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

Sébastien Labbé  Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400,
Talence, France<EMAIL_ADDRESS>
###### Abstract.

We propose a method for proving that a toral partition into polygons is a
Markov partition for a given toral $\mathbb{Z}^{2}$-rotation
($\mathbb{Z}^{2}$-action defined by rotations on a 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 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
ransformation (PETs). 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 (SFT). 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.

###### Key words and phrases:

Rauzy induction and Markov partition and self-induced and self-similar and SFT
and polyhedron exchange transformation and aperiodic tiling and Sturmian

###### 2010 Mathematics Subject Classification:

Primary 28D05; Secondary 37B50, 52C23

The author acknowledges financial support from the Laboratoire International
Franco-Québécois de Recherche en Combinatoire (LIRCO), the Agence Nationale de
la Recherche through the project CODYS (ANR-18-CE40-0007) and the Horizon 2020
European Research Infrastructure project OpenDreamKit (676541).

###### Contents

  1. 1 Introduction
  2. I Preliminaries
    1. 2 Dynamical systems and subshifts
    2. 3 Two-dimensional words and morphisms
    3. 4 Toral $\mathbb{Z}^{2}$-rotations and polygon exchange transformations (PETs)
    4. 5 Symbolic dynamical systems for toral $\mathbb{Z}^{2}$-rotations
  3. II Induction of $\mathbb{Z}^{2}$-actions and induced partitions of the 2-torus
    1. 6 Induced $\mathbb{Z}^{2}$-actions
    2. 7 Toral partitions induced by toral $\mathbb{Z}^{2}$-rotations
    3. 8 Renormalization schemes
    4. 9 Algorithms
  4. III Substitutive structure of $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$
    1. 10 Inducing the $\mathbb{Z}^{2}$-action $R_{0}$ and the partition $\mathcal{P}_{0}$ to get $\mathcal{X}_{\mathcal{P}_{1},R_{1}}$
    2. 11 Changing the base of the $\mathbb{Z}^{2}$-action to get $\mathcal{X}_{\mathcal{P}_{2},R_{2}}$
    3. 12 Inducing the symbolic dynamical system $\mathcal{X}_{\mathcal{P}_{2},R_{2}}$ to get $\mathcal{X}_{\mathcal{P}_{8},R_{8}}$ and $\mathcal{X}_{\mathcal{P}_{\mathcal{U}},R_{\mathcal{U}}}$
    4. 13 Proof of main results

###  1\. Introduction

A person is walking on a sidewalk made of alternating dark bricks of size 1
and light bricks of size $\alpha>0$ where step zero is made at position
$p\in\mathbb{R}$. They walk from left to right with steps of length 1 and uses
their right foot ($R$) on dark bricks and their left foot ($L$) on light
bricks,

Figure 1.  The shifted lattice $p+\mathbb{Z}$ is coded as an biinfinite binary
sequence $w\in\\{L,R\\}^{\mathbb{Z}}$ which is a symbolic representation of
$p$.

thus constructing a biinfinite binary sequence $w$ in $\\{L,R\\}^{\mathbb{Z}}$
(see Figure 1). For a fixed $\alpha$, let
$X_{\alpha}\subset\\{L,R\\}^{\mathbb{Z}}$ be the set of sequences obtained
when $p\in\mathbb{R}$. When the brick length $\alpha$ is irrational, the
partition of the circle $\mathbb{R}/(1+\alpha)\mathbb{Z}$ into the two colored
bricks yields a symbolic representation of the map $x\mapsto x+1$ having a
nice combinatorial interpretation known as _Sturmian sequences_ [Lot02, Chap.
2]:

  1. (C1)

every sequence $w\in X_{\alpha}$ is obtained from a unique starting point in
$\mathbb{R}/(1+\alpha)\mathbb{Z}$,

  2. (C2)

$X_{\alpha}$ is the set of sequences in $\\{L,R\\}^{\mathbb{Z}}$ whose
language has $n+1$ patterns of length $n$ for all $n\geq 0$ and whose letter
frequencies exist and satisfies $\lim_{n\to\infty}\frac{\\#\\{-n\leq i<n\mid
w_{i}=L\\}}{\\#\\{-n\leq i<n\mid w_{i}=R\\}}=\alpha$.

For example, the _patterns_ of length from 1 to 4 that we see in $w$ in the
figure are $\\{L,R\\}$ $\\{LL,LR,RL\\}$, $\\{LLL,LLR,LRL,RLL\\}$,
$\\{LLLL,LLLR,LLRL,LRLL,RLLL\\}$ and one can show in general that $w$ has
$n+1$ patterns of length $n$. The converse is the difficult part of the proof:
any biinfinite sequence whose letter frequencies exist and having $n+1$
distinct patterns of size $n$ belongs to $X_{\alpha}$ for some
$\alpha\in\mathbb{R}_{>0}/\mathbb{Q}$ and for some starting point
$p\in\mathbb{R}/(1+\alpha)\mathbb{Z}$.

This result is a fundamental result in symbolic dynamics known as Morse-
Hedlund’s theorem [MH40], although the connection with the number of fixed-
length patterns was done in [CH73]. Its proof uses important tools from
dynamical systems and number theory and is explained nowadays in terms of
$S$-adic developments, first return maps (Rauzy induction), continued fraction
expansion of real numbers, and Ostrowski numeration system [Arn02].

A generalization of the result of Morse and Hedlund was provided by Rauzy for
a single example [Rau82]. Based on the right-infinite sequence often called
the Tribonnacci word

$T=1213121121312121312112131213121121312121...$

which is fixed by $1\mapsto 12,2\mapsto 13,3\mapsto 1$, Rauzy proved that the
system $(X_{T},\sigma)$ is measurably conjugate to the toral rotation
$(\mathbb{T}^{2},x\mapsto x+(\beta^{-1},\beta^{-2}))$ where $\beta$ is the
real root of $x^{3}-x^{2}-x-1$, the characteristic polynomial of the incidence
matrix of the substitution. The coding of the toral translation is made
through the partition into three parts of a fundamental domain of
$\mathbb{T}^{2}$ known as the Rauzy fractal [Fog02, §7.4]. Proving that this
holds for all Pisot substitution is known as the Pisot Conjecture [ABB+15], an
important and still open question.

Finding further generalizations was coined the term of _Rauzy program_ in
[BFZ05], a survey divided into three parts: the _good_ coding of $k$-interval
exchange transformations (IET); the _bad_ coding of a rotation on
$\mathbb{T}^{k}$; and the _ugly_ coding of two rotations on $\mathbb{T}^{k}$
for $k=1$. The IET are the good part since they behave well with induced
transformations and admit continued fraction algorithms [Rau79, Vee82, Yoc06,
AF07]. The bad part was much improved since then with various recent results
using multidimensional continued fraction algorithms (MCFA) including Brun’s
algorithm [Bru58] which provides measurable-theoretic conjugacy with symbolic
systems for almost every toral rotations on $\mathbb{T}^{2}$ [BST19, Thu19].
As the authors wrote in [BFZ05], the term ugly “refers to some esthetic
difficulties in building two-dimensional sequences by iteration of patterns”.
Indeed, digital planes [ABEI01, ABI02, BFZ05, Ber09, BBJS13] are typical
objects that are described by the coding of two rotations on $\mathbb{T}^{1}$
and they are not built by rectangular shaped substitutions. In this article,
we want to add some shine on the ugly part by providing a method based on
induced transformations on $\mathbb{T}^{2}$ to prove that a toral partition is
a Markov partition for a toral $\mathbb{Z}^{2}$-rotation.

#### Markov partitions for automorphisms of the torus

While Morse-Hedlund’s theorem deals with the coding of irrational rotations,
other kinds of dynamical systems admit a symbolic representation. Hyperbolic
automorphisms of the torus are one example [LM95, Kit98]. Suppose that one
starts at some position $v\in\mathbb{R}^{2}$ and moves according to the
successive images under the application of the map $v\mapsto Mv$ with
$M=\left(\begin{smallmatrix}1&1\\\ 1&0\end{smallmatrix}\right)$ as shown in
Figure 2.

Figure 2.  The automorphism of $\mathbb{R}^{2}/\mathbb{Z}^{2}$ defined as
$v\mapsto Mv$ admits a Markov partition.

The map $v\mapsto Mv$ is an automorphism of $\mathbb{R}^{2}/\mathbb{Z}^{2}$
which is _hyperbolic_ since $M$ has no eigenvalue of module 1. It allows one
to code the orbit $(M^{k}v)_{k\in\mathbb{Z}}$ as a sequence in
$\\{A,B,C\\}^{\mathbb{Z}}$ according to a well-chosen partition $\mathcal{P}$
of a fundamental domain of $\mathbb{R}^{2}/\mathbb{Z}^{2}$ into three
rectangles indexed by letters in the set $\\{A,B,C\\}$. In Figure 2, the
positive orbit $(M^{k}v)_{k\geq 0}$ of the starting point
$v=\left(\frac{-7}{10},\frac{14}{10}\right)^{T}$ is coded by the sequence
$CBABABCB\ldots$ which avoids the patterns in $\mathcal{F}=\\{AA,BB,CC,CA\\}$.
We denote the set of obtained sequences as $\mathcal{X}_{\mathcal{P},M}$. The
partition of $\mathbb{R}^{2}/\mathbb{Z}^{2}$ is a Markov partition for the
automorphism because it has two important properties [LM95, §6.5]:

  1. (C1)

every sequence in $\mathcal{X}_{\mathcal{P},M}$ is obtained from a unique
starting point in $\mathbb{R}^{2}/\mathbb{Z}^{2}$,

  2. (C2’)

the set $\mathcal{X}_{\mathcal{P},M}$ is a shift of finite type (SFT), i.e.,
there exists a finite set $\mathcal{F}$ of patterns such that
$\mathcal{X}_{\mathcal{P},M}$ is the set of sequences in
$\\{A,B,C\\}^{\mathbb{Z}}$ which avoids the patterns in $\mathcal{F}$.

Such Markov Partitions exist for all hyperbolic automorphisms of the torus
[AW70, Sin68, Bow75] and various kinds of diffeomorphisms [Bow70], see also
[Kit95, Adl98, KV98]. It turns out that Markov partitions also exist for toral
$\mathbb{Z}^{2}$-rotations.

#### Results

In this article, we propose a method for proving that a 2-dimensional toral
partition is a Markov partition for a given toral $\mathbb{Z}^{2}$-rotation.
The method is inspired from the proof of Morse-Hedlund’s theorem and its link
with continued fraction expansion and induced transformations. We extend the
notion of Rauzy induction of IETs to the case of $\mathbb{Z}^{2}$-actions and
we introduce the notion of induced partitions. We apply the method on one
example related to the golden mean and Jeandel-Rao aperiodic Wang shift.

Figure 3.  For every starting point $p\in\mathbb{R}^{2}$, the coding of the
shifted lattice $p+\mathbb{Z}^{2}$ under the polygon partition
$\mathcal{P}_{0}$ yields a configuration which is a symbolic representation of
$p$. We show that the set of such configurations is a shift of finite type
(SFT) and hence that $\mathcal{P}_{0}$ is a Markov partition for the toral
$\mathbb{Z}^{2}$-rotation $R_{0}$.

Let

$\Gamma_{0}=\langle(\varphi,0),(1,\varphi+3)\rangle_{\mathbb{Z}}$

be a lattice in $\mathbb{R}^{2}$ with $\varphi=\frac{1+\sqrt{5}}{2}$. We
consider the dynamical system defined by the following toral
$\mathbb{Z}^{2}$-rotation, i.e., a $\mathbb{Z}^{2}$-action on the
$2$-dimensional torus $\mathbb{R}^{2}/\Gamma_{0}$:

$\begin{array}[]{rccl}R_{0}:&\mathbb{Z}^{2}\times\mathbb{R}^{2}/\Gamma_{0}&\to&\mathbb{R}^{2}/\Gamma_{0}\\\
&({\boldsymbol{n}},{\boldsymbol{x}})&\mapsto&{\boldsymbol{x}}+{\boldsymbol{n}}.\end{array}$

A polygon partition $\mathcal{P}_{0}$ of $\mathbb{R}^{2}/\Gamma_{0}$ indexed
by integers from the set $\\{0,1,2,\dots,10\\}$ was introduced in [Lab20a],
see Figure 3. It was shown that $\mathcal{P}_{0}$ gives a symbolic
representation of $R_{0}$ thus the set
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}\subset\llbracket
0,10\rrbracket^{\mathbb{Z}^{2}}$ of $2$-dimensional configurations satisfies
condition (C1). It was also shown that $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$
is a strict subset of the Jeandel-Rao Wang shift $\Omega_{0}$ [JR15], but the
proving that $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ is itself a SFT satisfying
condition (C2’) was left open.

######  Theorem 1.

$\mathcal{P}_{0}$ is a Markov partition for the dynamical system
$(\mathbb{R}^{2}/\Gamma_{0},\mathbb{Z}^{2},R_{0})$.

The fact that $\mathcal{X}_{\mathcal{P}_{0},R_{0}}\subset\Omega_{0}$
corresponds to the easy direction in the proof of Morse-Hedlund’s theorem,
namely that codings of irrational rotations have pattern complexity $n+1$.
Proving the converse, i.e., that a configuration in the Jeandel-Rao Wang shift
is obtained as the coding of a shifted lattice $p+\mathbb{Z}^{2}$ for some
$p\in\mathbb{R}^{2}/\Gamma_{0}$ is harder, for the same reason that proving
that a Sturmian sequence is a coding of an irrational rotation from some
starting point is more involved.

The proof of Theorem 1 is inspired from the proof of Morse-Hedlund’s theorem
and Rauzy induction of irrational rotations [Arn02]. It builds on top of
results proved previously on Jeandel-Rao aperiodic Wang shift and on the
substitutive structure of $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ which is
computed herein from the induction of $\mathbb{Z}^{2}$-actions and of toral
partitions. The substitutive structure of
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ is described in the next result.

######  Theorem 2.

Let $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ be the symbolic dynamical system
associated to $\mathcal{P}_{0}$, $R_{0}$. There exist lattices
$\Gamma_{i}\subset\mathbb{R}^{2}$, alphabets $\mathcal{A}_{i}$,
$\mathbb{Z}^{2}$-actions
$R_{i}:\mathbb{Z}^{2}\times\mathbb{R}^{2}/\Gamma_{i}\to\mathbb{R}^{2}/\Gamma_{i}$
and topological partitions $\mathcal{P}_{i}$ of $\mathbb{R}^{2}/\Gamma_{i}$
indexed by letters from the alphabet $\mathcal{A}_{i}$ that provide the
substitutive structure of $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$. More
precisely,

  1. (i)

There exists a $2$-dimensional morphism
$\beta_{0}:\mathcal{A}_{1}\to\mathcal{A}_{0}^{*^{2}}$

$\mathcal{X}_{\mathcal{P}_{0},R_{0}}\xleftarrow{\beta_{0}}\mathcal{X}_{\mathcal{P}_{1},R_{1}}$

that is onto up to a shift, i.e.,
$\overline{\beta_{0}(\mathcal{X}_{\mathcal{P}_{1},R_{1}})}^{\sigma}=\mathcal{X}_{\mathcal{P}_{0},R_{0}}$.

  2. (ii)

There exists a shear conjugacy

$\mathcal{X}_{\mathcal{P}_{1},R_{1}}\xleftarrow{\beta_{1}}\mathcal{X}_{\mathcal{P}_{2},R_{2}}$

shearing configurations by the action of the matrix
$M=\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$, i.e.,
satisfying
$\sigma^{M{\boldsymbol{k}}}\circ\beta_{1}=\beta_{1}\circ\sigma^{\boldsymbol{k}}$
for every ${\boldsymbol{k}}\in\mathbb{Z}^{2}$.

  3. (iii)

There exist $2$-dimensional morphisms $\beta_{2}$, $\beta_{3}$, $\beta_{4}$,
$\beta_{5}$, $\beta_{6}$ and $\beta_{7}$:

$\mathcal{X}_{\mathcal{P}_{2},R_{2}}\xleftarrow{\beta_{2}}\mathcal{X}_{\mathcal{P}_{3},R_{3}}\xleftarrow{\beta_{3}}\mathcal{X}_{\mathcal{P}_{4},R_{4}}\xleftarrow{\beta_{4}}\mathcal{X}_{\mathcal{P}_{5},R_{5}}\xleftarrow{\beta_{5}}\mathcal{X}_{\mathcal{P}_{6},R_{6}}\xleftarrow{\beta_{6}}\mathcal{X}_{\mathcal{P}_{7},R_{7}}\xleftarrow{\beta_{7}}\mathcal{X}_{\mathcal{P}_{8},R_{8}}$

that are onto up to a shift, i.e.,
$\overline{\beta_{i}(\mathcal{X}_{\mathcal{P}_{i+1},R_{i+1}})}^{\sigma}=\mathcal{X}_{\mathcal{P}_{i},R_{i}}$
for each $i\in\\{2,3,4,5,6,7\\}$.

  4. (iv)

The subshift $\mathcal{X}_{\mathcal{P}_{8},R_{8}}$ is self-similar as there
exist two $2$-dimensional morphisms $\beta_{8}$, $\beta_{9}$ and a permutation
$\tau:\mathcal{A}_{8}\to\mathcal{A}_{10}$

$\mathcal{X}_{\mathcal{P}_{8},R_{8}}\xleftarrow{\beta_{8}}\mathcal{X}_{\mathcal{P}_{9},R_{9}}\xleftarrow{\beta_{9}}\mathcal{X}_{\mathcal{P}_{10},R_{10}}\xleftarrow{\tau}\mathcal{X}_{\mathcal{P}_{8},R_{8}}$

that are onto up to a shift, i.e.,
$\overline{\beta_{8}(\mathcal{X}_{\mathcal{P}_{9},R_{9}})}^{\sigma}=\mathcal{X}_{\mathcal{P}_{8},R_{8}}$,
$\overline{\beta_{9}(\mathcal{X}_{\mathcal{P}_{10},R_{10}})}^{\sigma}=\mathcal{X}_{\mathcal{P}_{9},R_{9}}$
and
$\tau(\mathcal{X}_{\mathcal{P}_{8},R_{8}})=\mathcal{X}_{\mathcal{P}_{10},R_{10}}$
and the product $\beta_{8}\beta_{9}\tau$ is an expansive and primitive self-
similarity.

  5. (v)

The subshift $\mathcal{X}_{\mathcal{P}_{8},R_{8}}$ is topological conjugate to
the subshift $\mathcal{X}_{\mathcal{P}_{\mathcal{U}},R_{\mathcal{U}}}$
introduced in [Lab20a] as there exists a permutation
$\zeta:\mathcal{U}\to\mathcal{A}_{8}$ such that
$\zeta(\mathcal{X}_{\mathcal{P}_{\mathcal{U}},R_{\mathcal{U}}})=\mathcal{X}_{\mathcal{P}_{8},R_{8}}$.

Theorem 2 must be compared with the main result of [Lab19b], recalled herein
as Theorem 33, giving the substitutive structure of a minimal subshift $X_{0}$
of the Jeandel-Rao Wang shift $\Omega_{0}$. That result proved the existence
of sets of Wang tiles $\\{\mathcal{T}_{i}\\}_{1\leq i\leq 12}$ together with
their associated Wang shifts $\\{\Omega_{i}\\}_{1\leq i\leq 12}$ and
$2$-dimensional morphisms $\omega_{i}:\Omega_{i+1}\to\Omega_{i}$ that provide
the substitutive structure of Jeandel-Rao Wang shift, see Figure 4. In fact,
the consequence of the two theorems is that the subshifts $X_{0}$ and
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ have the exact same substitutive
structure given as the inverse limit of the same eventually periodic sequence
of $2$-dimensional morphisms.

$\Omega_{0}$$\Omega_{1}$$\Omega_{2}$$\Omega_{3}$$\Omega_{4}$$X_{0}$$X_{1}$$X_{2}$$X_{3}$$X_{4}$$\Omega_{5}$$\Omega_{6}$$\Omega_{7}$$\Omega_{12}$$\Omega_{\mathcal{U}}$$\mathcal{X}_{\mathcal{P}_{0},R_{0}}$$\mathcal{X}_{\mathcal{P}_{1},R_{1}}$$\mathcal{X}_{\mathcal{P}_{2},R_{2}}$$\mathcal{X}_{\mathcal{P}_{3},R_{3}}$$\mathcal{X}_{\mathcal{P}_{8},R_{8}}$$\mathcal{X}_{\mathcal{P}_{\mathcal{U}},R_{\mathcal{U}}}$$\subset$$\supset$$\supset$$\supset$$\supset$$=$$=$$=$$=$$=$$\beta_{0}$$\beta_{1}$$\beta_{2}$$\omega_{0}$$\omega_{1}$$\omega_{2}$$\omega_{3}$$\omega_{0}$$\omega_{1}$$\omega_{2}$$\omega_{3}$$\jmath$$\eta$$\omega_{6}$$\omega_{7}\omega_{8}\omega_{9}\omega_{10}\omega_{11}$$\beta_{3}\beta_{4}\beta_{5}\beta_{6}\beta_{7}$$\zeta$$\zeta^{-1}\beta_{8}\beta_{9}\tau\zeta$$\beta_{8}\beta_{9}\tau$$\rho$$\rho\,\omega_{\mathcal{U}}\rho^{-1}$$\omega_{\mathcal{U}}$
Figure 4. We prove that the subshifts $X_{0}\subset\Omega_{0}$ and
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ are equal since they have a common
substitutive structure. The substitutive structure of $X_{0}$ computed in
[Lab19b] and the substitutive structure of
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ satisfy
$\beta_{0}=\omega_{0}\omega_{1}\omega_{2}\omega_{3}$,
$\beta_{1}\beta_{2}=\jmath\,\eta\,\omega_{6}$,
$\beta_{3}\beta_{4}\beta_{5}\beta_{6}\beta_{7}=\omega_{7}\omega_{8}\omega_{9}\omega_{10}\omega_{11}$,
$\zeta=\rho$ and
$\beta_{8}\,\beta_{9}\,\tau=\rho\,\omega_{\mathcal{U}}\,\rho^{-1}$. We deduce
that $\mathcal{X}_{\mathcal{P}_{8},R_{8}}=\Omega_{12}$,
$\mathcal{X}_{\mathcal{P}_{3},R_{3}}=\Omega_{7}$,
$\mathcal{X}_{\mathcal{P}_{1},R_{1}}=X_{4}$ and finally
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}=X_{0}$.

######  Theorem 3.

The symbolic dynamical system $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ and the
minimal subshift $X_{0}\subset\Omega_{0}$ of the Jeandel-Rao Wang shift have
the same substitutive structure in the sense that the following equalities
hold:

$\begin{array}[]{lllll}\beta_{0}=\omega_{0}\,\omega_{1}\,\omega_{2}\,\omega_{3},&\beta_{1}\beta_{2}=\jmath\,\eta\,\omega_{6},&\beta_{3}=\omega_{7},&\beta_{4}=\omega_{8},\\\
\beta_{5}=\omega_{9},&\beta_{6}=\omega_{10},&\beta_{7}=\omega_{11},&\zeta=\rho,&\beta_{8}\,\beta_{9}\,\tau=\rho\,\omega_{\mathcal{U}}\,\rho^{-1},\end{array}$

where $\omega_{0}$, $\omega_{1}$, $\omega_{2}$, $\omega_{3}$, $\jmath$,
$\eta$, $\omega_{6}$, $\omega_{7}$, $\omega_{8}$, $\omega_{9}$, $\omega_{10}$,
$\omega_{11}$, $\rho$ were computed in [Lab19b] and $\omega_{\mathcal{U}}$ was
first defined in [Lab19a].

######  Remark 4.

To obtain equalities between substitutions computed from totally different
objects, we use a common convention for the definition of the $\beta_{i}$ from
the induction of toral partitions, and in [Lab19b] for the construction of the
$\omega_{i}$ from sets of Wang tiles: when constructing the substitutions, we
sort the images of letters first by length and then lexicographically.

The description of the symbolic dynamical system
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ and the minimal subshift $X_{0}$ of
Jeandel-Rao aperiodic subshift $\Omega_{0}$ by their substitutive structure
allows to prove their topological conjugacy.

######  Corollary 5.

The subshifts $\mathcal{X}_{\mathcal{P}_{\mathcal{U}},R_{\mathcal{U}}}$ and
$\Omega_{\mathcal{U}}$ are topologically conjugate and are equal to the
minimal aperiodic substitutive subshift $\mathcal{X}_{\omega_{\mathcal{U}}}$.
The subshifts $\mathcal{X}_{\mathcal{P}_{8},R_{8}}$ and $\Omega_{12}$ are
topologically conjugate. The subshifts $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$
and $X_{0}$ are topologically conjugate, and the same holds for intermediate
subshifts:

$\mathcal{X}_{\mathcal{P}_{1},R_{1}}=\jmath(\Omega_{5}),\;\mathcal{X}_{\mathcal{P}_{3},R_{3}}=\Omega_{7},\;\mathcal{X}_{\mathcal{P}_{4},R_{4}}=\Omega_{8},\;\mathcal{X}_{\mathcal{P}_{5},R_{5}}=\Omega_{9},\;\mathcal{X}_{\mathcal{P}_{6},R_{6}}=\Omega_{10}\text{
and }\mathcal{X}_{\mathcal{P}_{7},R_{7}}=\Omega_{11}.$

The fact that $X_{0}$ and $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$ are equal
implies Theorem 1 since it was proved in [Lab19b] that $X_{0}$ is a shift of
finite type. It also implies the following corollary which can be seen as a
generalization of what happens for Sturmian sequences.

######  Corollary 6.

The dynamical system $(X_{0},\mathbb{Z}^{2},\sigma)$ is strictly ergodic and
the measure-preserving dynamical system $(X_{0},\mathbb{Z}^{2},\sigma,\nu)$ is
isomorphic to the toral $\mathbb{Z}^{2}$-rotation
$(\mathbb{R}^{2}/\Gamma_{0},\mathbb{Z}^{2},R_{0},\lambda)$ where $\nu$ is the
unique shift-invariant probability measure on $X_{0}$ and $\lambda$ is the
Haar measure on $\mathbb{R}^{2}/\Gamma_{0}$.

Although Jeandel-Rao Wang shift $\Omega_{0}$ is not minimal as it contains the
proper minimal subshift $\mathcal{X}_{\mathcal{P}_{0},R_{0}}=X_{0}$, we
believe that it is uniquely ergodic.

######  Conjecture 7.

The Jeandel-Rao subshift $\Omega_{0}$ is uniquely ergodic.

That conjecture is equivalent to prove that $\Omega_{0}\setminus X_{0}$ has
measure 0 for any shift-invariant probability measure on $\Omega_{0}$ which
was stated as a conjecture in [Lab19b] and where some progress was done. That
would also imply the existence of an isomorphism of measure-preserving
dynamical systems between Jeandel-Rao Wang shift
$(\Omega_{0},\mathbb{Z}^{2},\sigma,\nu)$ and
$(\mathbb{R}^{2}/\Gamma_{0},\mathbb{Z}^{2},R_{0},\lambda)$ where $\nu$ would
be the unique shift-invariant probability measure on $\Omega_{0}$ and
$\lambda$ is the Haar measure on $\mathbb{R}^{2}/\Gamma_{0}$.

This calls for a general theory of $d$-dimensional subshifts of finite type
coded by Markov partitions of the $d$-dimensional torus and admitting induced
subsystems.

#### Structure of the article

The article is divided into three parts. Part I gathers the preliminary
notions including two-dimensional words and morphisms, toral
$\mathbb{Z}^{2}$-rotations and polyhedron exchange transformations (PETs).
Part II defines the induction of toral $\mathbb{Z}^{2}$-rotations, induced
toral partitions and renormalization schemes. It also provides algorithms to
compute induced partitions and induced PETs. In Part III, we induce the
partition $\mathcal{P}_{0}$ introduced in [Lab20a] to get a sequence of
substitutions until we reach a self-induced partition. The SageMath code to
reproduce the proofs is included. We prove that the substitutive structure
obtained from the induction procedure is the same as the substitutive
structure computed directly from the Jeandel-Rao Wang tiles in [Lab19b]. The
appendix contains a short section explaining informally and visually the
classical 1-dimensional induction.

#### Acknowledgements

I am thankful to Vincent Delecroix for many helpful discussions at LaBRI in
Bordeaux during the preparation of this article which allowed me to improve my
knowledge on dynamical systems and Rauzy induction of interval exchange
transformations.

##  Part I Preliminaries

This part is divided into four sections on dynamical systems, subshifts and
Wang shift; two-dimensional words and morphisms; toral
$\mathbb{Z}^{2}$-rotations and polygonal exchange transformations; symbolic
representation of toral $\mathbb{Z}^{2}$-rotations.

###  2\. Dynamical systems and subshifts

In this section, we introduce dynamical systems, subshifts and shifts of
finite type. We let $\mathbb{Z}=\\{\dots,-1,0,1,2,\dots\\}$ denote the
integers and $\mathbb{N}=\\{0,1,2,\dots\\}$ be the nonnegative integers.

####  2.1. Topological dynamical systems

Most of the notions introduced here can be found in [Wal82]. 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 $T(Y):=\cup_{g\in G}T^{g}(Y)$ denote the $T$-closure of $Y$.
Alternatively, we also use the notation $\overline{Y}^{T}=T(Y)$ to denote the
$T$-closure of $Y$. A subset $Y\subset X$ is $T$-invariant if $T(Y)=Y$. 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
$T^{g}\circ\theta=\theta\circ S^{g}$ 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 subshifts are topologically conjugate if
there is a topological conjugacy between them.

Let $(X,G,T)$ and $(Y,G,S)$ be two dynamical systems with the same topological
group $G=\mathbb{Z}^{d}$. we say that a
$\textrm{GL}_{d}(\mathbb{Z})$-homomorphism
$\theta:(X,\mathbb{Z}^{d},T)\to(Y,\mathbb{Z}^{d},S)$ is a continuous function
$\theta:X\to Y$ equipped with some matrix $M\in\textrm{GL}_{d}(\mathbb{Z})$
satisfying the commuting property that
$T^{M{\boldsymbol{k}}}\circ\theta=\theta\circ S^{\boldsymbol{k}}$ for every
${\boldsymbol{k}}\in\mathbb{Z}^{d}$. A
$\textrm{GL}_{d}(\mathbb{Z})$-homomorphism is called an
$\textrm{GL}_{d}(\mathbb{Z})$-conjugacy if it is both one-to-one and onto and
its inverse map is continuous. A $\textrm{GL}_{d}(\mathbb{Z})$-conjugacy is
called a shear conjugacy if the matrix $M\in\textrm{GL}_{d}(\mathbb{Z})$ is a
shear matrix. The notion of $\textrm{GL}_{d}(\mathbb{Z})$-conjugacy
corresponds to flip-conjugacy when $d=1$ [BT98] and to extended symmetry when
$X=Y$ [BRY18].

The set of all $T$-invariant probability measures of a dynamical system
$(X,G,T)$ is denoted by $\mathcal{M}^{T}(X)$. An 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, i.e., $|\mathcal{M}^{T}(X)|=1$. A dynamical
system $(X,G,T)$ is said strictly ergodic if it is uniquely ergodic and
minimal.

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, $\mathcal{B}$ is always the Borel
$\sigma$-algebra of subsets of $X$, so we omit $\mathcal{B}$ and write
$(X,G,T,\mu)$ when it is clear from the context.

Let $(X,G,T,\mu,\mathcal{B})$ and
$(X^{\prime},G,T^{\prime},\mu^{\prime},\mathcal{B}^{\prime})$ be two measure-
preserving dynamical systems. We say that the two systems are isomorphic if
there exist measurable sets $X_{0}\subset X$ and $X^{\prime}_{0}\subset
X^{\prime}$ of full measure (i.e., $\mu(X\setminus X_{0})=0$ and
$\mu^{\prime}(X^{\prime}\setminus X^{\prime}_{0})=0$) with
$T^{g}(X_{0})\subset X_{0}$, $T^{\prime g}(X^{\prime}_{0})\subset
X^{\prime}_{0}$ for all $g\in G$ and there exists a map $\phi:X_{0}\to
X^{\prime}_{0}$, called an isomorphism, that is one-to-one and onto and such
that for all $A\in\mathcal{B}^{\prime}(X^{\prime}_{0})$,

  * •

$\phi^{-1}(A)\in\mathcal{B}(X_{0})$,

  * •

$\mu(\phi^{-1}(A))=\mu^{\prime}(A)$, and

  * •

$\phi\circ T^{g}(x)=T^{\prime g}\circ\phi(x)$ 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.

####  2.2. Subshifts and shifts of finite type

We follow the notation of [Sch01]. 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
$x,x^{\prime}\in\mathcal{A}^{\mathbb{Z}^{d}}$ by
$\mathrm{dist}(x,x^{\prime})=2^{-\min\left\\{\|{\boldsymbol{n}}\|\,:\,x_{\boldsymbol{n}}\neq
x^{\prime}_{\boldsymbol{n}}\right\\}}$ where
$\|{\boldsymbol{n}}\|=|n_{1}|+\dots+|n_{d}|$. The shift action
$\sigma:{\boldsymbol{n}}\mapsto\sigma^{\boldsymbol{n}}$ of $\mathbb{Z}^{d}$ on
$\mathcal{A}^{\mathbb{Z}^{d}}$ is defined by

(1)
$(\sigma^{\boldsymbol{n}}(x))_{\boldsymbol{m}}=x_{{\boldsymbol{m}}+{\boldsymbol{n}}}$

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 (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 said 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
subshift $X\subset\mathcal{A}^{\mathbb{Z}^{d}}$ is a shift of finite type
(SFT) if there exists a finite set $\mathcal{F}$ of forbidden patterns such
that

(2)
$X=\\{x\in\mathcal{A}^{\mathbb{Z}^{d}}\mid\pi_{S}\circ\sigma^{\boldsymbol{n}}(x)\notin\mathcal{F}\text{
for every }{\boldsymbol{n}}\in\mathbb{Z}^{d}\text{ and
}S\subset\mathbb{Z}^{d}\\}.$

In this case, we write $X=\textsc{SFT}(\mathcal{F})$. In this article, we
consider shifts of finite type on $\mathbb{Z}\times\mathbb{Z}$, that is, the
case $d=2$.

###  3\. Two-dimensional words and morphisms

In this section, we reuse the notations from [Lab19a] and [Lab19b] on
$d$-dimensional words, morphisms and self-similar subshifts.

####  3.1. $d$-dimensional word

In this section, we recall the definition of $d$-dimensional word that
appeared in [CKR10] and we keep the notation $u\odot^{i}v$ they proposed for
the concatenation.

We denote by $\\{{\boldsymbol{e}}_{k}|1\leq k\leq d\\}$ the canonical basis of
$\mathbb{Z}^{d}$ where $d\geq 1$ is an integer. If $i\leq j$ are integers,
then $\llbracket i,j\rrbracket$ denotes the interval of integers
$\\{i,i+1,\dots,j\\}$. Let
${\boldsymbol{n}}=(n_{1},\dots,n_{d})\in\mathbb{N}^{d}$ and $\mathcal{A}$ be
an alphabet. We denote by $\mathcal{A}^{{\boldsymbol{n}}}$ the set of
functions

$u:{\llbracket 0,n_{1}-1\rrbracket}\times\cdots\times{\llbracket
0,n_{d}-1\rrbracket}\to\mathcal{A}.$

An element $u\in\mathcal{A}^{\boldsymbol{n}}$ is called a $d$-dimensional word
$u$ of shape ${\boldsymbol{n}}=(n_{1},\dots,n_{d})\in\mathbb{N}^{d}$ on the
alphabet $\mathcal{A}$. We use the notation
$\textsc{shape}(u)={\boldsymbol{n}}$ when necessary. The set of all finite
$d$-dimensional words is
$\mathcal{A}^{*^{d}}=\\{\mathcal{A}^{\boldsymbol{n}}\mid{\boldsymbol{n}}\in\mathbb{N}^{d}\\}$.
A $d$-dimensional word of shape
${\boldsymbol{e}}_{k}+\sum_{i=1}^{d}{\boldsymbol{e}}_{i}$ is called a domino
in the direction ${\boldsymbol{e}}_{k}$. When the context is clear, we write
$\mathcal{A}$ instead of $\mathcal{A}^{(1,\dots,1)}$. When $d=2$, we represent
a $d$-dimensional word $u$ of shape $(n_{1},n_{2})$ as a matrix with Cartesian
coordinates:

$u=\left(\begin{array}[]{ccc}u_{0,n_{2}-1}&\dots&u_{n_{1}-1,n_{2}-1}\\\
\dots&\dots&\dots\\\ u_{0,0}&\dots&u_{n_{1}-1,0}\end{array}\right).$

Let ${\boldsymbol{n}},{\boldsymbol{m}}\in\mathbb{N}^{d}$ and
$u\in\mathcal{A}^{\boldsymbol{n}}$ and $v\in\mathcal{A}^{\boldsymbol{m}}$. If
there exists an index $i$ such that the shapes ${\boldsymbol{n}}$ and
${\boldsymbol{m}}$ are equal except maybe at index $i$, then the concatenation
of $u$ and $v$ in the direction ${\boldsymbol{e}}_{i}$ is defined: it is the
$d$-dimensional word $u\odot^{i}v$ of shape
$(n_{1},\dots,n_{i-1},n_{i}+m_{i},n_{i+1},\dots,n_{d})\in\mathbb{N}^{d}$ given
as

$(u\odot^{i}v)({\boldsymbol{a}})=\begin{cases}u({\boldsymbol{a}})&\text{if}\quad
0\leq a_{i}<n_{i},\\\
v({\boldsymbol{a}}-n_{i}{\boldsymbol{e}}_{i})&\text{if}\quad n_{i}\leq
a_{i}<n_{i}+m_{i}.\end{cases}$

If the shapes ${\boldsymbol{n}}$ and ${\boldsymbol{m}}$ are not equal except
at index $i$, we say that the concatenation of
$u\in\mathcal{A}^{\boldsymbol{n}}$ and $v\in\mathcal{A}^{\boldsymbol{m}}$ in
the direction ${\boldsymbol{e}}_{i}$ is not defined. The following equation
illustrates the concatenation of words in the direction ${\boldsymbol{e}}_{2}$
when $d=2$:

$\left(\begin{array}[]{ccccc}4&5\\\
10&5\end{array}\right)\odot^{2}\left(\begin{array}[]{ccccc}3&10\\\ 9&9\\\
0&0\\\ \end{array}\right)=\left(\begin{array}[]{ccccc}3&10\\\ 9&9\\\ 0&0\\\
4&5\\\ 10&5\end{array}\right)$

in the direction ${\boldsymbol{e}}_{1}$ when $d=2$:

$\left(\begin{array}[]{ccccc}2&8&7\\\ 7&3&9\\\ 1&1&0\\\ 6&6&7\\\
7&4&3\end{array}\right)\odot^{1}\left(\begin{array}[]{ccccc}3&10\\\ 9&9\\\
0&0\\\ 4&5\\\ 10&5\end{array}\right)=\left(\begin{array}[]{ccccc}2&8&7&3&10\\\
7&3&9&9&9\\\ 1&1&0&0&0\\\ 6&6&7&4&5\\\ 7&4&3&10&5\end{array}\right).$

Let ${\boldsymbol{n}},{\boldsymbol{m}}\in\mathbb{N}^{d}$ and
$u\in\mathcal{A}^{\boldsymbol{n}}$ and $v\in\mathcal{A}^{\boldsymbol{m}}$. We
say that $u$ occurs in $v$ at position ${\boldsymbol{p}}\in\mathbb{N}^{d}$ if
$v$ is large enough, i.e.,
${\boldsymbol{m}}-{\boldsymbol{p}}-{\boldsymbol{n}}\in\mathbb{N}^{d}$ and

$v({\boldsymbol{a}}+{\boldsymbol{p}})=u({\boldsymbol{a}})$

for all ${\boldsymbol{a}}=(a_{1},\dots,a_{d})\in\mathbb{N}^{d}$ such that
$0\leq a_{i}<n_{i}$ with $1\leq i\leq d$. If $u$ occurs in $v$ at some
position, then we say that $u$ is a $d$-dimensional subword or factor of $v$.

####  3.2. $d$-dimensional language

A subset $L\subseteq\mathcal{A}^{*^{d}}$ is called a $d$-dimensional language.
The factorial closure of a language $L$ is

$\overline{L}^{Fact}=\\{u\in\mathcal{A}^{*^{d}}\mid u\text{ is a
$d$-dimensional subword of some }v\in L\\}.$

A language $L$ is factorial if $\overline{L}^{Fact}=L$. All languages
considered in this contribution are factorial. Given a configuration
$x\in\mathcal{A}^{\mathbb{Z}^{d}}$, the language $\mathcal{L}(x)$ defined by
$x$ is

$\mathcal{L}(x)=\\{u\in\mathcal{A}^{*^{d}}\mid u\text{ is a $d$-dimensional
subword of }x\\}.$

The language of a subshift $X\subseteq\mathcal{A}^{\mathbb{Z}^{d}}$ is
$\mathcal{L}_{X}=\cup_{x\in X}\mathcal{L}(x)$. Conversely, given a factorial
language $L\subseteq\mathcal{A}^{*^{d}}$ we define the subshift

$\mathcal{X}_{L}=\\{x\in\mathcal{A}^{\mathbb{Z}^{d}}\mid\mathcal{L}(x)\subseteq
L\\}.$

A $d$-dimensional subword $u\in\mathcal{A}^{*^{d}}$ is allowed in a subshift
$X\subset\mathcal{A}^{\mathbb{Z}^{d}}$ if $u\in\mathcal{L}_{X}$ and it is
forbidden in $X$ if $u\notin\mathcal{L}_{X}$. A language
$L\subseteq\mathcal{A}^{*^{d}}$ is forbidden in a subshift
$X\subset\mathcal{A}^{\mathbb{Z}^{d}}$ if $L\cap\mathcal{L}_{X}=\varnothing$.

####  3.3. $d$-dimensional morphisms

In this section, we generalize the definition of $d$-dimensional morphisms
[CKR10] to the case where the domain and codomain are different as for
$S$-adic systems [BD14].

Let $\mathcal{A}$ and $\mathcal{B}$ be two alphabets. Let
$L\subseteq\mathcal{A}^{*^{d}}$ be a factorial language. A function
$\omega:L\to\mathcal{B}^{*^{d}}$ is a $d$-dimensional morphism if for every
$i$ with $1\leq i\leq d$, and every $u,v\in L$ such that $u\odot^{i}v\in L$ is
defined we have that the concatenation $\omega(u)\odot^{i}\omega(v)$ in
direction ${\boldsymbol{e}}_{i}$ is defined and

$\omega(u\odot^{i}v)=\omega(u)\odot^{i}\omega(v).$

Note that the left-hand side of the equation is defined since $u\odot^{i}v$
belongs to the domain of $\omega$. A $d$-dimensional morphism
$L\to\mathcal{B}^{*^{d}}$ is thus completely defined from the image of the
letters in $\mathcal{A}$, so we sometimes denote a $d$-dimensional morphism as
a rule $\mathcal{A}\to\mathcal{B}^{*^{d}}$ when the language $L$ is
unspecified.

Given a language $L\subseteq\mathcal{A}^{*^{d}}$ of $d$-dimensional words and
a $d$-dimensional morphism $\omega:L\to\mathcal{B}^{*^{d}}$, we define the
image of the language $L$ under $\omega$ as the language

$\overline{\omega(L)}^{Fact}=\\{u\in\mathcal{B}^{*^{d}}\mid u\text{ is a
$d$-dimensional subword of }\omega(v)\text{ with }v\in
L\\}\subseteq\mathcal{B}^{*^{d}}.$

Let $L\subseteq\mathcal{A}^{*^{d}}$ be a factorial language and
$\mathcal{X}_{L}\subseteq\mathcal{A}^{\mathbb{Z}^{d}}$ be the subshift
generated by $L$. A $d$-dimensional morphism $\omega:L\to\mathcal{B}^{*^{d}}$
can be extended to a continuous map
$\omega:\mathcal{X}_{L}\to\mathcal{B}^{\mathbb{Z}^{d}}$ in such a way that the
origin of $\omega(x)$ is at zero position in the word
$\omega(x_{\boldsymbol{0}})$ for all $x\in\mathcal{X}_{L}$. More precisely,
the image under $\omega$ of the configuration $x\in\mathcal{X}_{L}$ is

$\omega(x)=\lim_{n\to\infty}\sigma^{f(n)}\omega\left((\sigma^{-n{\boldsymbol{1}}}x)|_{\llbracket{\boldsymbol{0}},2n{\boldsymbol{1}}\llbracket}\right)\in\mathcal{B}^{\mathbb{Z}^{d}}$

where ${\boldsymbol{1}}=(1,\dots,1)\in\mathbb{Z}^{d}$,
$f(n)=\textsc{shape}\left(\omega\left((\sigma^{-n{\boldsymbol{1}}}x)|_{\llbracket{\boldsymbol{0}},n{\boldsymbol{1}}\llbracket}\right)\right)$
for all $n\in\mathbb{N}$ and
$\llbracket{\boldsymbol{0}},{\boldsymbol{n}}\llbracket={\llbracket
0,n_{1}-1\rrbracket}\times\cdots\times{\llbracket 0,n_{d}-1\rrbracket}$.

In general, the closure under the shift of the image of a subshift
$X\subseteq\mathcal{A}^{\mathbb{Z}^{d}}$ under $\omega$ is the subshift

$\overline{\omega(X)}^{\sigma}=\\{\sigma^{\boldsymbol{k}}\omega(x)\in\mathcal{B}^{\mathbb{Z}^{d}}\mid{\boldsymbol{k}}\in\mathbb{Z}^{d},x\in
X\\}\subseteq\mathcal{B}^{\mathbb{Z}^{d}}.$

The next lemma states that $d$-dimensional morphisms preserve minimality of
subshifts.

######  Lemma 8.

[Lab19b] Let $\omega:X\to\mathcal{B}^{\mathbb{Z}^{d}}$ be a $d$-dimensional
morphism for some $X\subseteq\mathcal{A}^{\mathbb{Z}^{d}}$. If $X$ is a
minimal subshift, then $\overline{\omega(X)}^{\sigma}$ is a minimal subshift.

####  3.4. Self-similar subshifts

In this section, we consider languages and subshifts defined from morphisms
leading to self-similar structures. In this situation, the domain and codomain
of morphisms are defined over the same alphabet. Formally, we consider the
case of $d$-dimensional morphisms $\mathcal{A}\to\mathcal{B}^{*^{d}}$ where
$\mathcal{A}=\mathcal{B}$.

The definition of self-similarity depends on the notion of expansiveness. It
avoids the presence of lower-dimensional self-similar structure by having
expansion in all directions. We say that a $d$-dimensional morphism
$\omega:\mathcal{A}\to\mathcal{A}^{*^{d}}$ is expansive if for every
$a\in\mathcal{A}$ and $K\in\mathbb{N}$, there exists $m\in\mathbb{N}$ such
that $\min(\textsc{shape}(\omega^{m}(a)))>K$.

######  Definition 9.

A subshift $X\subseteq\mathcal{A}^{\mathbb{Z}^{d}}$ is self-similar if there
exists an expansive $d$-dimensional morphism
$\omega:\mathcal{A}\to\mathcal{A}^{*^{d}}$ such that
$X=\overline{\omega(X)}^{\sigma}$.

Respectively, a language $L\subseteq\mathcal{A}^{*^{d}}$ is self-similar if
there exists an expansive $d$-dimensional morphism
$\omega:\mathcal{A}\to\mathcal{A}^{*^{d}}$ such that
$L=\overline{\omega(L)}^{Fact}$. Self-similar languages and subshifts can be
constructed by iterative application of a morphism $\omega$ starting with the
letters. The language $\mathcal{L}_{\omega}$ defined by an expansive
$d$-dimensional morphism $\omega:\mathcal{A}\to\mathcal{A}^{*^{d}}$ is

$\mathcal{L}_{\omega}=\\{u\in\mathcal{A}^{*^{d}}\mid u\text{ is a
$d$-dimensional subword of }\omega^{n}(a)\text{ for some
}a\in\mathcal{A}\text{ and }n\in\mathbb{N}\\}.$

It satisfies
$\mathcal{L}_{\omega}=\overline{\omega(\mathcal{L}_{\omega})}^{Fact}$ and thus
is self-similar. The substitutive subshift
$\mathcal{X}_{\omega}=\mathcal{X}_{\mathcal{L}_{\omega}}$ defined from the
language of $\omega$ is a self-similar subshift since
$\mathcal{X}_{\omega}=\overline{\omega(\mathcal{X}_{\omega})}^{\sigma}$ holds.

Substitutive shift obtained from expansive and primitive morphisms are
interesting for their properties. As in the one-dimensional case, we say that
$\omega$ is primitive if there exists $m\in\mathbb{N}$ such that for every
$a,b\in\mathcal{A}$ the letter $b$ occurs in $\omega^{m}(a)$.

######  Lemma 10.

Let $\omega:\mathcal{A}\to\mathcal{A}^{*^{d}}$ be an expansive and primitive
$d$-dimensional morphism. Then the self-similar substitutive subshift
$\mathcal{X}_{\omega}$ is minimal.

###### Proof.

The substitutive shift of $\omega$ is well-defined since $\omega$ is expansive
and it is minimal since $\omega$ is primitive using standard arguments [Que10,
§5.2]. ∎

###  4\. Toral $\mathbb{Z}^{2}$-rotations and polygon exchange transformations
(PETs)

Let $\Gamma$ be a lattice in $\mathbb{R}^{2}$, i.e., a discrete subgroup of
the additive group $\mathbb{R}^{2}$ with $2$ linearly independent generators.
This defines a $2$-dimensional torus $\mathbf{T}=\mathbb{R}^{2}/\Gamma$. By
analogy with the rotation $x\mapsto x+\alpha$ on the circle
$\mathbb{R}/\mathbb{Z}$ for some $\alpha\in\mathbb{R}/\mathbb{Z}$, we use the
terminology of _rotation_ to denote the following $\mathbb{Z}^{2}$-action
defined on a 2-dimensional torus.

######  Definition 11.

For some ${\boldsymbol{\alpha}},{\boldsymbol{\beta}}\in\mathbf{T}$, we
consider the dynamical system $(\mathbf{T},\mathbb{Z}^{2},R)$ where
$R:\mathbb{Z}^{2}\times\mathbf{T}\to\mathbf{T}$ is the continuous
$\mathbb{Z}^{2}$-action on $\mathbf{T}$ defined by

$R^{\boldsymbol{n}}({\boldsymbol{x}}):=R({\boldsymbol{n}},{\boldsymbol{x}})={\boldsymbol{x}}+n_{1}{\boldsymbol{\alpha}}+n_{2}{\boldsymbol{\beta}}$

for every ${\boldsymbol{n}}=(n_{1},n_{2})\in\mathbb{Z}^{2}$. We say that the
$\mathbb{Z}^{2}$-action $R$ is a toral $\mathbb{Z}^{2}$-rotation or a
$\mathbb{Z}^{2}$-rotation on $\mathbf{T}$ which defines a dynamical system
$(\mathbf{T},\mathbb{Z}^{2},R)$.

When the $\mathbb{Z}^{2}$-action $R$ is a $\mathbb{Z}^{2}$-rotation on the
torus $\mathbf{T}$, the maps $R^{{\boldsymbol{e}}_{1}}$ and
$R^{{\boldsymbol{e}}_{2}}$ can be seen as polygon exchange transformations
[Hoo13, Sch14] on a fundamental domain of $\mathbf{T}$.

######  Definition 12.

[AKY19] Let $X$ be a polygon together with two topological partitions of $X$
into polygons

$X=\bigcup_{k=0}^{N}P_{k}=\bigcup_{k=0}^{N}Q_{k}$

such that for each $k$, $P_{k}$ and $Q_{k}$ are translation equivalent, i.e.,
there exists $v_{k}\in\mathbb{R}^{2}$ such that $P_{k}=Q_{k}+v_{k}$. A polygon
exchange transformation (PET) is the piecewise translation on $X$ defined for
$x\in P_{k}$ by $T(x)=x+v_{k}$. The map is not defined for points
$x\in\bigcup_{k=0}^{N}\partial P_{k}$.

The fact that a rotation on a circle can be seen as a exchange of two
intervals is well-known as noticed for example in [Rau77]. It generalizes in
higher dimension where a generic translation on a $d$-dimensional torus is a
polyhedron exchange transformation defined by the exchange of at most $2^{d}$
pieces on a fundamental domain having for shape a $n$-dimensional
parallelotope. We state a 2-dimensional version of this lemma restricted to
the case of rectangular fundamental domain because we use this connection
several times in the following sections to prove that induced
$\mathbb{Z}^{2}$-actions are again $\mathbb{Z}^{2}$-rotations on a torus.

######  Lemma 13.

Let $\Gamma=\ell_{1}\mathbb{Z}\times\ell_{2}\mathbb{Z}$ be a lattice in
$\mathbb{R}^{2}$ and its rectangular fundamental domain
$D=[0,\ell_{1})\times[0,\ell_{2})$. For every
$\alpha=(\alpha_{1},\alpha_{2})\in D$, the dynamical system
$(\mathbb{R}^{2}/\Gamma,\mathbb{Z},{\boldsymbol{x}}\mapsto{\boldsymbol{x}}+\alpha)$
is measurably conjugate to the dynamical system $(D,\mathbb{Z},T)$ where
$T:D\to D$ is the polygon exchange transformation shown in Figure 5.

$0$$\ell_{1}$$\ell_{1}-\alpha_{1}$$0$$\ell_{2}$$\ell_{2}-\alpha_{2}$$A$$B$$C$$D$$T$$0$$\ell_{1}$$\alpha_{1}$$0$$\ell_{2}$$\alpha_{2}$$T(D)$$T(C)$$T(B)$$T(A)$
Figure 5. The polygon exchange transformation $T$ of the rectangle
$[0,\ell_{1})\times[0,\ell_{2})$ as defined on the figure can be seen as a
toral translation by the vector $(\alpha_{1},\alpha_{2})$ on the torus
$\mathbb{R}^{2}/(\ell_{1}\mathbb{Z}\times\ell_{2}\mathbb{Z})$.

###### Proof.

It follows from the fact that toral rotations and such polygon exchange
transformations are the Cartesian product of circle rotations and exchange of
two intervals. ∎

###  5\. Symbolic dynamical systems for toral $\mathbb{Z}^{2}$-rotations

Let $\Gamma$ be a lattice in $\mathbb{R}^{2}$ and
$\mathbf{T}=\mathbb{R}^{2}/\Gamma$ be a $2$-dimensional torus. Let
$(\mathbf{T},\mathbb{Z}^{2},R)$ be the dynamical system given by a
$\mathbb{Z}^{2}$-rotation $R$ on $\mathbf{T}$. For some $\mathcal{A}$ finite
set, a topological partition of $\mathbf{T}$ is a finite collection
$\\{P_{a}\\}_{a\in\mathcal{A}}$ of disjoint open sets $P_{a}\subset\mathbf{T}$
such that $\mathbf{T}=\bigcup_{a\in\mathcal{A}}\overline{P_{a}}$. If
$S\subset\mathbb{Z}^{2}$ is a finite set, we say that a pattern
$w\in\mathcal{A}^{S}$ is allowed for $\mathcal{P},R$ if

(3) $\bigcap_{{\boldsymbol{k}}\in
S}R^{-{\boldsymbol{k}}}(P_{w_{\boldsymbol{k}}})\neq\varnothing.$

Let $\mathcal{L}_{\mathcal{P},R}$ be the collection of all allowed patterns
for $\mathcal{P},R$. It can be checked that $\mathcal{L}_{\mathcal{P},R}$ is
the language of a subshift. Hence, using standard arguments [LM95, Prop.
1.3.4], there is a unique subshift
$\mathcal{X}_{\mathcal{P},R}\subset\mathcal{A}^{\mathbb{Z}^{2}}$ whose
language is $\mathcal{L}_{\mathcal{P},R}$.

######  Definition 14.

We call $\mathcal{X}_{\mathcal{P},R}$ the symbolic dynamical system
corresponding to $\mathcal{P},R$.

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

$D_{n}(w)=\bigcap_{\|{\boldsymbol{k}}\|\leq
n}R^{-{\boldsymbol{k}}}(P_{w_{\boldsymbol{k}}})\subseteq M.$

The closures $\overline{D}_{n}(w)$ of these sets are compact and decrease with
$n$, so that
$\overline{D}_{0}(w)\supseteq\overline{D}_{1}(w)\supseteq\overline{D}_{2}(w)\supseteq\dots$.
It follows that $\cap_{n=0}^{\infty}\overline{D}_{n}(w)\neq\varnothing$. In
order for points in $\mathcal{X}_{\mathcal{P},R}$ to correspond to points in
$M$, this intersection should contain only one point. This leads to the
following definition.

A topological partition $\mathcal{P}$ of $M$ gives a symbolic representation
of $(M,\mathbb{Z}^{2},R)$ if for every $w\in\mathcal{X}_{\mathcal{P},R}$ the
intersection $\cap_{n=0}^{\infty}\overline{D}_{n}(w)$ consists of exactly one
point $m\in M$. We call $w$ a symbolic representation of $m$.

######  Definition 15.

A topological partition $\mathcal{P}$ of $M$ is a Markov partition for
$(M,\mathbb{Z}^{2},R)$ if

  * •

$\mathcal{P}$ gives a symbolic representation of $(M,\mathbb{Z}^{2},R)$ and

  * •

$\mathcal{X}_{\mathcal{P},R}$ is a shift of finite type (SFT).

The set

$\Delta_{\mathcal{P},R}:=\bigcup_{{\boldsymbol{n}}\in\mathbb{Z}^{2}}R^{\boldsymbol{n}}\left(\bigcup_{a\in\mathcal{A}}\partial
P_{a}\right)\subset\mathbf{T}$

is the set of points whose orbit under the $\mathbb{Z}^{2}$-action of $R$
intersect the boundary of the topological partition
$\mathcal{P}=\\{P_{a}\\}_{a\in\mathcal{A}}$. From Baire Category Theorem
[LM95, Theorem 6.1.24], the set $\mathbf{T}\setminus\Delta_{\mathcal{P},R}$ is
dense in $\mathbf{T}$.

A topological partition $\mathcal{P}=\\{P_{a}\\}_{a\in\mathcal{A}}$ of
$\mathbf{T}=\mathbb{R}^{2}/\Gamma$ is associated to a coding map

$\begin{array}[]{rccl}\textsc{Code}:&\mathbf{T}\setminus\left(\bigcup_{a\in\mathcal{A}}\partial
P_{a}\right)&\to&\mathcal{A}\\\ &{\boldsymbol{x}}&\mapsto&a\quad\text{ if and
only if }\quad{\boldsymbol{x}}\in P_{a}.\end{array}$

For every starting point
${\boldsymbol{x}}\in\mathbf{T}\setminus\Delta_{\mathcal{P},R}$, the coding of
its orbit under the $\mathbb{Z}^{2}$-action of $R$ is a configuration
$\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}\in\mathcal{A}^{\mathbb{Z}^{2}}$
defined by

$\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}({\boldsymbol{n}})=\textsc{Code}(R^{\boldsymbol{n}}({\boldsymbol{x}})).$

for every ${\boldsymbol{n}}\in\mathbb{Z}\times\mathbb{Z}$.

######  Lemma 16.

The symbolic dynamical system $\mathcal{X}_{\mathcal{P},R}$ corresponding to
$\mathcal{P},R$ is the topological closure of the set of configurations:

$\mathcal{X}_{\mathcal{P},R}=\overline{\left\\{\textsc{Config}^{\mathcal{P},R}_{\boldsymbol{x}}\mid{\boldsymbol{x}}\in\mathbf{T}\setminus\Delta_{\mathcal{P},R}\right\\}}.$

###### Proof.

($\supseteq$) Let
${\boldsymbol{x}}\in\mathbf{T}\setminus\Delta_{\mathcal{P},R}$. The patterns
appearing in the configuration
$\textsc{Config}^{\mathcal{P},R}_{\boldsymbol{x}}$ are in
$\mathcal{L}_{\mathcal{P},R}$. Thus
$\textsc{Config}^{\mathcal{P},R}_{\boldsymbol{x}}\in\mathcal{X}_{\mathcal{P},R}$.
The topological closure of such configurations is in
$\mathcal{X}_{\mathcal{P},R}$ since $\mathcal{X}_{\mathcal{P},R}$ is
topologically closed.

($\subseteq$) Let $w\in\mathcal{A}^{S}$ be a pattern with finite support
$S\subset\mathbb{Z}^{2}$ appearing in $\mathcal{X}_{\mathcal{P},R}$. Then
$w\in\mathcal{L}_{\mathcal{P},R}$ and from Equation (3) there exists
${\boldsymbol{x}}\in\mathbf{T}\setminus\Delta_{\mathcal{P},R}$ such that
${\boldsymbol{x}}\in\bigcap_{{\boldsymbol{k}}\in
S}R^{-{\boldsymbol{k}}}(P_{w_{\boldsymbol{k}}})$. The pattern $w$ appears in
the configuration $\textsc{Config}^{\mathcal{P},R}_{\boldsymbol{x}}$. Any
configuration in $\mathcal{X}_{\mathcal{P},R}$ is the limit of a sequence
$(w_{n})_{n\in\mathbb{N}}$ of patterns covering a ball of radius $n$ around
the origin, thus equal to some limit
$\lim_{n\to\infty}\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}_{n}}$ with
${\boldsymbol{x}}_{n}\in\mathbf{T}\setminus\Delta_{\mathcal{P},R}$ for every
$n\in\mathbb{N}$. ∎

######  Lemma 17.

[Lab20a] Let $\mathcal{P}$ give a symbolic representation of the dynamical
system $(\mathbf{T},\mathbb{Z}^{2},R)$. Then

  1. (i)

if $(\mathbf{T},\mathbb{Z}^{2},R)$ is minimal, then
$(\mathcal{X}_{\mathcal{P},R},\mathbb{Z}^{2},\sigma)$ is minimal,

  2. (ii)

if $R$ is a free $\mathbb{Z}^{2}$-action on $\mathbf{T}$, then
$\mathcal{X}_{\mathcal{P},R}$ aperiodic.

##  Part II Induction of $\mathbb{Z}^{2}$-actions and induced partitions of
the 2-torus

This part is divided into four sections on induced $\mathbb{Z}^{2}$-actions;
toral partitions induced by $\mathbb{Z}^{2}$-rotations, renormalization
schemes and algorithms.

###  6\. Induced $\mathbb{Z}^{2}$-actions

Renormalization schemes also known as _Rauzy induction_ were originally
defined for dynamical system including interval exchange transformations (IET)
[Rau79]. A natural way to generalize it to higher dimension is to consider
polygon exchange transformations [Hoo13, AKY19] or even polytope exchange
transformations [Sch14, Sch11] where only one map is considered. But more
dimensions also allows to induce two or more (commuting) maps at the same
time.

In this section, we define the induction of $\mathbb{Z}^{2}$-actions on a sub-
domain. We consider the torus $\mathbf{T}=\mathbb{R}^{2}/\Gamma$ where
$\Gamma$ is a lattice in $\mathbb{R}^{2}$. Let $(\mathbf{T},\mathbb{Z}^{2},R)$
be a minimal dynamical system given by a $\mathbb{Z}^{2}$-action $R$ on
$\mathbf{T}$. For every ${\boldsymbol{n}}\in\mathbb{Z}^{2}$, the toral
translation $R^{{\boldsymbol{n}}}$ can be seen as polygon exchange
transformations on a fundamental domain of $\mathbf{T}$.

The set of return times of ${\boldsymbol{x}}\in\mathbf{T}$ to
$W\subset\mathbf{T}$ under the $\mathbb{Z}^{2}$-action $R$ is the subset of
$\mathbb{Z}\times\mathbb{Z}$ defined as:

$\delta_{W}({\boldsymbol{x}})=\\{{\boldsymbol{n}}\in\mathbb{Z}\times\mathbb{Z}\mid
R^{\boldsymbol{n}}({\boldsymbol{x}})\in W\\}.$

######  Definition 18.

Let $W\subset\mathbf{T}$. We say that the $\mathbb{Z}^{2}$-action $R$ is
Cartesian on $W$ if the set of return times $\delta_{W}({\boldsymbol{x}})$ can
be expressed as a Cartesian product, that is, for all
${\boldsymbol{x}}\in\mathbf{T}$ there exist two strictly increasing sequences
$r_{\boldsymbol{x}}^{(1)},r_{\boldsymbol{x}}^{(2)}:\mathbb{Z}\to\mathbb{Z}$
such that

$\delta_{W}({\boldsymbol{x}})=r_{\boldsymbol{x}}^{(1)}(\mathbb{Z})\times
r_{\boldsymbol{x}}^{(2)}(\mathbb{Z}).$

We always assume that the sequences are shifted in such a way that

$r_{\boldsymbol{x}}^{(i)}(n)\geq 0\iff n\geq 0\qquad\text{ for
}i\in\\{1,2\\}.$

In particular, if ${\boldsymbol{x}}\in W$, then
$(0,0)\in\delta_{W}({\boldsymbol{x}})$, so that
$r_{\boldsymbol{x}}^{(1)}(0)=r_{\boldsymbol{x}}^{(2)}(0)=0$.

When the $\mathbb{Z}^{2}$-action $R$ is Cartesian on $W\subset\mathbf{T}$, we
say that the tuple

(4) $(r_{\boldsymbol{x}}^{(1)}(1),r_{\boldsymbol{x}}^{(2)}(1))$

is the first return time of a starting point ${\boldsymbol{x}}\in\mathbf{T}$
to $W\subset\mathbf{T}$ under the action $R$. When the $\mathbb{Z}^{2}$-action
$R$ is Cartesian on $W\subset\mathbf{T}$, we may consider its return map on
$W$ and we prove in the next lemma that this induces a $\mathbb{Z}^{2}$-action
on $W$.

######  Lemma 19.

If the $\mathbb{Z}^{2}$-action $R$ is Cartesian on $W\subset\mathbf{T}$, then
the map $\widehat{R}|_{W}:\mathbb{Z}^{2}\times W\to W$ defined by

$(\widehat{R}|_{W})^{\boldsymbol{n}}({\boldsymbol{x}}):=\widehat{R}|_{W}({\boldsymbol{n}},{\boldsymbol{x}})=R^{(r_{\boldsymbol{x}}^{(1)}(n_{1}),r_{\boldsymbol{x}}^{(2)}(n_{2}))}({\boldsymbol{x}})$

for every ${\boldsymbol{n}}=(n_{1},n_{2})\in\mathbb{Z}^{2}$ is a well-defined
$\mathbb{Z}^{2}$-action.

We say that $\widehat{R}|_{W}$ is the induced $\mathbb{Z}^{2}$-action of the
$\mathbb{Z}^{2}$-action $R$ on $W$.

###### Proof.

Let ${\boldsymbol{x}}\in W$. We have that

$\widehat{R}|_{W}({\boldsymbol{0}},{\boldsymbol{x}})=R^{(r_{\boldsymbol{x}}^{(1)}(0),r_{\boldsymbol{x}}^{(2)}(0))}({\boldsymbol{x}})=R^{(0,0)}({\boldsymbol{x}})={\boldsymbol{x}}.$

Firstly, using

$r_{\boldsymbol{x}}^{(i)}(k+n)=r_{R^{{\boldsymbol{e}}_{i}\cdot
r_{\boldsymbol{x}}^{(i)}(n)}({\boldsymbol{x}})}(k)+r_{\boldsymbol{x}}^{(i)}(n),$

and skipping few details, we get

$\widehat{R}|_{W}(k{\boldsymbol{e}}_{i}+n{\boldsymbol{e}}_{i},{\boldsymbol{x}})=\widehat{R}|_{W}\left(k{\boldsymbol{e}}_{i},\left(\widehat{R}|_{W}(n{\boldsymbol{e}}_{i},{\boldsymbol{x}})\right)\right).$

Secondly, using the fact that

$r_{\boldsymbol{x}}^{(1)}(k_{1})=r_{\boldsymbol{y}}^{(1)}(k_{1})$

whenever
$y=R^{(0,r_{\boldsymbol{x}}^{(2)}(k_{2}))}{\boldsymbol{x}}=\widehat{R}|_{W}(k_{2}{\boldsymbol{e}}_{2},{\boldsymbol{x}})$,
we obtain

$\displaystyle\widehat{R}|_{W}({\boldsymbol{k}},{\boldsymbol{x}})$
$\displaystyle=R^{(r_{\boldsymbol{x}}^{(1)}(k_{1}),r_{\boldsymbol{x}}^{(2)}(k_{2}))}({\boldsymbol{x}})=R^{(r_{\boldsymbol{x}}^{(1)}(k_{1}),0)}R^{(0,r_{\boldsymbol{x}}^{(2)}(k_{2}))}({\boldsymbol{x}})$
$\displaystyle=R^{(r_{\boldsymbol{x}}^{(1)}(k_{1}),0)}\widehat{R}|_{W}(k_{2}{\boldsymbol{e}}_{2},{\boldsymbol{x}})=\widehat{R}|_{W}\left(k_{1}{\boldsymbol{e}}_{1},\widehat{R}|_{W}(k_{2}{\boldsymbol{e}}_{2},{\boldsymbol{x}})\right).$

Therefore, for every ${\boldsymbol{k}},{\boldsymbol{n}}\in\mathbb{Z}^{2}$, we
have

$\displaystyle(\widehat{R}|_{W})^{{\boldsymbol{k}}+{\boldsymbol{n}}}({\boldsymbol{x}})$
$\displaystyle=(\widehat{R}|_{W})^{(k_{1}+n_{1}){\boldsymbol{e}}_{1}}(\widehat{R}|_{W})^{(k_{2}+n_{2}){\boldsymbol{e}}_{2}}({\boldsymbol{x}})$
$\displaystyle=(\widehat{R}|_{W})^{k_{1}{\boldsymbol{e}}_{1}}(\widehat{R}|_{W})^{n_{1}{\boldsymbol{e}}_{1}}(\widehat{R}|_{W})^{k_{2}{\boldsymbol{e}}_{2}}(\widehat{R}|_{W})^{n_{2}{\boldsymbol{e}}_{2}}({\boldsymbol{x}})$
$\displaystyle=(\widehat{R}|_{W})^{k_{1}{\boldsymbol{e}}_{1}}(\widehat{R}|_{W})^{(n_{1},k_{2})}(\widehat{R}|_{W})^{n_{2}{\boldsymbol{e}}_{2}}({\boldsymbol{x}})$
$\displaystyle=(\widehat{R}|_{W})^{k_{1}{\boldsymbol{e}}_{1}}(\widehat{R}|_{W})^{k_{2}{\boldsymbol{e}}_{2}}(\widehat{R}|_{W})^{n_{1}{\boldsymbol{e}}_{1}}(\widehat{R}|_{W})^{n_{2}{\boldsymbol{e}}_{2}}({\boldsymbol{x}})$
$\displaystyle=(\widehat{R}|_{W})^{\boldsymbol{k}}(\widehat{R}|_{W})^{\boldsymbol{n}}({\boldsymbol{x}}),$

which shows that $\widehat{R}|_{W}$ is a $\mathbb{Z}^{2}$-action on $W$. ∎

A consequence of the lemma is that the induced $\mathbb{Z}^{2}$-action
$\widehat{R}|_{W}$ is generated by two commutative maps

$(\widehat{R}|_{W})^{{\boldsymbol{e}}_{1}}({\boldsymbol{x}})=R^{(r_{\boldsymbol{x}}^{(1)}(1),0)}({\boldsymbol{x}})\quad\text{
and
}\quad(\widehat{R}|_{W})^{{\boldsymbol{e}}_{2}}({\boldsymbol{x}})=R^{(0,r_{\boldsymbol{x}}^{(2)}(1))}({\boldsymbol{x}})$

which are the first return maps of $R^{{\boldsymbol{e}}_{1}}$ and
$R^{{\boldsymbol{e}}_{2}}$ to $W$:

$(\widehat{R}|_{W})^{{\boldsymbol{e}}_{1}}({\boldsymbol{x}})=\widehat{R^{{\boldsymbol{e}}_{1}}}|_{W}({\boldsymbol{x}})\quad\text{
and
}\quad(\widehat{R}|_{W})^{{\boldsymbol{e}}_{2}}({\boldsymbol{x}})=\widehat{R^{{\boldsymbol{e}}_{2}}}|_{W}({\boldsymbol{x}}).$

Recall that the first return map $\widehat{T}|_{W}$ of a dynamical system
$(X,T)$ maps a point ${\boldsymbol{x}}\in W\subset X$ to the first point in
the forward orbit of $T$ lying in $W$, i.e.

$\widehat{T}|_{W}({\boldsymbol{x}})=T^{r({\boldsymbol{x}})}({\boldsymbol{x}})\quad\text{
where
}r({\boldsymbol{x}})=\min\\{k\in\mathbb{Z}_{>0}:T^{k}({\boldsymbol{x}})\in
W\\}.$

From Poincaré’s recurrence theorem, if $\mu$ is a finite $T$-invariant measure
on $X$, then the first return map $\widehat{T}|_{W}$ is well defined for
$\mu$-almost all ${\boldsymbol{x}}\in W$. When $T$ is a translation on a
torus, if the subset $W$ is open, then the first return map is well-defined
for every point ${\boldsymbol{x}}\in W$. Moreover if $W$ is a polygon, then
the first return map $\widehat{T}|_{W}$ is a polygon exchange transformation.
An algorithm to compute the induced transformation
$\widehat{T}|_{W}=\widehat{R^{{\boldsymbol{e}}_{i}}}|_{W}$ of the sub-action
$R^{{\boldsymbol{e}}_{i}}$ is provided in Section 9.

###  7\. Toral partitions induced by toral $\mathbb{Z}^{2}$-rotations

For IETs, the interval on which we define the Rauzy induction is usually given
by one of the atom of the partition which defines the IET itself. In our
setting, it is not the case. The partition that we use carries more
information than the natural partition which allows to define the
$\mathbb{Z}^{2}$-rotation $R$ as a pair of polygon exchange transformations.
The partition is a refinement of the natural partition involving well-chosen
sloped lines.

Let $\Gamma$ be a lattice in $\mathbb{R}^{2}$ and
$\mathbf{T}=\mathbb{R}^{2}/\Gamma$ be a $2$-dimensional torus. Let
$(\mathbf{T},\mathbb{Z}^{2},R)$ be the dynamical system given by a
$\mathbb{Z}^{2}$-rotation $R$ on $\mathbf{T}$. Assuming the
$\mathbb{Z}^{2}$-rotation $R$ is Cartesian on a window $W\subset\mathbf{T}$,
then there exist two strictly increasing sequences
$r_{\boldsymbol{x}}^{(1)},r_{\boldsymbol{x}}^{(2)}:\mathbb{Z}\to\mathbb{Z}$
are such that

$(r,s)=\left(r_{\boldsymbol{x}}^{(1)}(1),r_{\boldsymbol{x}}^{(2)}(1)\right)$

is the first return time of a starting point ${\boldsymbol{x}}\in\mathbf{T}$
to the window $W$ under the action $R$, see Equation (4). It allows to define
the return word map as

$\begin{array}[]{rccl}\textsc{ReturnWord}:&W&\to&\mathcal{A}^{*^{2}}\\\
&{\boldsymbol{x}}&\mapsto&\left(\begin{array}[]{ccc}\textsc{Code}(R^{(0,s-1)}{\boldsymbol{x}})&\cdots&\textsc{Code}(R^{(r-1,s-1)}{\boldsymbol{x}})\\\
\cdots&\cdots&\cdots\\\
\textsc{Code}(R^{(0,0)}{\boldsymbol{x}})&\cdots&\textsc{Code}(R^{(r-1,0)}{\boldsymbol{x}})\end{array}\right)\end{array},$

where $r,s\geq 1$ both obviously depend on ${\boldsymbol{x}}$.

The image $\mathcal{L}=\textsc{ReturnWord}(W)\subset\mathcal{A}^{*^{2}}$ is a
language called the set of return words. When the return time to $W$ is
bounded, the set of return words $\mathcal{L}$ is finite. Let
$\mathcal{L}=\\{w_{b}\\}_{b\in\mathcal{B}}$ be an enumeration of $\mathcal{L}$
for some finite set $\mathcal{B}$.

######  Remark 20.

The way the enumeration of $\mathcal{L}$ is done influences the substitutions
which is obtained afterward. To obtain a canonical ordering when the words are
1-dimensional, we use the total order $(\mathcal{L},\prec)$ defined by $u\prec
v$ if $|u|<|v|$ or $|u|=|v|$ and $u<_{lex}v$. See Line 12 of Algorithm 1.

The induced partition of $\mathcal{P}$ by the action of $R$ on the sub-domain
$W$ is a topological partition of $W$ defined as the set of preimage sets
under ReturnWord:

$\widehat{\mathcal{P}}|_{W}=\\{\textsc{ReturnWord}^{-1}(w_{b})\\}_{b\in\mathcal{B}}.$

This yields the induced coding on $W$

$\begin{array}[]{rccl}\textsc{Code}|_{W}:&W&\to&\mathcal{B}\\\
&{\boldsymbol{y}}&\mapsto&b\quad\text{ if and only if
}\quad{\boldsymbol{y}}\in\textsc{ReturnWord}^{-1}(w_{b}).\end{array}$

A natural substitution comes out of this induction procedure:

(5) $\begin{array}[]{rccl}\omega:&\mathcal{B}&\to&\mathcal{A}^{*^{2}}\\\
&b&\mapsto&w_{b}.\end{array}$

The partition $\widehat{\mathcal{P}}|_{W}$ of $W$ can be effectively computed
by the refinement of the partition $\mathcal{P}$ with translated copies of the
sub-domain $W$ under the action of $R$. In Section 9. we propose Algorithm 1
to compute the induced partition $\widehat{\mathcal{P}}|_{W}$ and substitution
$\omega$. The next result shows that the coding of the orbit under the
$\mathbb{Z}^{2}$-rotation $R$ is the image under the 2-dimensional
substitution $\omega$ of the coding of the orbit under the
$\mathbb{Z}^{2}$-action $\widehat{R}|_{W}$.

######  Lemma 21.

If the $\mathbb{Z}^{2}$-action $R$ is Cartesian on a window
$W\subset\mathbf{T}$, then $\omega$ is a $2$-dimensional morphism, and for
every ${\boldsymbol{x}}\in W$ we have

$\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}=\omega\left(\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}}\right).$

###### Proof.

Let ${\boldsymbol{x}}\in W$. By hypothesis, there exists
$P,Q\subset\mathbb{Z}$ such that the set of returns times satisfies
$\delta_{W}({\boldsymbol{x}})=P\times Q$ and we may write
$P=\\{r_{i}\\}_{i\in\mathbb{Z}}$ and $Q=\\{s_{j}\\}_{j\in\mathbb{Z}}$ as
increasing sequences such that $r_{0}=s_{0}=0$. Therefore,
$\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}$ may be decomposed into a
lattice of rectangular blocks. More precisely, for every $i,j\in\mathbb{Z}$,
the following block is the image of a letter under $\omega$:

$\displaystyle\left(\begin{array}[]{ccc}\textsc{Code}(R^{(r_{i},s_{j+1}-1)}{\boldsymbol{x}})&\cdots&\textsc{Code}(R^{(r_{i+1}-1,s_{j+1}-1)}{\boldsymbol{x}})\\\
\cdots&\cdots&\cdots\\\
\textsc{Code}(R^{(r_{i},s_{j})}{\boldsymbol{x}})&\cdots&\textsc{Code}(R^{(r_{i+1}-1,s_{j})}{\boldsymbol{x}})\end{array}\right)$
$\displaystyle\qquad=\textsc{ReturnWord}(R^{(r_{i},s_{j})}{\boldsymbol{x}})=w_{b_{ij}}=\omega(b_{ij})$

for some letter $b_{ij}\in\mathcal{B}$. Moreover,

$b_{ij}=\textsc{Code}|_{W}(R^{(r_{i},s_{j})}{\boldsymbol{x}})=\textsc{Code}|_{W}\left((\widehat{R}|_{W})^{(i,j)}{\boldsymbol{x}}\right).$

Since the adjacent blocks have matching dimensions, for every
$i,j\in\mathbb{Z}$, the following concatenations

$\displaystyle\omega\left(b_{ij}\odot^{1}b_{(i+1)j}\right)=\omega\left(b_{ij}\right)\odot^{1}\omega\left(b_{(i+1)j}\right)\qquad\text{
and }$
$\displaystyle\omega\left(b_{ij}\odot^{2}b_{i(j+1)}\right)=\omega\left(b_{ij}\right)\odot^{2}\omega\left(b_{i(j+1)}\right)$

are well defined. Thus $\omega$ is a $2$-dimensional morphism on the set
$\left\\{\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}}\,\middle|\,{\boldsymbol{x}}\in
W\right\\}$ and we have

$\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}=\omega\left(\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}}\right)$

which ends the proof. Note that the domain of $\omega$ can be extended to its
topological closure. ∎

######  Proposition 22.

Let $\mathcal{P}$ be a topological partition of $\mathbf{T}$. If the
$\mathbb{Z}^{2}$-action $R$ is Cartesian on a window $W\subset\mathbf{T}$,
then
$\mathcal{X}_{\mathcal{P},R}=\overline{\omega(\mathcal{X}_{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}})}^{\sigma}$.

###### Proof.

Let

$Y=\left\\{\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}\,\middle|\,{\boldsymbol{x}}\in\mathbf{T}\right\\}\quad\text{
and }\quad
Z=\left\\{\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}}\,\middle|\,{\boldsymbol{x}}\in
W\right\\},$

($\supseteq$). Let ${\boldsymbol{x}}\in W$. From Lemma 21,
$\omega\left(\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}}\right)=\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}$
with ${\boldsymbol{x}}\in W\subset\mathbf{T}$. ($\subseteq$). Let
${\boldsymbol{x}}\in\mathbf{T}$. There exists $k_{1},k_{2}\in\mathbb{N}$ such
that ${\boldsymbol{x}}^{\prime}=R^{-(k_{1},k_{2})}({\boldsymbol{x}})\in W$.
Therefore, we have
${\boldsymbol{x}}=R^{(k_{1},k_{2})}({\boldsymbol{x}}^{\prime})$ where $0\leq
k_{1}<r({\boldsymbol{x}}^{\prime})$ and $0\leq
k_{2}<s({\boldsymbol{x}}^{\prime})$. Thus the shift
${\boldsymbol{k}}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ is bounded by the maximal
return time of $R^{{\boldsymbol{e}}_{1}}$ and $R^{{\boldsymbol{e}}_{2}}$ to
$W$. We have

$\displaystyle\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}$
$\displaystyle=\textsc{Config}^{\mathcal{P},R}_{R^{\boldsymbol{k}}{\boldsymbol{x}}^{\prime}}=\sigma^{{\boldsymbol{k}}}\circ\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}^{\prime}}$
$\displaystyle=\sigma^{{\boldsymbol{k}}}\circ\omega\left(\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}^{\prime}}\right)$

where we used Lemma 21 with ${\boldsymbol{x}}^{\prime}\in W$. We conclude that
$Y=\overline{\omega(Z)}^{\sigma}$. The result follows from Lemma 16 by taking
the topological closure on both sides. ∎

#### Including the first vs the last letter

We finish with a remark on the definition of the return word map. There is a
choice to be made in its definition whether we include the first or the last
letter of the orbit that comes back to the window (vertically and
horizontally). For example, another option is to define return words that
includes the last letter instead of the first letter. In general, for each
value of $\epsilon_{1},\epsilon_{2}\in\\{0,1\\}$, we may define

$\begin{array}[]{rccl}\textsc{ReturnWord}_{\epsilon_{1},\epsilon_{2}}:&W&\to&\mathcal{A}^{*^{2}}\\\
&{\boldsymbol{x}}&\mapsto&\left(\begin{array}[]{ccc}\textsc{Code}(R^{(\epsilon_{1},\epsilon_{2}+s-1)}{\boldsymbol{x}})&\cdots&\textsc{Code}(R^{(\epsilon_{1}+r-1,\epsilon_{2}+s-1)}{\boldsymbol{x}})\\\
\cdots&\cdots&\cdots\\\
\textsc{Code}(R^{(\epsilon_{1},\epsilon_{2})}{\boldsymbol{x}})&\cdots&\textsc{Code}(R^{(\epsilon_{1}+r-1,\epsilon_{2})}{\boldsymbol{x}})\end{array}\right)\end{array}$

where $r=r_{\boldsymbol{x}}^{(1)}({\boldsymbol{x}})$ and
$s=r_{\boldsymbol{x}}^{(2)}({\boldsymbol{x}})$. An alternative is to consider
ReturnWord for the $\mathbb{Z}^{2}$-actions $R^{(-n_{1},n_{2})}$,
$R^{(n_{1},-n_{2})}$ or $R^{(-n_{1},-n_{2})}$ and taking horizontal and/or
vertical mirror images of the obtained return words.

###  8\. Renormalization schemes

In what follows, we define a renormalization scheme on a topological partition
of the domain $X$ together with a dynamical systems $(X,\mathbb{Z}^{2},R)$
defined by a $\mathbb{Z}^{2}$-action $R$ on $X$.

Sometimes, when $R$ is a $\mathbb{Z}^{2}$-action on $X$, the induced
$\mathbb{Z}^{2}$-action $\widehat{R}|_{W}$ can be renormalized as a
$\mathbb{Z}^{2}$-action on the original domain $X$. This leads to the
definition of renormalization schemes. The following definition is inspired
from [AKY19] and adapted to the case dynamical systems defined by
$\mathbb{Z}^{2}$-actions.

######  Definition 23.

A dynamical system $(X,\mathbb{Z}^{2},R)$ has a renormalization scheme if
there exists a proper subset $W\subset X$, a homeomorphism $\phi:W\to X$, a
renormalized dynamical system $(X,\mathbb{Z}^{2},R^{\prime})$ such that

$(R^{\prime})^{\boldsymbol{n}}=\phi\circ(\widehat{R}|_{W})^{\boldsymbol{n}}\circ\phi^{-1}$

for all ${\boldsymbol{n}}\in\mathbb{Z}^{2}$, meaning that the following
diagram commute:

${W}$${X}$${W}$${X}$$\scriptstyle{\phi}$$\scriptstyle{(\widehat{R}|_{W})^{\boldsymbol{n}}}$$\scriptstyle{(R^{\prime})^{\boldsymbol{n}}}$$\scriptstyle{\phi}$

The dynamical system $(X,\mathbb{Z}^{2},R)$ is self-induced if it is equal to
the renormalized one, that is, if $R=R^{\prime}$.

Now we define the renormalization of a topological partition
$\mathcal{P}=\\{P_{a}\\}_{a\in\mathcal{A}}$ of $X$. If a dynamical system
$(X,\mathbb{Z}^{2},R)$ has a renormalization scheme with the same notations as
above, then the induced partition
$\widehat{\mathcal{P}}|_{W}=\\{Q_{b}\\}_{b\in\mathcal{B}}$ of $W$ can be
renormalized as a partition

$\phi(\widehat{\mathcal{P}}|_{W})=\left\\{\phi(Q_{b})\right\\}_{b\in\mathcal{B}}$

of $X$. If the dynamical system $(X,\mathbb{Z}^{2},R)$ is self-induced and
there exists a permutation $\pi:\mathcal{B}\to\mathcal{A}$ such that
$P_{\pi(b)}=\phi(Q_{b})$ for all $b\in\mathcal{B}$, then we say that the
topological partition $\mathcal{P}$ of $X$ is self-induced.

######  Proposition 24.

Let $\mathcal{P}$ be a topological partition of $\mathbf{T}$ and suppose that
the $\mathbb{Z}^{2}$-action $R$ is Cartesian on a window $W\subset\mathbf{T}$.
If $\mathcal{P}$ is self-induced with permutation
$\pi:\mathcal{B}\to\mathcal{A}$, then $\mathcal{X}_{\mathcal{P},R}$ is self-
similar satisfying
$\mathcal{X}_{\mathcal{P},R}=\overline{\omega\pi(\mathcal{X}_{\mathcal{P},R})}^{\sigma}$
where $\omega$ is the $2$-dimensional morphism defined in Eq. (5).

###### Proof.

From Proposition 22, we have
$\mathcal{X}_{\mathcal{P},R}=\overline{\omega(\mathcal{X}_{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}})}^{\sigma}$.
It remains to show
$\mathcal{X}_{\mathcal{P},R}=\pi\left(\mathcal{X}_{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}\right)$.
Since $\mathcal{P}$ is self-induced, we have
$\pi\circ\textsc{Code}|_{W}=\textsc{Code}\circ\phi$. Thus for every
$m,n\in\mathbb{Z}$ and every ${\boldsymbol{x}}\in W$, we have

$\displaystyle\pi\left(\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}}({\boldsymbol{n}})\right)$
$\displaystyle=\pi\circ\textsc{Code}|_{W}\circ(\widehat{R}|_{W})^{\boldsymbol{n}}({\boldsymbol{x}})=\textsc{Code}\circ\phi\circ(\widehat{R}|_{W})^{\boldsymbol{n}}({\boldsymbol{x}})$
$\displaystyle=\textsc{Code}\circ
R^{\boldsymbol{n}}\circ\phi({\boldsymbol{x}})=\textsc{Config}^{\mathcal{P},R}_{\phi({\boldsymbol{x}})}({\boldsymbol{n}}).$

Since $\phi:W\to\mathbf{T}$ is a homeomorphism, we obtain

$\displaystyle\mathcal{X}_{\mathcal{P},R}$
$\displaystyle=\overline{\left\\{\textsc{Config}^{\mathcal{P},R}_{{\boldsymbol{x}}}\,\middle|\,{\boldsymbol{x}}\in\mathbf{T}\right\\}}=\overline{\left\\{\textsc{Config}^{\mathcal{P},R}_{\phi({\boldsymbol{x}})}\,\middle|\,{\boldsymbol{x}}\in
W\right\\}}$
$\displaystyle=\overline{\left\\{\pi\left(\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}}\right)\,\middle|\,{\boldsymbol{x}}\in
W\right\\}}$
$\displaystyle=\pi\left(\overline{\left\\{\textsc{Config}^{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}_{{\boldsymbol{x}}}\,\middle|\,{\boldsymbol{x}}\in
W\right\\}}\right)=\pi\left(\mathcal{X}_{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}\right)$

and the conclusion follows. ∎

###  9\. Algorithms

Algorithm 1 Compute the induced partition $\widehat{\mathcal{P}}|_{W}$ and
substitution $\omega$ associated to the induced transformation
$\widehat{T}|_{W}$ (we use it when $T=R^{{\boldsymbol{e}}_{i}}$ for some $i$.)

1:$T$ is a polytope exchange transformation (PET) on a convex domain
$D\subset\mathbb{R}^{d}$ and $\mathcal{G}$ is a partition of $D$ into convex
polytopes such that the restriction of $T$ on each atom of $\mathcal{G}$ is
continuous; ${\boldsymbol{v}}\in\mathbb{R}^{d+1}$ defines a half space
$H_{\boldsymbol{v}}=\\{{\boldsymbol{x}}\in\mathbb{R}^{d}\mid
v_{0}+\sum_{i=1}^{d}v_{i}x_{i}\geq 0\\}$ such that $D\cap
H_{\boldsymbol{v}}=W$; $\mathcal{P}$ is a list of pairs $(a,p)$ such that
$\\{p\mid(a,p)\in\mathcal{P}\\}$ is a partition of $D$ into convex polytopes
indexed by the alphabet $\mathcal{A}=\\{a\mid(a,p)\in\mathcal{P}\\}$.

2:function InducedPartition($T$, ${\boldsymbol{v}}$, $\mathcal{P}$)

3: $\mathcal{Q}\leftarrow\\{(\varepsilon,W)\\}$ $\triangleright$
$\varepsilon\in\mathcal{A}^{*}$ is the empty word and $W=D\cap
H_{\boldsymbol{v}}$

4: $\mathcal{K}\leftarrow T(\mathcal{P}\wedge\mathcal{G})=\\{(a,T(p\cap
g))\mid(a,p)\in\mathcal{P}\text{ and }g\in\mathcal{G}\text{ and }p\cap
g\neq\varnothing\\}$

5: $S\leftarrow\textsc{EmptyList}($)

6: while $\mathcal{Q}$ not empty do

7: $\mathcal{Q}\leftarrow
T^{-1}(\mathcal{Q}\wedge\mathcal{K})=\\{(au,T^{-1}(q\cap
k))\mid(u,q)\in\mathcal{Q}\text{ and }(a,k)\in\mathcal{K}\text{ and }q\cap
k\neq\varnothing\\}$

8: $S\leftarrow S\cup\left(\mathcal{Q}\wedge
H_{\boldsymbol{v}}\right)=S\cup\left\\{\left(u,q\cap
H_{\boldsymbol{v}}\right)\mid(u,q)\in\mathcal{Q}\text{ and }q\cap
H_{\boldsymbol{v}}\neq\varnothing\right\\}$

9: $\mathcal{Q}\leftarrow\mathcal{Q}\wedge\left(\mathbb{R}^{d}\setminus
H_{\boldsymbol{v}}\right)=\left\\{\left(u,q\cap\left(\mathbb{R}^{d}\setminus
H_{\boldsymbol{v}}\right)\right)\mid(u,q)\in\mathcal{Q}\text{ and
}q\cap\left(\mathbb{R}^{d}\setminus
H_{\boldsymbol{v}}\right)\neq\varnothing\right\\}$

10: $\mathcal{L}\leftarrow\\{u\in\mathcal{A}^{*}\mid(u,q)\in S\\}$
$\triangleright$ the set of return words

11: $\mathcal{B}\leftarrow\\{0,1,\dots,\\#\mathcal{L}-1\\}$ $\triangleright$
the new alphabet

12: $\omega\leftarrow$ bijection $\mathcal{B}\to\mathcal{L}$ $\triangleright$
such that $i<j$ if and only if $\omega(i)\prec\omega(j)$

13: $\mathcal{P}^{\prime}\leftarrow\\{(\omega^{-1}(u),q)\mid(u,q)\in S\\}$
$\triangleright$ the induced partition labeled by $\mathcal{B}$

14: return $(\mathcal{P}^{\prime},\omega)$

15:$\\{q\mid(b,q)\in\mathcal{P}^{\prime}\\}=\widehat{\mathcal{P}}|_{W}$ is the
induced partition of $W$ into convex polytopes; $\mathcal{P}^{\prime}$ is a
list of pairs $(b,q)$ such that $\textsc{Code}|_{W}({\boldsymbol{x}})=b$ for
every ${\boldsymbol{x}}\in q$; the map $\omega:\mathcal{B}\to\mathcal{A}^{*}$
extends to a morphism of monoid $\omega:\mathcal{B}^{*}\to\mathcal{A}^{*}$
satisfying the following equation for one-dimensional subshifts:
$\mathcal{X}_{\mathcal{P},T}=\overline{\omega\left(\mathcal{X}_{\widehat{\mathcal{P}}|_{W},\widehat{T}|_{W}}\right)}^{\sigma}$;
if $T=R^{{\boldsymbol{e}}_{i}}$ for some $i$ and the $\mathbb{Z}^{2}$-action
$R$ is Cartesian on the window $W$, then
$\omega:\mathcal{B}\to\mathcal{A}^{*}$ defines a $d$-dimensional morphism in
the direction ${\boldsymbol{e}}_{i}$ satisfying
$\mathcal{X}_{\mathcal{P},R}=\overline{\omega\left(\mathcal{X}_{\widehat{\mathcal{P}}|_{W},\widehat{R}|_{W}}\right)}^{\sigma}$.

Algorithm 2 Compute the induced transformation $\widehat{T}|_{W}$ (alternate
definition)

1:$T$ is a polytope exchange transformation (PET) on a convex domain
$D\subset\mathbb{R}^{d}$ given as a pair $(\mathcal{P},h)$ where $\mathcal{P}$
is a list of pairs $(a,p)$ such that $\\{p\mid(a,p)\in\mathcal{P}\\}$ is a
partition of $D$ into convex polytopes,
$\mathcal{A}=\\{a\mid(a,p)\in\mathcal{P}\\}$ is some alphabet and
$h:\mathcal{A}\to\mathbb{R}^{d}$ is a map such that $(a,p)\in\mathcal{P}$
implies that $T({\boldsymbol{x}})={\boldsymbol{x}}+h(a)$ for all
${\boldsymbol{x}}\in p$, the map $h$ extends to a morphism of monoids
$h:\mathcal{A}^{*}\to\mathbb{R}^{d}$ satisfying $h(u\cdot v)=h(u)+h(v)$;
${\boldsymbol{v}}\in\mathbb{R}^{d+1}$ defines a half space
$H_{\boldsymbol{v}}=\\{{\boldsymbol{x}}\in\mathbb{R}^{d}\mid
v_{0}+\sum_{i=1}^{d}v_{i}x_{i}\geq 0\\}$ such that $D\cap
H_{\boldsymbol{v}}=W$.

2:function InducedTransformation($T$, ${\boldsymbol{v}}$)

3: $(\mathcal{P},h)\leftarrow T$

4:
$(\mathcal{P}^{\prime},\omega)\leftarrow\textsc{InducedPartition}(T,{\boldsymbol{v}},\mathcal{P})$

5: $T^{\prime}\leftarrow(\mathcal{P}^{\prime},h\circ\omega)$ $\triangleright$
the induced transformation

6: return $(T^{\prime},\omega)$

7:$T^{\prime}$ is a PET equal to the induced transformation $\widehat{T}|_{W}$
given as a pair $(\mathcal{P}^{\prime},h\circ\omega)$ where
$\\{q\mid(b,q)\in\mathcal{P}^{\prime}\\}$ is a partition of $W$ into convex
polytopes, $\mathcal{B}=\\{b\mid(b,q)\in\mathcal{P}^{\prime}\\}$ is some
alphabet, $h\circ\omega:\mathcal{B}^{*}\to\mathbb{R}^{d}$ is a morphism of
monoids such that $(b,q)\in\mathcal{P}^{\prime}$ implies that
$T^{\prime}({\boldsymbol{x}})={\boldsymbol{x}}+h\circ\omega(b)$ for all
${\boldsymbol{x}}\in q$; $\omega:\mathcal{B}^{*}\to\mathcal{A}^{*}$ is a
morphism of monoid satisfying the following equation for one-dimensional
subshifts:
$\mathcal{X}_{\mathcal{P},T}=\overline{\omega\left(\mathcal{X}_{\widehat{\mathcal{P}}|_{W},\widehat{T}|_{W}}\right)}^{\sigma}$.

In this section, we provide two algorithms to compute the induced partition
and induced transformation of a polytope exchange transformation on a sub-
domain. More precisely, Algorithm 1 computes the induced partition
$\widehat{\mathcal{P}}|_{W}$ of a partition $\mathcal{P}$ of $D$ by a polytope
exchange transformation $T:D\to D$ on a sub-domain $W\subset D$. It also
computes the substitution $\omega$ allowing to express configuration coded by
the partition $\mathcal{P}$ as the image of configurations coded by the
induced partition $\widehat{\mathcal{P}}|_{W}$. Algorithm 2 computes the
induced transformation $\widehat{T}|_{W}$ of a polytope exchange
transformation $T:D\to D$ on a sub-domain $W\subset D$. We first present the
algorithm computing the induced partition since the other one can be deduced
from it.

Algorithms are written for domain $D\subset\mathbb{R}^{d}$, with $d\geq 1$, as
they work in arbitrary dimension. All polytopes manipulated in the algorithms
are assumed to be open so that the intersection of two polytopes is nonempty
if and only if it of positive volume and convex so that they can be
represented as a set of linear inequalities. It is possible to deal with
polytope partitions where some polytopes are not convex, but it is to the user
responsability to split the non-convex atoms into many convex polytopes.
Partitions are represented as a list of pairs $(a,p)$ where $a$ is the index
associated to the convex polytope $p$. The same index can be used for
different polytopes.

We assume that the polytope exchange transformation $T:D\to D$ is defined on
domain $D\subset\mathbb{R}^{d}$ which is a convex polytope and that the sub-
domain $W$ on which the induction is being done is such that $W=D\cap
H_{\boldsymbol{v}}$ where ${\boldsymbol{v}}\in\mathbb{R}^{d+1}$ and
$H_{\boldsymbol{v}}$ is a half-space given by the part of $\mathbb{R}^{d}$ on
one side of a hyperplane:

$H_{\boldsymbol{v}}=\left\\{{\boldsymbol{x}}\in\mathbb{R}^{d}\;\middle|\;v_{0}+\sum_{i=1}^{d}v_{i}x_{i}\geq
0\right\\}.$

The reason is that in the algorithms, polytopes are intersected sometimes with
$H_{\boldsymbol{v}}$ and sometimes with the complement
$\mathbb{R}^{d}\setminus H_{\boldsymbol{v}}$, that is, both sides of the
hyperplane given by ${\boldsymbol{v}}$ and we want the result to be convex in
both cases. Inducing on more general polytope $W$ must be done in many steps
(once for each inequalities defining $W$).

When considering $\mathbb{Z}^{d}$-action $R$ on a polytope
$D\subset\mathbb{R}^{d}$ that is Cartesian on a window $W\subset D$, it is
possible to compute the induced $\mathbb{Z}^{d}$-action $\widehat{R}|_{W}$ by
considering each subaction $\widehat{R^{{\boldsymbol{e}}_{i}}}|_{W}$
individually for $i\in\\{1,\dots,d\\}$. In the next sections, we use the
algorithms when the return times to $W\subset D$ under
$R^{{\boldsymbol{e}}_{j}}$ is $1$ for all ${\boldsymbol{x}}\in W$ and for
every direction ${\boldsymbol{e}}_{j}$ except for some $j=i$. In that case,
the set of return words $\mathcal{L}=\textsc{ReturnWord}(W)$ is a set of one-
dimensional words in the direction ${\boldsymbol{e}}_{i}$ for some $i$ and the
substitution $\omega$ is a $d$-dimensional morphism.

Algorithm 1 and Algorithm 2 are implemented in the module on polyhedron
exchange transformations of the optional package slabbe [Lab20b] for SageMath
[Sag20].

##  Part III Substitutive structure of $\mathcal{X}_{\mathcal{P}_{0},R_{0}}$

In this part, we start with the partition introduced in [Lab20a] shown in
Figure 3 and construct a sequence of induced partition until the induction
process loops. The proofs can be verified with a computer as we provide the
SageMath code to reproduce the computations of all induced
$\mathbb{Z}^{2}$-actions, induced partitions and 2-dimensional morphisms.

###  10\. Inducing the $\mathbb{Z}^{2}$-action $R_{0}$ and the partition
$\mathcal{P}_{0}$ to get $\mathcal{X}_{\mathcal{P}_{1},R_{1}}$

Let $\Gamma_{0}=\langle(\varphi,0),(1,\varphi+3)\rangle_{\mathbb{Z}}$ be a
lattice with $\varphi=\frac{1+\sqrt{5}}{2}$. We consider the dynamical system
$(\mathbb{R}^{2}/\Gamma_{0},\mathbb{Z}^{2},R_{0})$ defined by the
$\mathbb{Z}^{2}$-action

$\begin{array}[]{rccl}R_{0}:&\mathbb{Z}^{2}\times\mathbb{R}^{2}/\Gamma_{0}&\to&\mathbb{R}^{2}/\Gamma_{0}\\\
&({\boldsymbol{n}},{\boldsymbol{x}})&\mapsto&{\boldsymbol{x}}+{\boldsymbol{n}}.\end{array}$

We consider the topological partition $\mathcal{P}_{0}$ of
$\mathbb{R}^{2}/\Gamma_{0}$ illustrated in Figure 6 on a fundamental domain
where each atom is associated to a letter in the alphabet
$\mathcal{A}_{0}=\llbracket 0,10\rrbracket$. The partition $\mathcal{P}_{0}$
defines a coding map
$\textsc{Code}_{0}:\mathbb{R}^{2}/\Gamma_{0}\to\mathcal{A}_{0}$ and it was
proved in [Lab20a] that the coding of orbits under the $\mathbb{Z}^{2}$-action
$R_{0}$ are valid Wang configuration using the 11 Wang tiles from Jeandel-Rao
tile set.

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{{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}\qquad\begin{array}[]{l}R_{0}^{\boldsymbol{n}}({\boldsymbol{x}})={\boldsymbol{x}}+{\boldsymbol{n}}\quad\bmod\Gamma_{0}\end{array}$
Figure 6. The partition $\mathcal{P}_{0}$ of $\mathbb{R}^{2}/\Gamma_{0}$ into
atoms associated to letters in $\mathcal{A}_{0}$ illustrated on a rectangular
fundamental domain.

######  Proposition 25.

Let $(\mathbb{R}^{2}/\Gamma_{1},\mathbb{Z}^{2},R_{1})$ be a dynamical system
with lattice $\Gamma_{1}=\varphi\mathbb{Z}\times\mathbb{Z}$ and
$\mathbb{Z}^{2}$-action

$\begin{array}[]{rccl}R_{1}:&\mathbb{Z}^{2}\times\mathbb{R}^{2}/\Gamma_{1}&\to&\mathbb{R}^{2}/\Gamma_{1}\\\
&({\boldsymbol{n}},{\boldsymbol{x}})&\mapsto&{\boldsymbol{x}}+n_{1}{\boldsymbol{e}}_{1}+n_{2}(\varphi^{-1},\varphi^{-2}).\end{array}$

Let $\mathcal{P}_{1}$ be the topological partition illustrated in Figure 9
where each atom is associated to a letter in $\mathcal{A}_{1}=\llbracket
0,27\rrbracket$. Then
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}=\overline{\beta_{0}(\mathcal{X}_{\mathcal{P}_{1},R_{1}})}^{\sigma}$
where $\beta_{0}:\mathcal{A}_{1}^{*^{2}}\to\mathcal{A}_{0}^{*^{2}}$ is the
$2$-dimensional morphism defined by

(6)
$\beta_{0}:\left\\{\footnotesize\begin{array}[]{lllllll}0\mapsto\left(\begin{array}[]{r}7\\\
3\\\ 9\\\ 0\end{array}\right),&1\mapsto\left(\begin{array}[]{r}5\\\ 7\\\ 9\\\
0\end{array}\right),&2\mapsto\left(\begin{array}[]{r}4\\\ 10\\\ 9\\\
0\end{array}\right),&3\mapsto\left(\begin{array}[]{r}7\\\ 3\\\ 3\\\
1\end{array}\right),&4\mapsto\left(\begin{array}[]{r}6\\\ 7\\\ 3\\\
1\end{array}\right),&5\mapsto\left(\begin{array}[]{r}7\\\ 8\\\ 3\\\
1\end{array}\right),&6\mapsto\left(\begin{array}[]{r}5\\\ 2\\\ 7\\\
1\end{array}\right),\\\ 7\mapsto\left(\begin{array}[]{r}5\\\ 5\\\ 7\\\
1\end{array}\right),&8\mapsto\left(\begin{array}[]{r}6\\\ 5\\\ 7\\\
1\end{array}\right),&9\mapsto\left(\begin{array}[]{r}5\\\ 7\\\ 8\\\
1\end{array}\right),&10\mapsto\left(\begin{array}[]{r}4\\\ 10\\\ 8\\\
1\end{array}\right),&11\mapsto\left(\begin{array}[]{r}5\\\ 4\\\ 10\\\
1\end{array}\right),&12\mapsto\left(\begin{array}[]{r}6\\\ 4\\\ 10\\\
1\end{array}\right),&13\mapsto\left(\begin{array}[]{r}7\\\ 3\\\ 3\\\ 9\\\
0\end{array}\right),\\\ 14\mapsto\left(\begin{array}[]{r}6\\\ 7\\\ 3\\\ 9\\\
0\end{array}\right),&15\mapsto\left(\begin{array}[]{r}7\\\ 8\\\ 3\\\ 9\\\
0\end{array}\right),&16\mapsto\left(\begin{array}[]{r}5\\\ 2\\\ 7\\\ 9\\\
0\end{array}\right),&17\mapsto\left(\begin{array}[]{r}6\\\ 2\\\ 7\\\ 9\\\
0\end{array}\right),&18\mapsto\left(\begin{array}[]{r}5\\\ 5\\\ 7\\\ 9\\\
0\end{array}\right),&19\mapsto\left(\begin{array}[]{r}6\\\ 5\\\ 7\\\ 9\\\
0\end{array}\right),&20\mapsto\left(\begin{array}[]{r}5\\\ 7\\\ 8\\\ 9\\\
0\end{array}\right),\\\ 21\mapsto\left(\begin{array}[]{r}4\\\ 10\\\ 8\\\ 9\\\
0\end{array}\right),&22\mapsto\left(\begin{array}[]{r}6\\\ 4\\\ 10\\\ 9\\\
0\end{array}\right),&23\mapsto\left(\begin{array}[]{r}6\\\ 7\\\ 3\\\ 3\\\
1\end{array}\right),&24\mapsto\left(\begin{array}[]{r}6\\\ 7\\\ 8\\\ 3\\\
1\end{array}\right),&25\mapsto\left(\begin{array}[]{r}6\\\ 5\\\ 2\\\ 7\\\
1\end{array}\right),&26\mapsto\left(\begin{array}[]{r}6\\\ 4\\\ 10\\\ 8\\\
1\end{array}\right),&27\mapsto\left(\begin{array}[]{r}6\\\ 5\\\ 4\\\ 10\\\
1\end{array}\right).\end{array}\right.$

###### Proof.

Let $W=(0,\varphi)\times(0,1)$. The $\mathbb{Z}^{2}$-action $R_{0}$ is
Cartesian on the window $W$. Thus from Lemma 19, the induced
$\mathbb{Z}^{2}$-action $\widehat{R_{0}}|_{W}:\mathbb{Z}^{2}\times W\to W$
given by

$(\widehat{R_{0}}|_{W})^{\boldsymbol{n}}({\boldsymbol{x}})=(\widehat{R_{0}^{{\boldsymbol{e}}_{1}}}|_{W})^{n_{1}}\circ(\widehat{R_{0}^{{\boldsymbol{e}}_{2}}}|_{W})^{n_{2}}({\boldsymbol{x}}),$

for every ${\boldsymbol{n}}=(n_{1},n_{2})\in\mathbb{Z}^{2}$, is well-defined.
The first return time to $W$ under $R_{0}$ is $(1,4)$ or $(1,5)$. The orbits
of points in $W$ until the first return to $W$ under
$R_{0}^{{\boldsymbol{e}}_{2}}$ is shown in Figure 7.

$0$$1$$2$$3$$4$$\varphi-1$$\varphi$$\varphi+1$$\varphi+2$$\varphi+3$$0$$1$$\frac{1}{\varphi}$$\varphi$$A$$B$$C$$D$$R_{0}^{(0,1)}(A)$$R_{0}^{(0,1)}(B)$$R_{0}^{(0,1)}(C)$$R_{0}^{(0,1)}(D)$$R_{0}^{(0,2)}(A)$$R_{0}^{(0,2)}(B)$$R_{0}^{(0,2)}(C)$$R_{0}^{(0,2)}(D)$$R_{0}^{(0,3)}(A)$$R_{0}^{(0,3)}(B)$$R_{0}^{(0,3)}(C)$$R_{0}^{(0,3)}(D)$$R_{0}^{(0,4)}(C)$$R_{0}^{(0,4)}(D)$$0$$1$$\frac{1}{\varphi^{2}}$$0$$1$$\frac{1}{\varphi}$$\varphi$$R_{0}^{(0,4)}(B)$$R_{0}^{(0,4)}(A)$$R_{0}^{(0,5)}(D)$$R_{0}^{(0,5)}(C)$
Figure 7. The set $\\{A,B,C,D\\}$ forms a partition of the rectangle
$W=(0,\varphi)\times(0,1)$. The return time to $W$ under the map
$R_{0}^{{\boldsymbol{e}}_{2}}$ is 4 or 5. The orbit of $A$, $B$, $C$ and $D$
under $R_{0}^{{\boldsymbol{e}}_{2}}$ before it returns to $W$ yields a
partitions of $\mathbb{R}^{2}/\Gamma_{0}$. The first return map
$\widehat{R_{0}^{{\boldsymbol{e}}_{2}}}|_{W}$ is equivalent to a toral
translation by the vector $(\frac{1}{\varphi},\frac{1}{\varphi^{2}})$ on
$\mathbb{R}^{2}/\Gamma_{1}$.

The maximal subsets of $W$ on which the first return map
$\widehat{R_{0}^{{\boldsymbol{e}}_{2}}}|_{W}$ is continuous yield a partition
$\\{A,B,C,D\\}$ of $W$ defined as

$\displaystyle A=(0,1)\times(\varphi^{-1},1),$ $\displaystyle
B=(1,\varphi)\times(\varphi^{-1},1),$ $\displaystyle
C=(0,1)\times(0,\varphi^{-1}),$ $\displaystyle
D=(1,\varphi)\times(0,\varphi^{-1}).$

The return time of $A\cup B$ to $W$ under $R_{0}^{{\boldsymbol{e}}_{2}}$ is 4.
The return time of $C\cup D$ to $W$ under $R_{0}^{{\boldsymbol{e}}_{2}}$ is 5.
The first return maps $\widehat{R_{0}^{{\boldsymbol{e}}_{2}}}|_{W}$ is defined
as exchanges of rectangles $\\{A,B,C,D\\}$ which, from Lemma 13, is equivalent
to a toral translation by $(\varphi^{-1},\varphi^{-2})$ on
$\mathbb{R}^{2}/\Gamma_{1}$.

The first return map $\widehat{R_{0}^{{\boldsymbol{e}}_{1}}}|_{W}$ is defined
as an exchange of two rectangles, see Figure 8, which, from Lemma 13, is
equivalent to a toral translation by ${\boldsymbol{e}}_{1}$ on
$\mathbb{R}^{2}/\Gamma_{1}$. Thus, we may identify $W$ with the torus
$\mathbb{R}^{2}/\Gamma_{1}$ and define the induced $\mathbb{Z}^{2}$-action as
toral rotations on $\mathbb{R}^{2}/\Gamma_{1}$:

$(\widehat{R_{0}}|_{W})^{\boldsymbol{n}}({\boldsymbol{x}})={\boldsymbol{x}}+n_{1}{\boldsymbol{e}}_{1}+n_{2}(\varphi^{-1},\varphi^{-2})\quad\bmod\Gamma_{1}.$

We observe that $R_{1}=\widehat{R_{0}}|_{W}$.

$0$$1$$\frac{1}{\varphi^{2}}$$0$$1$$\frac{1}{\varphi}$$\varphi$$L$$R$$0$$1$$\frac{1}{\varphi^{2}}$$0$$1$$\frac{1}{\varphi}$$\varphi$$R_{0}^{{\boldsymbol{e}}_{1}}(R)$$R_{0}^{{\boldsymbol{e}}_{1}}(L)$
Figure 8.  The return time to $W$ under map $R_{0}^{{\boldsymbol{e}}_{1}}$ is
always 1. The first return map $\widehat{R_{0}^{{\boldsymbol{e}}_{1}}}|_{W}$
is equivalent to a toral translation by the vector $(1,0)$ on
$\mathbb{R}^{2}/\Gamma_{1}$.

Thus we have that the return word function $\textsc{ReturnWord}:W\to\llbracket
0,10\rrbracket^{*^{2}}$ is

${\boldsymbol{x}}\mapsto\left(\begin{array}[]{c}\textsc{Code}_{0}(R_{0}^{(0,3)}{\boldsymbol{x}})\\\
\textsc{Code}_{0}(R_{0}^{(0,2)}{\boldsymbol{x}})\\\
\textsc{Code}_{0}(R_{0}^{(0,1)}{\boldsymbol{x}})\\\
\textsc{Code}_{0}(R_{0}^{(0,0)}{\boldsymbol{x}})\end{array}\right)\text{if
}{\boldsymbol{x}}\in A\cup B\quad\text{ or
}\quad{\boldsymbol{x}}\mapsto\left(\begin{array}[]{c}\textsc{Code}_{0}(R_{0}^{(0,4)}{\boldsymbol{x}})\\\
\textsc{Code}_{0}(R_{0}^{(0,3)}{\boldsymbol{x}})\\\
\textsc{Code}_{0}(R_{0}^{(0,2)}{\boldsymbol{x}})\\\
\textsc{Code}_{0}(R_{0}^{(0,1)}{\boldsymbol{x}})\\\
\textsc{Code}_{0}(R_{0}^{(0,0)}{\boldsymbol{x}})\end{array}\right)\text{if
}{\boldsymbol{x}}\in C\cup D.$

We get the set of return words $\mathcal{L}=\\{w_{b}\\}_{b\in\llbracket
0,27\rrbracket}=\textsc{ReturnWord}(W)$ as listed in (6). The partition
induced by $R_{0}$ on $W$ is

$\mathcal{P}_{1}:=\widehat{\mathcal{P}_{0}}|_{W}=\\{\textsc{ReturnWord}^{-1}(w_{b})\\}_{b\in\llbracket
0,27\rrbracket}$

which is a topological partition of $W$ made of 28 atoms (the atoms
$\textsc{ReturnWord}^{-1}(w_{19})$ and $\textsc{ReturnWord}^{-1}(w_{22})$ are
both the union of two triangles), see Figure 9.

$\begin{array}[]{l}\mathcal{P}_{1}=\end{array}\begin{array}[]{cc}\includegraphics{SageCreation/partition_JR_P1.pdf}\\\
\end{array}\quad\begin{array}[]{l}R_{1}^{\boldsymbol{n}}({\boldsymbol{x}})={\boldsymbol{x}}+n_{1}{\boldsymbol{e}}_{1}+n_{2}(\varphi^{-1},\varphi^{-2})\quad\bmod\Gamma_{1}\end{array}$
Figure 9. The partition $\mathcal{P}_{1}:=\widehat{\mathcal{P}_{0}}|_{W}$ of
$\mathbb{R}^{2}/\Gamma_{1}$ into 30 convex atoms each associated to one of the
28 letters in $\mathcal{A}_{1}$ (indices 19 and 22 are both used twice).

The induced coding is

$\begin{array}[]{rccl}\textsc{Code}_{1}:=\textsc{Code}_{0}|_{W}:&W&\to&\llbracket
0,27\rrbracket\\\ &{\boldsymbol{y}}&\mapsto&b\quad\text{ if and only if
}{\boldsymbol{y}}\in\textsc{ReturnWord}^{-1}(w_{b}).\end{array}$

A natural substitution comes out of this construction:

$\begin{array}[]{rccc}\beta_{0}:&\llbracket 0,27\rrbracket&\to&\llbracket
0,10\rrbracket^{*}\\\ &b&\mapsto&w_{b}\end{array}$

which corresponds to the one defined in Equation (6). From Proposition 22, we
have
$\mathcal{X}_{\mathcal{P}_{0},R_{0}}=\overline{\beta_{0}(\mathcal{X}_{\mathcal{P}_{1},R_{1}})}^{\sigma}$
which ends the proof. ∎

######  Remark 26 (SageMath Code).

The following allows to reproduce the proof of Proposition 25 using SageMath
[Sag20] with the optional package slabbe [Lab20b]. First we construct the
golden mean as a element of a quadratic number field because it is more
efficient for arithmetic operations and comparisons:

    sage: z = polygen(QQ, ’z’)
    sage: K = NumberField(z**2-z-1, ’phi’, embedding=RR(1.6))
    sage: phi = K.gen()

We import the polygon partition $\mathcal{P}_{0}$ of
$\mathbb{R}^{2}/\Gamma_{0}$ which is predefined in slabbe:

    sage: from slabbe.arXiv_1903_06137 import jeandel_rao_wang_shift_partition
    sage: P0 = jeandel_rao_wang_shift_partition()

We import polyhedron exchange transformations from the package:

    sage: from slabbe import PolyhedronExchangeTransformation as PET

We define the lattice $\Gamma_{0}$ and the maps
$R_{0}^{{\boldsymbol{e}}_{1}}$, $R_{0}^{{\boldsymbol{e}}_{2}}$ which can be
seen as a polygon exchange transformations on a rectangular fundamental domain
of $\mathbb{R}^{2}/\Gamma_{0}$:

    sage: Gamma0 = matrix.column([(phi,0), (1,phi+3)])
    sage: fundamental_domain = polytopes.parallelotope([(phi,0), (0,phi+3)])
    sage: R0e1 = PET.toral_translation(Gamma0, vector((1,0)), fundamental_domain)
    sage: R0e2 = PET.toral_translation(Gamma0, vector((0,1)), fundamental_domain)

We compute the induced partition $\mathcal{P}_{1}$ of
$\mathbb{R}^{2}/\Gamma_{1}$, the substitution $\beta_{0}$ and the induced
$\mathbb{Z}^{2}$-action $R_{1}$:

    sage: y_le_1 = [1, 0, -1]   # syntax for the inequality y <= 1
    sage: P1,beta0 = R0e2.induced_partition(y_le_1, P0, substitution_type=’column’)
    sage: R1e1,_ = R0e1.induced_transformation(y_le_1)
    sage: R1e2,_ = R0e2.induced_transformation(y_le_1)

######  Remark 27 (SageMath Code).

To observe and verify the computed induced partitions, substitutions and
induced $\mathbb{Z}^{2}$-action, one can do:

    sage: P1.plot()                     # or P1.tikz().pdf()
    sage: show(beta0)
    sage: R1e1
    Polyhedron Exchange Transformation of Polyhedron partition of 2 atoms with 2 letters
    with translations {0: (1, 0), 1: (-phi + 1, 0)}
    sage: R1e1.plot()
    sage: R1e2
    Polyhedron Exchange Transformation of Polyhedron partition of 4 atoms with 4 letters
    with translations {0: (phi - 1, -phi + 1), 1: (-1, -phi + 1),
                       2: (phi - 1, -phi + 2), 3: (-1, -phi + 2)}
    sage: R1e2.plot()

###  11\. Changing the base of the $\mathbb{Z}^{2}$-action to get
$\mathcal{X}_{\mathcal{P}_{2},R_{2}}$

Recall that $\Gamma_{1}=\varphi\mathbb{Z}\times\mathbb{Z}$ and that the coding
on the torus $\mathbb{R}^{2}/\Gamma_{1}$ is given by the
$\mathbb{Z}^{2}$-action $R_{1}$ of
$({\boldsymbol{n}},{\boldsymbol{x}})\mapsto{\boldsymbol{x}}+n_{1}{\boldsymbol{e}}_{1}+n_{2}(\varphi^{-1},\varphi^{-2})$.
We remark that $R_{1}^{{\boldsymbol{e}}_{1}}$ is a horizontal translation
while $R_{1}^{{\boldsymbol{e}}_{2}}$ is not. But we observe that
$R_{1}^{{\boldsymbol{e}}_{1}+{\boldsymbol{e}}_{2}}=R_{1}^{{\boldsymbol{e}}_{1}}\circ
R_{1}^{{\boldsymbol{e}}_{2}}$ is a vertical translation
${\boldsymbol{x}}\mapsto{\boldsymbol{x}}+(0,\varphi^{-2})$. So it is natural
to change the base of the $\mathbb{Z}^{2}$-action $R_{1}$ by using
$R_{1}^{{\boldsymbol{e}}_{1}+{\boldsymbol{e}}_{2}}$ instead of
$R_{1}^{{\boldsymbol{e}}_{2}}$. Therefore we let

$R_{2}^{(n_{1},n_{2})}({\boldsymbol{x}})=R_{1}^{(n_{1}+n_{2},n_{2})}({\boldsymbol{x}})={\boldsymbol{x}}+n_{1}{\boldsymbol{e}}_{1}+\varphi^{-2}n_{2}{\boldsymbol{e}}_{2}$

for every ${\boldsymbol{n}}=(n_{1},n_{2})\in\mathbb{Z}^{2}$ which defines an
action
$\mathbb{Z}^{2}\times\mathbb{R}^{2}/\Gamma_{2}\to\mathbb{R}^{2}/\Gamma_{2}$
where $\Gamma_{2}=\Gamma_{1}$. We keep the partition as is, that is
$\mathcal{A}_{2}=\mathcal{A}_{1}$, $\mathcal{P}_{2}=\mathcal{P}_{1}$ and
$\textsc{Code}_{2}=\textsc{Code}_{1}$. We get the setup illustrated in Figure
10. We show in this section that this base change corresponds to a shear
conjugacy
$\beta_{1}:\mathcal{X}_{\mathcal{P}_{2},R_{2}}\to\mathcal{X}_{\mathcal{P}_{1},R_{1}}$.

######  Remark 28 (SageMath Code).

The following defines $\mathcal{P}_{2}$, $R_{2}^{{\boldsymbol{e}}_{1}}$ and
$R_{2}^{{\boldsymbol{e}}_{2}}$ in SageMath:

    sage: P2 = P1
    sage: R2e1 = R1e1
    sage: R2e2 = (R1e1 * R1e2).merge_atoms_with_same_translation()

$\begin{array}[]{l}\mathcal{P}_{2}=\end{array}\begin{array}[]{cc}\includegraphics{SageCreation/partition_JR_P1.pdf}\\\
\end{array}\begin{array}[]{l}R_{2}^{\boldsymbol{n}}({\boldsymbol{x}})={\boldsymbol{x}}+n_{1}{\boldsymbol{e}}_{1}+\varphi^{-2}n_{2}{\boldsymbol{e}}_{2}\quad\bmod\Gamma_{2}\end{array}$
Figure 10. The partition $\mathcal{P}_{2}$ of $\mathbb{R}^{2}/\Gamma_{2}$ into
atoms associated to letters in $\mathcal{A}_{2}$.

In general we have the following result.

######  Lemma 29.

Let $\mathcal{A}$ be some finite alphabet. For every
$M\in\textrm{GL}_{d}(\mathbb{Z})$, the map

$\begin{array}[]{rccl}\theta_{M}:&\mathcal{A}^{\mathbb{Z}^{d}}&\to&\mathcal{A}^{\mathbb{Z}^{d}}\\\
&w&\mapsto&({\boldsymbol{n}}\mapsto w({M^{-1}{\boldsymbol{n}}}))\end{array}$

is a $\textrm{GL}_{d}(\mathbb{Z})$-conjugacy of the full shift
$(\mathcal{A}^{\mathbb{Z}^{d}},\mathbb{Z}^{d},\sigma)$ satisfying
$\sigma^{M{\boldsymbol{k}}}\circ\theta_{M}=\theta_{M}\circ\sigma^{\boldsymbol{k}}$
for all ${\boldsymbol{k}}\in\mathbb{Z}^{d}$.

###### Proof.

The map $\theta_{M}$ is continuous and admits an inverse $\theta_{M^{-1}}$
which is also continuous. If $w\in\mathcal{A}^{\mathbb{Z}^{d}}$ and
${\boldsymbol{k}},{\boldsymbol{n}}\in\mathbb{Z}^{d}$, then

$\displaystyle\left[\sigma^{M{\boldsymbol{k}}}\circ\theta_{M}(w)\right]({\boldsymbol{n}})$
$\displaystyle=\left[\theta_{M}(w)\right]({\boldsymbol{n}}+M{\boldsymbol{k}})=w(M^{-1}({\boldsymbol{n}}+M{\boldsymbol{k}}))=w(M^{-1}{\boldsymbol{n}}+{\boldsymbol{k}})$
$\displaystyle=\left[\sigma^{{\boldsymbol{k}}}w\right](M^{-1}{\boldsymbol{n}})=\left[\theta_{M}(\sigma^{{\boldsymbol{k}}}w)\right]({\boldsymbol{n}})=\left[\theta_{M}\circ\sigma^{{\boldsymbol{k}}}(w)\right]({\boldsymbol{n}}).$

Hence $\theta_{M}$ is a $\textrm{GL}_{d}(\mathbb{Z})$-conjugacy of
$(\mathcal{A}^{\mathbb{Z}^{d}},\mathbb{Z}^{d},\sigma)$. ∎

Let
$\beta_{1}=\theta_{M}:\mathcal{A}_{2}^{\mathbb{Z}^{2}}\to\mathcal{A}_{1}^{\mathbb{Z}^{2}}$
be the map defined by the shear matrix $M=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$ where $\mathcal{A}_{1}=\mathcal{A}_{2}=\llbracket
0,27\rrbracket$.

######  Lemma 30.

The map $\beta_{1}$ is a shear conjugacy
$\beta_{1}:\mathcal{X}_{\mathcal{P}_{2},R_{2}}\to\mathcal{X}_{\mathcal{P}_{1},R_{1}}$
satisfying
$\sigma^{M{\boldsymbol{k}}}\circ\beta_{1}=\beta_{1}\circ\sigma^{\boldsymbol{k}}$
for every ${\boldsymbol{k}}\in\mathbb{Z}^{d}$ with
$M=\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$.

###### Proof.

From Lemma 29, $\beta_{1}$ is a shear-conjugacy
$(\mathcal{A}_{2}^{\mathbb{Z}^{2}},\mathbb{Z}^{2},\sigma)\to(\mathcal{A}_{1}^{\mathbb{Z}^{2}},\mathbb{Z}^{2},\sigma)$
satisfying
$\sigma^{M{\boldsymbol{k}}}\circ\beta_{1}=\beta_{1}\circ\sigma^{\boldsymbol{k}}$
for all ${\boldsymbol{k}}\in\mathbb{Z}^{d}$. The restriction of the map
$\beta_{1}$ on the domain $\mathcal{X}_{\mathcal{P}_{2},R_{2}}$ is a shear
conjugacy onto its image. It remains to show that the image satisfies
$\beta_{1}(\mathcal{X}_{\mathcal{P}_{2},R_{2}})=\mathcal{X}_{\mathcal{P}_{1},R_{1}}$.

For every
${\boldsymbol{x}}\in\mathbb{R}^{2}/\Gamma_{1}=\mathbb{R}^{2}/\Gamma_{2}$ and
$m,n\in\mathbb{Z}$, we have

$\displaystyle\beta_{1}\left(\textsc{Orbit}^{\mathcal{P}_{2},R_{2}}_{{\boldsymbol{x}}}(m,n)\right)$
$\displaystyle=\textsc{Orbit}^{\mathcal{P}_{2},R_{2}}_{{\boldsymbol{x}}}(m-n,n)=\textsc{Code}_{2}(R_{2}^{(m-n,n)}({\boldsymbol{x}}))$
$\displaystyle=\textsc{Code}_{1}(R_{1}^{(m,n)}({\boldsymbol{x}}))=\textsc{Orbit}^{\mathcal{P}_{1},R_{1}}_{{\boldsymbol{x}}}(m,n)$

Therefore

$\displaystyle\beta_{1}(\mathcal{X}_{\mathcal{P}_{2},R_{2}})$
$\displaystyle=\beta_{1}\left(\overline{\left\\{\textsc{Config}^{\mathcal{P}_{2},R_{2}}_{\boldsymbol{x}}\mid{\boldsymbol{x}}\in\mathbb{R}^{2}/\Gamma_{2}\setminus\Delta_{\mathcal{P}_{2},R_{2}}\right\\}}\right)$
$\displaystyle=\overline{\left\\{\beta_{1}\left(\textsc{Config}^{\mathcal{P}_{2},R_{2}}_{\boldsymbol{x}}\right)\mid{\boldsymbol{x}}\in\mathbb{R}^{2}/\Gamma_{2}\setminus\Delta_{\mathcal{P}_{2},R_{2}}\right\\}}$
$\displaystyle=\overline{\left\\{\textsc{Config}^{\mathcal{P}_{1},R_{1}}_{\boldsymbol{x}}\mid{\boldsymbol{x}}\in\mathbb{R}^{2}/\Gamma_{1}\setminus\Delta_{\mathcal{P}_{1},R_{1}}\right\\}}=\mathcal{X}_{\mathcal{P}_{1},R_{1}}.$

Thus $\beta_{1}$ is a shear conjugacy
$\mathcal{X}_{\mathcal{P}_{2},R_{2}}\to\mathcal{X}_{\mathcal{P}_{1},R_{1}}$. ∎

###  12\. Inducing the symbolic dynamical system
$\mathcal{X}_{\mathcal{P}_{2},R_{2}}$ to get
$\mathcal{X}_{\mathcal{P}_{8},R_{8}}$ and
$\mathcal{X}_{\mathcal{P}_{\mathcal{U}},R_{\mathcal{U}}}$

In this section, we induce the topological partition $\mathcal{P}_{2}$ until
the process loops. We need six induction steps before obtaining a topological
partition $\mathcal{P}_{8}$ which is self-induced, see Figure 11.