theta :(X,G,T)\to (Y,G,S)$ is a continuous function $\theta :X\to Y$ satisfying the commuting property that $S^g\circ \theta =\theta \circ T^g$ for every $g\in G$ . A homomorphism $\theta :(X,G,T)\to (Y,G,S)$ is called an embedding if it is one-to-one, a factor map if it is onto, and a topological conjugacy if it is both one-to-one and onto and its inverse map is continuous. If $\theta :(X,G,T)\to (Y,G,S)$ is a factor map, then $(Y,G,S)$ is called a factor of $(X,G,T)$ and $(X,G,T)$ is called an extension of $(Y,G,S)$ . Two dynamical systems are topologically conjugate if there is a topological conjugacy between them.

A measure-preserving dynamical system is defined as a system $(X,G,T,\mu ,\mathcal {B})$ , where $\mu $ is a probability measure defined on the Borel $\sigma $ -algebra $\mathcal {B}$ of subsets of X, and $T^g:X\to X$ is a measurable map which preserves the measure $\mu $ for all $g\in G$ , that is, $\mu (T^g(B))=\mu (B)$ for all $B\in \mathcal {B}$ . The measure $\mu $ is said to be T -invariant. In what follows, when it is clear from the context, we omit the Borel $\sigma $ -algebra $\mathcal {B}$ of subsets of X and write $(X,G,T,\mu )$ to denote a measure-preserving dynamical system.

The set of all T-invariant probability measures of a dynamical system $(X,G,T)$ is denoted by $\mathcal {M}^T(X)$ . A T-invariant probability measure on X is called ergodic if for every set $B\in \mathcal {B}$ such that $T^{g}(B)=B$ for all $g\in G$ , we have that B has either zero or full measure. A dynamical system $(X,G,T)$ is uniquely ergodic if it has only one invariant probability measure, that is, $|\mathcal {M}^T(X)|=1$ . One can prove that a uniquely ergodic dynamical system is ergodic. A dynamical system $(X,G,T)$ is said strictly ergodic if it is uniquely ergodic and minimal.

Let $(X,G,S,\mu ,\mathcal {A})$ and $(Y,G,T,\nu ,\mathcal {B})$ be two measure-preserving dynamical systems. We say that the two systems are isomorphic (mod 0) if there exist measurable sets $X_0\subset X$ and $Y_0\subset Y$ of full measure (i.e., $\mu (X_0)=1$ and $\nu (Y_0)=1$ ) with $S^g(X_0)\subset X_0$ , $T^g(Y_0)\subset Y_0$ for all $g\in G$ and there exists a bi-measurable bijection $\phi _0:X_0\to Y_0$ ,

• ○ which is measure-preserving, that is, $\mu (\phi _0^{-1}(B))=\nu (B)$ for all measurable sets $B\subset Y_0$ ,

• ○ satisfying $\phi _0\circ S^g(x)=T^g\circ \phi _0(x)$ for all $x\in X_0$ and $g\in G$ .

The role of the set $X_0$ is to make precise the fact that the properties of the isomorphism need to hold only on a set of full measure. In this case, we call $\phi _0$ an isomorphism (mod 0) with respect to $\mu $ and $\nu $ . We also refer to an everywhere defined measurable map $\phi :X\to Y$ as an isomorphism (mod 0) with respect to $\mu $ and $\nu $ if $\phi (x)=\phi _0(x)$ with $x\in X$ for some $\phi _0$ and $X_0$ as above. When $\phi $ is also a factor map, some authors say that $\phi $ is a topo-isomorphism in order to express both its topological and measurable nature [Reference Fuhrmann, Gröger and Lenz18].

3.2 Maximal equicontinuous factor

A metrizable dynamical system $(X,G,T)$ is called equicontinuous if the family of homeomorphisms $\{T^{g}\}{}_{g\in G}$ is equicontinuous, that is, if for all ${\varepsilon }>0$ there exists ${\delta }>0$ such that

$$\begin{align*}{\mathrm{dist}}(T^{g}(x), T^{g}(y)) {<} {\varepsilon} \end{align*}$$

for all $g\in G$ and all $x,y\in X$ with ${\mathrm {dist}}(x,y)<{\delta }$ . According to a well-known theorem [Reference Aujogue, Barge, Kellendonk and Lenz4, Theorem 3.2], equicontinuous minimal systems defined by the action of an Abelian group are rotations on groups.

We say that $\theta :(X,G,T)\to (Y,G,S)$ is an equicontinuous factor if $\theta $ is a factor map and $(Y,G,S)$ is equicontinuous. We say that $(X_{\mathrm {max}}, G, T_{\mathrm {max}})$ is the maximal equicontinuous factor of $(X,G,T)$ if there exists an equicontinuous factor $\pi _{\mathrm {max}}:(X,G,T)\to (X_{\mathrm {max}}, G, T_{\mathrm {max}})$ , such that for any equicontinuous factor $\theta :(X,G,T)\to (Y,G,S)$ , there exists a unique factor map $\psi :(X_{\mathrm {max}}, G, T_{\mathrm {max}})\to (Y,G,S)$ with $\psi \circ \pi _{\mathrm {max}}=\theta $ . The maximal equicontinuous factor exists and is unique (up to topological conjugacy), see [Reference Aujogue, Barge, Kellendonk and Lenz4, Theorem 3.8] and [Reference Kurka31, Theorem 2.44].

Let $\theta :(X,G,T)\to (Y,G,S)$ be a factor map. We call the preimage set $\theta ^{-1}(y)$ of a point $y\in Y$ the fiber of $\theta $ over y. The cardinality of the fiber $\theta ^{-1}(y)$ for some $y\in Y$ has an important role and is related to the definition of other notions, see [Reference Aujogue, Barge, Kellendonk and Lenz4]. In particular, the factor map $\theta $ is almost one-to-one if $\{y\in Y:\mathrm {card}(\theta ^{-1}(y))=1\}$ is a $G_\delta $ -dense set in Y (that is a countable intersection of open sets which is dense in Y). In that case, $(X,G,T)$ is an almost one-to-one extension of $(Y,G,S)$ . The set of fiber cardinalities of a factor map $\theta :(X,G,T)\to (Y,G,S)$ is the set $\{\mathrm {card}(\theta ^{-1}(y)) : y \in Y\}\subset \mathbb {N}\cup \{\infty \}$ , see [Reference Fiebig17]. The set of fiber cardinalities of the maximal equicontinuous factor of a minimal dynamical system is invariant under topological conjugacy, see for instance [Reference Labbé33, Lemma 2.2].

3.3 Subshifts and shifts of finite type

In this section, we introduce multidimensional subshifts, a particular type of dynamical systems [Reference Lind and Marcus40, §13.10], [Reference Schmidt51, Reference Lind39, Reference Hochman20]. Let $\mathcal {A}$ be a finite set, $d\geq 1$ , and let $\mathcal {A}^{\mathbb {Z}^d}$ be the set of all maps $x:\mathbb {Z}^d\to \mathcal {A}$ , equipped with the compact product topology. An element $x\in \mathcal {A}^{\mathbb {Z}^d}$ is called configuration and we write it as $x=(x_{\boldsymbol{m}})=(x_{\boldsymbol{m}}:{\boldsymbol{m}}\in \mathbb {Z}^d)$ , where $x_{\boldsymbol{m}}\in \mathcal {A}$ denotes the value of x at ${\boldsymbol{m}}$ . The topology on $\mathcal {A}^{\mathbb {Z}^d}$ is compatible with the metric defined for all configurations $x,x'\in {\mathcal {A}}^{\mathbb {Z}^d}$ by ${\mathrm {dist}}(x,x')=2^{-\min \left \{\Vert {\boldsymbol{n}}\Vert \,:\, x_{\boldsymbol{n}}\neq x^{\prime }_{\boldsymbol{n}}\right \}}$ where $\Vert {\boldsymbol{n}}\Vert = |n_1| + \dots + |n_d|$ . The shift action $\sigma :{\boldsymbol{n}}\mapsto \sigma ^{\boldsymbol{n}}$ of the additive group $\mathbb {Z}^d$ on $\mathcal {A}^{\mathbb {Z}^d}$ is defined by

(3.1) $$ \begin{align} (\sigma^{\boldsymbol{n}}(x))_{\boldsymbol{m}} = x_{{\boldsymbol{m}}+{\boldsymbol{n}}} \end{align} $$

for every $x=(x_{\boldsymbol{m}})\in \mathcal {A}^{\mathbb {Z}^d}$ and ${\boldsymbol{n}}\in \mathbb {Z}^d$ . If $X\subset \mathcal {A}^{\mathbb {Z}^d}$ , let $\overline {X}$ denote the topological closure of X and let ${\overline {X}^{\sigma }}:=\{\sigma ^{\boldsymbol{n}}(x)\mid x\in X, {\boldsymbol{n}}\in \mathbb {Z}^d\}$ denote the shift-closure of X. A subset $X\subset \mathcal {A}^{\mathbb {Z}^d}$ is shift-invariant if ${\overline {X}^{\sigma }}=X$ . A closed, shift-invariant subset $X\subset \mathcal {A}^{\mathbb {Z}^d}$ is a subshift. If $X\subset \mathcal {A}^{\mathbb {Z}^d}$ is a subshift we write $\sigma =\sigma ^X$ for the restriction of the shift action (3.1) to X. When X is a subshift, the triple $(X,\mathbb {Z}^d,\sigma )$ is a dynamical system and the notions presented in the previous section hold.

A configuration $x\in X$ is periodic if there is a nonzero vector ${\boldsymbol{n}}\in \mathbb {Z}^d\setminus \{{\boldsymbol{0}}\}$ such that $x=\sigma ^{\boldsymbol{n}}(x)$ and otherwise it is nonperiodic. We say that a nonempty subshift X is aperiodic if the shift action $\sigma $ on X is free.

For any subset $S\subset \mathbb {Z}^d$ let $\pi _S:\mathcal {A}^{\mathbb {Z}^d}\to \mathcal {A}^S$ denote the projection map which restricts every $x\in \mathcal {A}^{\mathbb {Z}^d}$ to S. A pattern is a function $p\in \mathcal {A}^S$ for some finite subset $S\subset \mathbb {Z}^d$ . To every pattern $p\in \mathcal {A}^S$ corresponds a subset $\pi _S^{-1}(p)\subset \mathcal {A}^{\mathbb {Z}^d}$ called cylinder. A nonempty set $X\subset \mathcal {A}^{\mathbb {Z}^d}$ is a subshift if and only if there exists a set $\mathcal {F}$ of forbidden patterns such that

(3.2) $$ \begin{align} X = \{x\in\mathcal{A}^{\mathbb{Z}^d} \mid \pi_S\circ\sigma^{\boldsymbol{n}}(x)\notin\mathcal{F} \text{ for every } {\boldsymbol{n}}\in\mathbb{Z}^d \text{ and } S\subset\mathbb{Z}^d\}, \end{align} $$

see [Reference Hochman20, Prop. 9.2.4]. A subshift $X\subset \mathcal {A}^{\mathbb {Z}^d}$ is a subshift of finite type (SFT) if there exists a finite set $\mathcal {F}$ such that (3.2) holds. In this article, we consider shifts of finite type on $\mathbb {Z}\times \mathbb {Z}$ , that is, the case $d=2$ .

3.4 Wang shifts

A Wang tile is a tuple of four colors $(a,b,c,d)\in I\times J\times I\times J$ where I is a finite set of vertical colors and J is a finite set of horizontal colors, see [Reference Wang62, Reference Robinson49]. A Wang tile is represented as a unit square with colored edges:

For each Wang tile $\tau =(a,b,c,d)$ , let , , , denote respectively the colors of the right, top, left and bottom edges of $\tau $ .

Figure 9

The set of 3 Wang tiles introduced in [Reference Wang62] using letters $\{A,B,C,D,E\}$ instead of numbers from the set $\{1,2,3,4,5\}$ for labeling the edges. Each tile is identified uniquely by an index from the set $\{0,1,2\}$ written at the center each tile.

Figure 10

A finite $3\times 3$ pattern on the left is valid with respect to the Wang tiles since it respects Equations (3.3) and (3.4). Validity can be verified on the tiling shown on the right.

Let $\mathcal {T}=\{t_0,\dots ,t_{m-1}\}$ be a set of Wang tiles such as the one shown in Figure 9. A configuration $x:\mathbb {Z}^2\to \{0,\dots ,m-1\}$ is valid with respect to $\mathcal {T}$ if it assigns a tile in $\mathcal {T}$ to each position of $\mathbb {Z}^2$ so that contiguous edges of adjacent tiles have the same color, that is,

(3.3)

(3.4)

for every ${\boldsymbol{n}}\in \mathbb {Z}^2$ where ${\boldsymbol{e}}_1=(1,0)$ and ${\boldsymbol{e}}_2=(0,1)$ . A finite pattern which is valid with respect to $\mathcal {U}$ is shown in Figure 10.

Let $\Omega _{\mathcal {T}}\subset \{0,\dots ,m-1\}^{\mathbb {Z}^2}$ denote the set of all valid configurations with respect to $\mathcal {T}$ . Together with the shift action $\sigma $ of $\mathbb {Z}^2$ , $\Omega _{\mathcal {T}}$ is a subshift that we call a Wang shift. Furthermore, $\Omega _{\mathcal {T}}$ is a subshift of finite type (SFT) of the form (3.2) since $\Omega _{\mathcal {T}}$ is the subshift defined from the finite set of forbidden patterns made of all horizontal and vertical dominoes of two tiles that do not share an edge of the same color. Reciprocally, every subshift of finite type can be encoded into a Wang shift following a well-known construction (see [Reference Mozes41, p. 141-142]).

To a configuration $x\in \Omega _{\mathcal {T}}$ corresponds a tiling of the plane $\mathbb {R}^2$ by the tiles $\mathcal {T}$ where the unit square Wang tile $t_{x({\boldsymbol{n}})}$ is placed at position ${\boldsymbol{n}}$ for every ${\boldsymbol{n}}\in \mathbb {Z}^2$ , as in Figure 10. In this article, we consider tilings from the symbolic point of view. In particular, we represent tilings of plane by Wang tiles symbolically by configurations $\mathbb {Z}^2\to \mathcal {T}$ .

A configuration $x\in \Omega _{\mathcal {T}}$ is periodic if there exists ${\boldsymbol{n}}\in \mathbb {Z}^2\setminus \{0\}$ such that $x=\sigma ^{\boldsymbol{n}}(x)$ . A set of Wang tiles $\mathcal {T}$ is periodic if there exists a periodic configuration $x\in \Omega _{\mathcal {T}}$ . Originally, Wang thought that every set of Wang tiles $\mathcal {T}$ is periodic as soon as $\Omega _{\mathcal {T}}$ is nonempty [Reference Wang62]. This statement is equivalent to the existence of an algorithm solving the domino problem, that is, taking as input a set of Wang tiles and returning yes or no whether there exists a valid configuration with these tiles. Berger, a student of Wang, later proved that the domino problem is undecidable and he also provided a first example of an aperiodic set of Wang tiles [Reference Berger7]. A set of Wang tiles $\mathcal {T}$ is aperiodic if the Wang shift $\Omega _{\mathcal {T}}$ is a nonempty aperiodic subshift. This means that in general one cannot decide the emptiness of a Wang shift $\Omega _{\mathcal {T}}$ .

4.1 The tiles

For every integer $n\geq 1$ , let

$$\begin{align*}V_n = \{(v_0,v_1,v_2)\in\mathbb{Z}^3\colon 0\leq v_0\leq v_1\leq 1\text{ and }v_1\leq v_2\leq n+1\}. \end{align*}$$

be a set of vectors having nondecreasing entries with an upper bound of 1 on the middle entry and an upper bound of $n+1$ on the last entry. The label of the edges of the Wang tiles considered in this article are vectors in $V_n$ . To lighten the figures and the presentation of the Wang tiles, it is convenient to denote the vector $(v_0,v_1,v_2)\in V_n$ more compactly as a word $v_0v_1v_2$ . For instance the vector $(1,1,1)$ is represented as $111$ .

To help the reading of the tiles and tilings, we assign a color to the vectors according to the following rule: a vector $v\in 00\mathbb {N}$ is drawn in blue, a vector $v\in 01\mathbb {N}$ is drawn in yellow and a vector $v\in 11\mathbb {N}$ is drawn in white. Overlap between blue and yellow regions will be shown in green.

For every integer $n\geq 1$ and for every $i,j\in \mathbb {N}$ such that $0\leq i\leq n$ and $0\leq j\leq n$ , we have the following white tiles:

For every $i,n\in \mathbb {N}$ such that $0\leq i\leq n$ , we have the following blue, yellow, green and antigreen tiles:

For every $n\in \mathbb {N}$ and $k,\ell ,r,s\in \{0,1\}$ such that $k\leq \ell $ and $r\leq s$ , we have the following junction tiles (the gray region will be drawn in a blue or yellow color depending on the specific values of $k,\ell ,r,s$ according to the same rule as above):

Junction tiles play a similar role as junction tiles in [Reference Mozes41].

4.2 The extended set $\mathcal {T}_n'$ of metallic mean Wang tiles

In this section, we give the definition of the family of extended sets of Wang tiles $(\mathcal {T}_n')_{n\geq 1}$ .

From the above, we define the following sets of tiles:

$$ \begin{align*} W_n &= \left\{ w_n^{i,j} \mid 1\leq i\leq n, 1\leq j\leq n \right\} &&(n^2\text{ white tiles}),\\ B^{\prime}_n &= \left\{ b_n^i \mid 0\leq i \leq n \right\} &&(n+1\text{ horizontal blue tiles}),\\ Y_n &= \left\{ y_n^i \mid 1\leq i\leq n \right\} &&(n\text{ horizontal yellow tiles}),\\ G_n &= \left\{ g_n^i\mid 0\leq i\leq n \right\} &&(n+1\text{ horizontal green tiles}),\\ A_n &= \left\{ a_n^i \mid 1\leq i \leq n \right\} &&(n \text{ horizontal antigreen tiles}). \end{align*} $$

Finally, we have a set of 9 junction tiles:

We may observe that $\widehat {W_n}=W_n$ and $\widehat {J^{\prime }_n}=J^{\prime }_n$ are closed under reflection. Also, $\widehat {B^{\prime }_n}$ are $n+1$ vertical blue tiles, $\widehat {Y_n}$ are n vertical yellow tiles, $\widehat {G_n}$ are $n+1$ vertical green tiles and $\widehat {A_n}$ are n vertical antigreen tiles.

The extended set of metallic mean Wang tiles $\mathcal {T}_n'$ can be described in terms of the white, yellow, green, blue, antigreen and junction tiles seen before.

Definition 4.1 (Extended set of metallic mean Wang tiles [Reference Labbé37])

Let

$$\begin{align*}\mathcal{T}_n'= W_n\cup Y_n \cup \widehat{Y_n} \cup G_n \cup \widehat{G_n} \cup B^{\prime}_n \cup \widehat{B^{\prime}_n} \cup A_n \cup \widehat{A_n} \cup J^{\prime}_n. \end{align*}$$

The set $\mathcal {T}_n'$ defines the extended metallic mean Wang shift $\Omega ^{\prime }_n=\Omega _{\mathcal {T}_n'}$ .

The set $\mathcal {T}_n'$ contains $n^2+2(n+1+n+n+1+n)+9=n^2+8n+13$ Wang tiles. The set of Wang tiles $\mathcal {T}_n'$ for $n=4$ is shown in Figure 11.

Figure 11

Extended metallic mean Wang tile sets $\mathcal {T}_n'$ for $n=4$ . The junction tiles $j_n^{0,0,1,1}$ and $j_n^{1,1,0,0}$ are shown with a $\times $ -mark in their center.

4.3 The family $\mathcal {T}_n$ of $(n+3)^2$ Wang tiles

In this section, we give the definition of the family of sets of Wang tiles $(\mathcal {T}_n)_{n\geq 1}$ . The set $\mathcal {T}_n$ is a subset of $\mathcal {T}_n'$ defined as follows. Let

$$ \begin{align*} B_n &= B_n' \setminus \left\{ b_n^n\right\} &&(\text{subset of}\ n \text{ horizontal blue tiles}),\\ J_n &= J_n' \setminus \left\{ j_n^{1,1,0,0}, j_n^{0,0,1,1} \right\} &&(\text{subset of 7 junction tiles}). \end{align*} $$

Definition 4.2 (Metallic mean Wang tiles[Reference Labbé37])

For every positive integer n, we construct the set of Wang tiles

$$\begin{align*}\mathcal{T}_n= W_n\cup Y_n \cup \widehat{Y_n} \cup G_n \cup \widehat{G_n} \cup B_n \cup \widehat{B_n} \cup J_n. \end{align*}$$

The set of tiles defines the Metallic mean Wang shift $\Omega _n=\Omega _{\mathcal {T}_n}$ .

The subset $\mathcal {T}_n$ contains $n^2+2(n+n+1+n)+7=(n+3)^2$ Wang tiles. They are shown in Figure 12 for $n=1,2,3,4,5$ .

Figure 12

Metallic mean Wang tile sets $\mathcal {T}_n$ for $n=1,2,3,4,5$ .

9.1 Symbolic dynamical system $\mathcal {X}_{\mathcal {P}_n,R_n}$

We now define the symbolic dynamical system associated with the toral $\mathbb {Z}^2$ -rotation $R_n$ generated by the partition $\mathcal {P}_n$ . We adapt [Reference Lind and Marcus40] to the 2-dimensional setting as it was done in [Reference Hochman20] and [Reference Labbé33].

If $S\subset \mathbb {Z}^2$ is a finite set, we say that a pattern $w\in \mathcal {A}^S$ is allowed for $\mathcal {P}_n,R_n$ if

(9.1) $$ \begin{align} \bigcap_{{\boldsymbol{k}}\in S} R_n^{-{\boldsymbol{k}}}(P_{w_{\boldsymbol{k}}}) \neq \varnothing. \end{align} $$

Let $\mathcal {L}_{\mathcal {P}_n,R_n}$ be the collection of all allowed patterns for $\mathcal {P}_n,R_n$ . The set $\mathcal {L}_{\mathcal {P}_n,R_n}$ is the language of a subshift $\mathcal {X}_{\mathcal {P}_n,R_n}\subseteq \mathcal {A}^{\mathbb {Z}^2}$ defined as follows, see [Reference Hochman20, Prop. 9.2.4],

$$\begin{align*}\mathcal{X}_{\mathcal{P}_n,R_n} = \{x\in\mathcal{A}^{\mathbb{Z}^2} \mid \pi_S\circ\sigma^{\boldsymbol{n}}(x)\in\mathcal{L}_{\mathcal{P}_n,R_n} \text{ for every } {\boldsymbol{n}}\in\mathbb{Z}^2 \text{ and finite subset } S\subset\mathbb{Z}^2\}. \end{align*}$$

We say that $\mathcal {X}_{\mathcal {P}_n,R_n}$ is the symbolic dynamical system corresponding to $\mathcal {P}_n,R_n$ .

For each $w\in \mathcal {X}_{\mathcal {P}_n,R_n}\subset \mathcal {A}^{\mathbb {Z}^2}$ and $m\geq 0$ there is a corresponding nonempty open set

$$\begin{align*}D_m(w) = \bigcap_{\Vert{\boldsymbol{k}}\Vert\leq m} R_n^{-{\boldsymbol{k}}}(P_{w_{\boldsymbol{k}}}) \subseteq \mathbb{T}^2. \end{align*}$$

The closures $\overline {D}_m(w)$ of these sets are compact and decrease with m, so that $\overline {D}_0(w)\supseteq \overline {D}_1(w)\supseteq \overline {D}_2(w)\supseteq \dots $ . It follows that $\cap _{m=0}^{\infty }\overline {D}_m(w)\neq \varnothing $ . In order for points in $\mathcal {X}_{\mathcal {P}_n,R_n}$ to correspond to points in $\mathbb {T}^2$ , this intersection should contain only one point. This leads to the following definition. A topological partition $\mathcal {P}_n$ of $\mathbb {T}^2$ gives a symbolic representation of $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ if for every $w\in \mathcal {X}_{\mathcal {P}_n,R_n}$ the intersection $\cap _{m=0}^{\infty }\overline {D}_m(w)$ consists of exactly one point ${\boldsymbol{x}}\in \mathbb {T}^2$ . We call w a symbolic representation of ${\boldsymbol{x}}$ .

Markov partitions were originally defined for one-dimensional dynamical systems $\mathbb {Z}\overset {T}{\curvearrowright }\mathbb {T}^2$ and were extended to $\mathbb {Z}^d$ -actions by automorphisms of compact Abelian group in [Reference Einsiedler and Schmidt16]. Following [Reference Labbé33, Reference Labbé34], we use the same terminology and extend the definition proposed in [Reference Lind and Marcus40, §6.5] for dynamical systems defined by higher-dimensional actions by rotations.

9.2 Proofs of main results

First, we have the following result.

Each atom of the partition $\mathcal {P}_n$ is invariant only under the trivial translation. Therefore, from [Reference Labbé33, Lemma 3.4], $\mathcal {P}_n$ gives a symbolic representation of the dynamical system $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ . Thus, we can define the following function:

(9.2) $$ \begin{align} f_n:\mathcal{X}_{\mathcal{P}_n,R_n}\to\mathbb{T}^2 \end{align} $$

be such that $f_n(w)$ is the unique point in the intersection $\cap _{m=0}^{\infty }\overline {D}_m(w)$ .

Proposition 9.3. Let $n\geq 1$ be an integer. The map $f_n:\mathcal {X}_{\mathcal {P}_n,R_n}\to \mathbb {T}^2$ is a factor map satisfying

$$\begin{align*}f_n\circ\sigma^k = R_n^k\circ f_n \end{align*}$$

for every $k\in \mathbb {Z}^2$ .

From the minimality of the Wang shift $\Omega _n$ proved separately in [Reference Labbé37], we may now prove Theorem E using the same method as in [Reference Labbé33].

Theorem E

For every integer $n\geq 1$ , the symbolic dynamical system $\mathcal {X}_{\mathcal {P}_n,R_n}$ corresponding to $\mathcal {P}_n,R_n$ is equal to the metallic mean Wang shift $\Omega _n$ :

$$\begin{align*}\Omega_n = \mathcal{X}_{\mathcal{P}_n,R_n}. \end{align*}$$

In particular, $\mathcal {P}_n$ is a Markov partition for the dynamical system $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ .

In fact, we can show that the factor map $f_n$ is equal to the map $\Phi _n$ explicitly defined in Section 8 from the average of the labels of Wang tiles on the row and column containing the origin. It follows from the next lemma.

Proof. Let $v_1,v_2,v_3,v_4\in V_n$ . Observe that

For every $k\in \mathbb {Z}^2$ , we have

so that

Therefore, for every $m\in \mathbb {N}$ , we have

$$\begin{align*}(x,y) \in \bigcap_{\Vert k\Vert\leq m} R_n^{-k}(\overline{P_{c_{(x,y)}(k)}}) = \overline{D_m}(c_{(x,y)}). \end{align*}$$

Since $\mathcal {P}_n$ gives a symbolic representation of the dynamical system $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ , we have that $\cap _{m=0}^{\infty }\overline {D}_m(c_{(x,y)})$ is a singleton and

$$\begin{align*}\cap_{m=0}^{\infty}\overline{D}_m(c_{(x,y)})=\{(x,y)\}. \end{align*}$$

Therefore, $f(c_{(x,y)})=(x,y)$ .

Proposition 9.5. The factor map $f_n:\Omega _n\to \mathbb {T}^2$ is equal to the factor map $\Phi _n:\Omega _n\to \mathbb {T}^2$ explicitly defined in Equation (8.2):

$$\begin{align*}f_n=\Phi_n. \end{align*}$$

Proof. From Lemma 9.4, we have $ f_n(c_{(0,0)}) =(0,0)$ . Also, observe that the configuration $c_{(0,0)}$ is symmetric: $\widehat {c_{(0,0)}}=c_{(0,0)}$ . Thus, we have

$$\begin{align*}\Phi_n(c_{(0,0)}) =(\phi_n(\widehat{c_{(0,0)}}), \phi_n(c_{(0,0)})) =(\phi_n(c_{(0,0)}), \phi_n(c_{(0,0)}))=(0,0). \end{align*}$$

Let $w\in \Omega _n$ be any configuration. Since $\Omega _n$ is minimal [Reference Labbé37], there exists a sequence $(k_\ell )_{\ell \in \mathbb {N}}$ such that $k_\ell \in \mathbb {Z}^2$ such that $w=\lim _{\ell \to \infty }\sigma ^{k_\ell }(c_{(0,0)})$ . From Proposition 9.3 and Theorem D, $f_n$ and $\Phi _n$ are factor maps commuting the shift map with the $\mathbb {Z}^2$ -action $R_n$ on the torus $\mathbb {T}^2$ . Thus, we obtain

$$ \begin{align*} \Phi_n(w) &= \Phi_n\left(\lim_{\ell\to\infty}\sigma^{k_\ell}(c_{(0,0)})\right)\\ &= \lim_{\ell\to\infty}\Phi_n\circ\sigma^{k_\ell}(c_{(0,0)})\\ &= \lim_{\ell\to\infty}R_n^{k_\ell}\circ\Phi_n(c_{(0,0)})\\ &= \lim_{\ell\to\infty}R_n^{k_\ell}\left((0,0)\right)\\ &= \lim_{\ell\to\infty}R_n^{k_\ell}\circ f_n(c_{(0,0)})\\ &= \lim_{\ell\to\infty}f_n\circ\sigma^{k_\ell}(c_{(0,0)})\\ &= f_n\left(\lim_{\ell\to\infty}\sigma^{k_\ell}(c_{(0,0)})\right) = f_n(w).\\[-42pt] \end{align*} $$

The factor map $\Phi _n$ between the dynamical system $\mathbb {Z}^2\overset {\sigma }{\curvearrowright }\Omega _n$ and the $\mathbb {Z}^2$ -action $R_n$ on the torus $\mathbb {T}^2$ satisfies additional properties. In particular, $\Phi _n$ is an isomorphism of measure-preserving dynamical systems. Their proofs follow the structure of similar results proved in [Reference Labbé33] for Jeandel–Rao tilings.

Proof. From Theorem E, we have $\mathcal {X}_{\mathcal {P}_n,R_n}=\Omega _n$ .

(i) From Proposition 9.3, the factor map $f_n:\mathcal {X}_{\mathcal {P}_n,R_n}\to \mathbb {T}^2$ commutes the actions $\mathbb {Z}^2\overset {\sigma }{\curvearrowright }\mathcal {X}_{\mathcal {P}_n,R_n}$ and $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ . From [Reference Labbé33, Proposition 5.1], $f_n$ is one-to-one on $f_n^{-1}(\mathbb {T}^2\setminus \Delta _{\mathcal {P}_n,R_n})$ where

$$ \begin{align*} \Delta_{\mathcal{P}_n,R_n}:=\bigcup_{{\boldsymbol{k}}\in\mathbb{Z}^2}R_n^{\boldsymbol{k}} \left(\bigcup_{\tau\in\mathcal{T}_n}\partial P_\tau\right) \subset \mathbb{T}^2 \end{align*} $$

is the set of points whose orbit under the $\mathbb {Z}^2$ -action $R_n$ intersect the boundary of the topological partition $\mathcal {P}_n=\{P_\tau \}_{\tau \in \mathcal {T}_n}$ . From [Reference Labbé33, Corollary 5.3] (which is a consequence of [Reference Aujogue, Barge, Kellendonk and Lenz4, Lemma 3.11]), $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ is the maximal equicontinuous factor of $\mathbb {Z}^2\overset {\sigma }{\curvearrowright }\mathcal {X}_{\mathcal {P}_n,R_n}$ .

(ii) We have that $\{y\in \mathbb {T}^2:\mathrm {card}(f_n^{-1}(y))=1\}=\mathbb {T}^2\setminus \Delta _{\mathcal {P}_n,R_n}$ is a countable intersection of open sets and is dense in $\mathbb {T}^2$ . Thus, it is a $G_\delta $ -dense set in $\mathbb {T}^2$ . Therefore, the factor map $f_n:\mathcal {X}_{\mathcal {P}_n,R_n}\to \mathbb {T}^2$ is almost one-to-one. From Proposition 9.5, we have $f_n=\Phi _n$ .

Suppose that ${\boldsymbol{x}}\in \Delta _{\mathcal {P}_n,R_n}$ . We have $\mathrm {card}(f_n^{-1}({\boldsymbol{x}}))\geq 2$ . If $\mathrm {card}(f_n^{-1}({\boldsymbol{x}}))>2$ , then we may show that there exists ${\boldsymbol{n}}\in \mathbb {Z}^2$ such that ${\boldsymbol{x}}=R_n^{\boldsymbol{n}}({\boldsymbol{0}})$ . If ${\boldsymbol{x}}=R_n^{{\boldsymbol{n}}}({\boldsymbol{0}})$ for some ${\boldsymbol{n}}\in \mathbb {Z}^2$ , then the set $f_n^{-1}({\boldsymbol{x}})$ contains 8 different configurations of the form $\lim _{\varepsilon \to 0}c_{\varepsilon {\mathbf{v}}}$ for some ${\mathbf{v}}\in \mathbb {R}^2\setminus \Theta ^{\mathcal {P}_n}$ where $\Theta ^{\mathcal {P}_n}=\mathbb {R}\cdot \{(1,0),(0,1),(1,-\beta ),(1,\beta ^*)\}$ . If ${\boldsymbol{x}}\in \Delta _{\mathcal {P}_n,R_n}$ but not in the orbit of ${\boldsymbol{0}}$ under $R_n$ , then $\mathrm {card}(f_n^{-1}({\boldsymbol{x}}))=2$ . We conclude that $\{\mathrm {card}(f_n^{-1}({\boldsymbol{x}}))\mid {\boldsymbol{x}}\in \mathbb {T}^2\}=\{1,2,8\}$ .

(iii) The dynamical system $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ is minimal. We have that $\lambda (\partial P)=0$ for each atom $P\in \mathcal {P}_n$ where $\lambda $ is the Haar measure on $\mathbb {T}^2$ . The partition $\mathcal {P}_n$ gives a symbolic representation of the dynamical system $\mathbb {Z}^2\overset {R_n}{\curvearrowright }\mathbb {T}^2$ . Thus, from [Reference Labbé33, Proposition 6.1], the dynamical system $\mathbb {Z}^2\overset {\sigma }{\curvearrowright }\mathcal {X}_{\mathcal {P}_n,R_n}$ is uniquely ergodic.

(iv) Since the dynamical system $\mathbb {Z}^2\overset {\sigma }{\curvearrowright }\mathcal {X}_{\mathcal {P}_n,R_n}$ is uniquely ergodic, it admits a unique shift-invariant probability measure $\nu $ on $\Omega _n$ . From [Reference Labbé33, Proposition 6.1], the measure-preserving dynamical system $(\Omega _n,\mathbb {Z}^2,\sigma ,\nu )$ is isomorphic to $(\mathbb {T}^2,\mathbb {Z}^2,R_n,\lambda )$ where $\lambda $ is the Haar measure on $\mathbb {T}^2$ .