# Multidimensional continued fractions and symbolic codings of toral
translations

Valérie Berthé  ,  Wolfgang Steiner  Université de Paris, CNRS, IRIF, F-75013
Paris, France<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Jörg M. Thuswaldner  Chair
of Mathematics and Statistics, University of Leoben, A-8700 Leoben, AUSTRIA
<EMAIL_ADDRESS>
###### Abstract.

It has been a long standing problem to find good symbolic codings for
translations on the $d$-dimensional torus that enjoy the beautiful properties
of Sturmian sequences like low complexity and good local discrepancy
properties (i.e., bounded remainder sets of any scale). Inspired by Rauzy’s
approach we construct such codings in terms of multidimensional continued
fraction algorithms that are realized by sequences of substitutions. In
particular, given any strongly convergent continued fraction algorithm, these
sequences lead to renormalization schemes which produce symbolic codings and
bounded remainder sets at all scales in a natural way.

As strong convergence of a continued fraction algorithm results in a Pisot
type property of the attached symbolic dynamical systems, our approach
provides a systematic way to confirm purely discrete spectrum results for wide
classes of dynamical systems. Indeed, as our examples illustrate, we are able
to confirm the Pisot conjecture for many well-known families of sequences of
substitutions. These examples comprise classical algorithms like the Jacobi-
Perron, Brun, Cassainge-Selmer, and Arnoux-Rauzy algorithms.

As a consequence, we gain symbolic codings of almost all translations of the
$2$-dimensional torus having subword complexity $2n+1$ that are balanced on
all factors (and hence, bounded remainder sets at all scales). Using the Brun
algorithm, we also give symbolic codings of almost all $3$-dimensional torus
translations with multi-scale bounded remainder sets.

###### Key words and phrases:

Symbolic dynamics; nonstationary dynamics; multidimensional continued fraction
algorithms; $S$-adic dynamical systems; substitutions; tilings; Pisot numbers;
Lyapunov exponents

###### 2010 Mathematics Subject Classification:

37B10, 37A30, 11K50, 28A80

This work was supported by the Agence Nationale de la Recherche through the
project CODYS (ANR-18-CE40-0007). The third author was supported by the
projects FWF P27050 and FWF P29910 granted by the Austrian Science Fund.

##  1\. Introduction

One of the classical motivations of symbolic dynamics is to provide
representations of dynamical systems as subshifts made of infinite sequences
which code itineraries through suitable choices of partitions. In the present
paper we focus on symbolic models for toral translations. More precisely, for
a given toral translation, we provide symbolic realizations based on
multidimensional continued fraction algorithms. These realizations have strong
dynamical and arithmetic properties. In particular, they define bounded
remainder sets for toral translations with a natural subdivision structure
governed by the underlying continued fraction algorithm. We recall that
bounded remainder sets are defined as sets having bounded local discrepancy.
In ergodic terms, these are sets for which the Birkhoff sums of their
characteristic function have bounded deviations. Their study started with the
work of W. M. Schmidt in his series of papers on irregularities of
distributions (see for instance [Sch74]) and has generated an important
literature (see e.g. [GL15] for the according references).

Our approach is inspired by the seminal example of Sturmian dynamical systems,
introduced by M. Morse and G. Hedlund in [MH40]. There is an impressive
literature devoted to their study and to possible generalizations in word
combinatorics, and also in digital geometry [Fog02]. This is due to several
reasons. Sturmian dynamical systems provide symbolic codings for the simplest
arithmetic dynamical systems, namely for irrational translations on the
circle, they also code discrete lines, and they are one-dimensional models of
quasicrystals [BG13]. The scale invariance of Sturmian dynamical systems
allows them to be described by using a renormalization scheme governed by
usual continued fractions which in turn can be interpreted as Poincaré
sections of the geodesic flow acting on the modular surface. This admits
important generalizations in the study of interval exchange transformations in
relation with the Teichmüller flow and renormalization schemes that can often
be interpreted as continued fractions [Yoc06]. The basic combinatorial
elements for their understanding are substitutions which are symbolic versions
of induction steps (i.e., of first return maps).

In order to get symbolic models, in the present work we rely on substitutions
and so-called $S$-adic systems. A substitution is a rule, either combinatorial
or geometric, that replaces a letter by a word, or a tile by a patch of tiles.
Substitutions play a fundamental role in symbolic dynamics, such as emphasized
e.g. in the monographs [BG13, Fog02, Que10]. In prticular, Pisot substitutions
are of importance in this context since they create hierarchical structures
with a significant amount of long range order [ABB+15]. Substitutive dynamical
systems defined in terms of Pisot substitutions are conjectured to have pure
discrete spectrum, that is, to be isomorphic (in the measure-theoretic sense)
to a translation on a compact abelian group. The still open Pisot substitution
conjecture, even if solved for beta-numeration [Bar18], shows that important
parts of the picture are still to be developed.

More generally, $S$-adic dynamical systems are defined in terms of words that
are generated by iterating sequences of substitutions rather than iterating
just a single substitution [BD14] much the same way like multidimensional
continued fraction algorithms in general produce sequences of matrices (and
not just powers of a single one). This formalism offers representations
similar to the Bratteli–Vershik systems related to Markov compacta, and to
representations by Rohlin towers (see e.g. [BR10, Chapter 6]). In [BST19a], we
extend the Pisot conjecture to the $S$-adic setting, which enables us to go
beyond algebraicity. The associated $S$-adic systems are defined as sequences
of substitutions which can be regarded as non-abelian versions of
multidimensional continued fraction algorithms. The Pisot condition is
replaced by the requirement that the second Lyapunov exponent of the system is
negative. In [BST19a] we prove that the extended Pisot conjecture holds for
large families of $S$-adic dynamical systems based on well-known continued
fraction algorithms, such as the Brun or the Arnoux–Rauzy algorithm. As a
striking outcome, this yields symbolic codings for almost every translation of
${\mathbb{T}}^{2}$ [BST19a], paving the way for the development of
equidistribution results for the associated two-dimensional Kronecker
sequences.

In the present paper we extend this study to higher dimensions and handle many
well-known continued fraction algorithms. Our new theory works for many
generalized continued fraction algorithms including classical ones like the
Brun, Selmer, and Jacobi–Perron algorithm. To each strongly convergent
continued fraction algorithm we attach a shift of $S$-adic sequences which
generically leads to $S$-adic dynamical systems having pure discrete spectrum.
This shows that $S$-adic systems are measurably conjugate to minimal
translations on the torus under quite mild assumptions (of Pisot type). In
other words, we provide symbolic representations of toral translations, i.e.,
symbolic dynamical systems that code in the measure-theoretic sense toral
translations, as well as symbolic representations for multidimensional
continued fractions.

We use two main ingredients, namely a Pisot type property, that can be seen as
a strong convergence property in the setting of continued fractions, and the
existence of particular substitutive dynamical systems that “behave well”. We
mention that some of our results on the purely discrete spectrum of $S$-adic
dynamical systems do not require “coincidence type” conditions which so far
were commonly needed in this context in order to get purely discrete spectrum.
In particular, we can prove that each algorithm that satisfies the Pisot
condition has an acceleration that leads to toral translations almost surely.

In our proofs we also heavily rely on the theory on $S$-adic Rauzy fractals
which has been developed in [BST19a]. For an illustration of such a Rauzy
fractal, see Figure 1. This allows us to verify the Pisot conjecture on
sequences of substitutions for wide families of systems of Pisot type thereby
extending the results in [BST19a, FN20]. Already in [BST19a] for the Brun as
well as the Arnoux–Rauzy algorithm discrete spectrum results have been shown.
Parallel to our work, [FN20] proved results on pure discrete spectrum of
$S$-adic systems coming from continued fraction algorithms with special
emphasis on the Cassaigne–Selmer algorithm. The conditions we have to assume
in our main results are easy to check effectively and our present results
(stated in Section 3) are more general than the ones in [BST19a, FN20]. This
allows us to treat the Arnoux–Rauzy algorithm in arbitrary dimensions as well
as multiplicative continued fraction algorithms as the Jacobi–Perron algorithm
(which requires to work with $S$-adic systems based on infinitely many
substitutions).

Our results can be used to define multiscale bounded remainder sets and
natural codings for almost all translations on the torus. Note that the
constructions of bounded remainder sets given in [GL15, HKK17] do not offer
such a scalability. As applications and motivation for the present results, we
are currently considering higher-dimensional versions of the three-distance
theorem in [ABK+20] where the involved shapes are generated by symbolic and
geometric versions of continued fractions algorithms (related again to
$S$-adic Rauzy fractals). In [ABM+20] we also consider Markov partitions for
nonstationary hyperbolic toral automorphisms related to continued fraction
algorithms. We thereby develop symbolic models as nonstationary subshifts of
finite type and Markov partitions for sequences of toral automorphisms. The
pieces of the corresponding Markov partitions are fractal sets (and more
precisely $S$-adic Rauzy fractals) obtained by associating substitutions to
(incidence) matrices, or in terms of Bratteli diagrams, obtained by
constructing suspensions via two-sided Markov compacta [Buf14].

Figure 1. An $S$-adic Rauzy fractal and its subdivision (cf. Section 2.4)
whose directive sequence $\boldsymbol{\sigma}=(\sigma_{n})_{n\in\mathbb{N}}$
starts with $\sigma_{0}=\dots=\sigma_{7}$ and $\sigma_{8}=\dots=\sigma_{15}$,
where $\sigma_{0}=\tau$ with $\tau$ as in (6.3) and $\sigma_{8}$ is the
classical Tribonacci substitution $1\mapsto 12$, $2\mapsto 13$, $3\mapsto 1$.

Let us sketch the contents of the paper. After recalling all basic notation in
Section 2 we give the main results in Section 3. All concepts needed in the
proofs of our results are provided in Section 4. In particular, we recall the
required background on Rauzy fractals. Proofs of the theorems are given in
Section 5. We also discuss consequences of our main results including natural
codings of translations and bounded remainder sets. Section 6 is devoted to
the detailed discussion of some examples.

##  2\. Mise en scène

###  2.1. Multidimensional continued fraction algorithms

There is a diversity of formalisms to defining multidimensional continued
fractions, see e.g. [AL18, Bre81, BAG01, KLDM06, Lag93, Lag94, Sch00]. In this
paper, a _$(d{-}1)$ -dimensional continued fraction algorithm_ $(\Delta,T,A)$
is defined on a $d$-dimensional set $\Delta$ with

$\Delta\subseteq\Delta_{d}=\\{\mathbf{x}\in[0,1]^{d}\,:\,\|\mathbf{x}\|_{1}=1\\}$

by a map usually assumed here to be piecewise constant

$A:\,\Delta\to\mathrm{GL}(d,\mathbb{Z})$

satisfying
$\vphantom{\\!}{}^{t}\\!A(\mathbf{x})^{-1}\mathbf{x}\in\mathbb{R}_{\geq
0}^{d}$ for all $\mathbf{x}\in\Delta$ (where $\vphantom{\\!M}{}^{t}\\!M$
denotes the transpose of a matrix $M$), together with the associated
(piecewise continuous) transformation

(2.1)
$T:\,\Delta\to\Delta,\quad\mathbf{x}\mapsto\frac{\vphantom{\\!A}{}^{t}\\!A(\mathbf{x})^{-1}\mathbf{x}}{\|\vphantom{\\!}^{t}\\!A(\mathbf{x})^{-1}\mathbf{x}\|_{1}}.$

This class of algorithms is called _Markovian_ in [Lag93, Section 2]. It
contains prominent examples like the classical algorithms of Brun [Bru58],
Jacobi–Perron [Ber71, Per07, Sch73], and Selmer [Sel61], which are detailed in
Section 6. For more on multidimensional continued fractions, see also [AL18,
Bre81, BAG01, KLDM06, Lag93, Lag94, Sch00]. A Markovian multidimensional
continued fraction algorithm $(\Delta,T,A)$ is called _positive_ if
$A(\mathbf{x})$ is a nonnegative matrix for all $\mathbf{x}\in\Delta$, i.e.,
if $A(\Delta)$ is contained in

$\mathcal{M}_{d}=\\{M\in\mathbb{N}^{d\times d}\,:\,|\\!\det M|=1\\},$

with $\mathbb{N}=\\{0,1,2,\dots\\}$. Setting

$A^{(n)}(\mathbf{x})=A(T^{n-1}\mathbf{x})\cdots\,A(T\mathbf{x})\,A(\mathbf{x}),$

$A$ is a _linear cocycle_ for $T$, i.e., it fulfills the cocycle property
$A^{(m+n)}(\mathbf{x})=A^{(m)}(T^{n}\mathbf{x})A^{(n)}(\mathbf{x})$; this is
the reason for defining $T$ by the transpose of $A$.

The column vectors $\mathbf{y}^{(n)}_{i}$ of
$\vphantom{\\!}{}^{t}\\!A^{(n)}(\mathbf{x})$, $1\leq i\leq d$, produce $d$
sequences of _rational convergents_
$(\mathbf{y}^{(n)}_{i}/\|\mathbf{y}^{(n)}_{i}\|_{1})_{n\in\mathbb{N}}$ that
are supposed to converge to $\mathbf{x}$. More precisely, we say that

  * •

$T$ converges _weakly_ at $\mathbf{x}\in\Delta$ if
$\lim_{n\to\infty}\mathbf{y}^{(n)}_{i}/\|\mathbf{y}^{(n)}_{i}\|_{1}=\mathbf{x}$
holds for all $i\in\\{1,\ldots,d\\}$;

  * •

$T$ converges _strongly_ at $\mathbf{x}\in\Delta$ if
$\lim_{n\to\infty}\|\mathbf{y}^{(n)}_{i}-\|\mathbf{y}^{(n)}_{i}\|_{1}\,\mathbf{x}\|=0$
holds for all $i\in\\{1,\ldots,d\\}$;

  * •

$T$ converges _exponentially_ at $\mathbf{x}\in\Delta$ if there are positive
constants $\kappa,\delta\in\mathbb{R}$ such that
$\|\mathbf{y}^{(n)}_{i}-\|\mathbf{y}^{(n)}_{i}\|_{1}\,\mathbf{x}\|<\kappa
e^{-\delta n}$ holds for all $i\in\\{1,\ldots,d\\}$ and all $n\in\mathbb{N}$.

An important role is played by the following condition, which entails almost
everywhere strong (and even exponential) convergence of the algorithm; see
[Lag93, equation (4.21)].

######  Definition 2.1 (Pisot condition, cf. [BD14, BST19a]).

Let $(X,T,\nu)$ be a dynamical system with ergodic invariant probability
measure $\nu$, and let $C:\,X\to\mathcal{M}_{d}$ be a log-integrable linear
cocycle for $T$; here log-integrable means that
$\int_{X}\log\max(1,\|C(x)\|)\,\mathrm{d}\nu(x)<\infty$. Then the Lyapunov
exponents $\theta_{k}(C)$ of $C$ exist and are given for $k\in\\{1,\dots,d\\}$
by

$\theta_{1}(C)+\dots+\theta_{k}(C)=\lim_{n\to\infty}\frac{1}{n}\log\|\wedge^{k}C^{(n)}(x)\|\quad\mbox{for
$\nu$-almost all $x\in X$}.$

We say that $(X,T,C,\nu)$ satisfies the _Pisot condition_ if
$\theta_{1}(C)>0>\theta_{2}(C)$.

We always assume that the continued fraction algorithm $(\Delta,T,A)$ is
endowed with an ergodic $T$-invariant probability measure $\nu$ such that the
map $A$ is $\nu$-measurable; here $\mathrm{GL}(d,\mathbb{Z})$ carries the
discrete topology. Then the Pisot condition together with the Oseledets
theorem implies that there is a constant $\delta<0$ such that, for
$\nu$-almost all $\mathbf{x}\in\Delta$, there is a hyperplane $V$ of
$\mathbb{R}^{d}$ with

$\lim_{n\to\infty}\frac{1}{n}\log\|A^{(n)}(\mathbf{x})\,\mathbf{v}\|\leq\delta\quad\mbox{for
all}\ \mathbf{v}\in V.$

###  2.2. Substitutive and $S$-adic dynamical systems, shifts of directive
sequences

Let $\mathcal{A}=\\{1,2,\ldots,d\\}$ be a finite ordered alphabet and let
$\sigma:\mathcal{A}^{*}\to\mathcal{A}^{*}$ be an endomorphism of the free
monoid $\mathcal{A}^{*}$ of words over $\mathcal{A}$, which is equipped with
the operation of concatenation. If $\sigma$ is non-erasing, i.e., if $\sigma$
does not map a non-empty word to the empty word, then we call $\sigma$ a
_substitution_ over the alphabet $\mathcal{A}$. With $\sigma$, we associate
the language

$L_{\sigma}=\big{\\{}w\in\mathcal{A}^{*}\,:\,w\hbox{ is a factor of
}\sigma^{n}(i)\hbox{ for some }i\in\mathcal{A},\,n\in\mathbb{N}\big{\\}}$

of words that occur as subwords in iterations of $\sigma$ on a letter of
$\mathcal{A}$. Here, a word $w$ is a _factor_ of a word $v$ if there exist
words $p,s$ such that $v=pws$. Moreover, if $p$ is the empty word, then $w$ is
a _prefix_ of $v$. Using the language $L_{\sigma}$, the _substitutive
dynamical system_ $(X_{\sigma},\Sigma)$ is defined by

$X_{\sigma}=\\{\omega\in\mathcal{A}^{\mathbb{N}}\,:\,\mbox{each factor of
$\omega$ is contained in $L_{\sigma}$}\\},$

with $\Sigma$ being the _shift map_
$(\omega_{n})_{n\in\mathbb{N}}\mapsto(\omega_{n+1})_{n\in\mathbb{N}}$111We
denote the shift map on any space of sequences by $\Sigma$; this should not
cause any confusion.; $X_{\sigma}$ is obviously $\Sigma$-invariant. The nature
of a substitution $\sigma$ very much depends on its abelianized counterpart,
its so-called _incidence matrix_

$M_{\sigma}=(|\sigma(j)|_{i})_{1\leq i,j\leq d},$

where $|w|_{i}$ denotes the number of occurrences of a letter
$i\in\mathcal{A}$ in the word $w\in\mathcal{A}^{*}$. We assume that the
incidence matrix of $\sigma$ is unimodular, i.e., we consider the set of
substitutions

$\mathcal{S}_{d}=\\{\sigma\,:\,\mbox{$\sigma$ is a substitution over
$\mathcal{A}=\\{1,\ldots,d\\}$},\,M_{\sigma}\in\mathcal{M}_{d}\\}.$

The _abelianization_ of a word $w\in\mathcal{A}^{*}$ is
$\mathbf{l}(w)=\vphantom{}^{t}(|w|_{1},\ldots,|w|_{d})$, so that
$\mathbf{l}(\sigma(w))=M_{\sigma}\mathbf{l}(w)$.

Substitutive dynamical systems (and related tiling flows) have been studied
extensively in the literature; see for instance [BG19, BS18, Fog02, Que10].
So-called _unit Pisot substitutions_ received particular interest; a unit
Pisot substitution is a substitution $\sigma$ whose incidence matrix
$M_{\sigma}$ has a characteristic polynomial which is the minimal polynomial
of a Pisot unit. Recall that a Pisot number is an algebraic integer greater
than $1$ whose Galois conjugates are all contained in the open unit disk. This
class of substitutions is of importance for several reasons; one of them is
their relation to strongly convergent multidimensional continued fraction
algorithms, a relation that will be important in the present paper. Note also
that they are primitive in a sense defined in Section 4.1, which implies that
the associated symbolic dynamical systems are minimal. The main conjecture in
this context, the so-called _Pisot substitution conjecture_ , claims that, for
each unit Pisot substitution $\sigma$, the substitutive dynamical system
$(X_{\sigma},\Sigma)$ is measurably conjugate to a minimal translation on the
torus $\mathbb{T}^{d-1}$, and, hence, has purely discrete spectrum. Although
there are many partial results (see e.g. [ABB+15, Bar16, Bar18, HS03, MA18]),
this conjecture is still open. However, given a single unit Pisot substitution
$\sigma$, there are many algorithms that can be used to show that
$(X_{\sigma},\Sigma)$ has purely discrete spectrum; see [BST10, MA18, SS02].
Thus, for each single unit Pisot substitution $\sigma$, this property is easy
to check, which is important for us. In the present paper, we show that wide
classes of symbolic dynamical systems of Pisot type are measurably conjugate
to minimal translations on the torus, provided that the same is true for a
particular substitutive element of the class.

The $S$-adic dynamical systems constitute generalizations of substitutive
dynamical systems; see for instance [AMS14, ABM+20, BD14, BST19a, Thu19],
where $S$-adic dynamical systems are studied in a similar context as in the
present paper. They are defined in terms of a sequence
$\boldsymbol{\sigma}=(\sigma_{n})_{n\in\mathbb{N}}$ of substitutions over a
given alphabet $\mathcal{A}$ in a way that is analogous to the definition of
substitutive dynamical systems. The language associated with
$\boldsymbol{\sigma}$ is defined to be

$L_{\boldsymbol{\sigma}}=\big{\\{}w\in\mathcal{A}^{*}\,:\,w\hbox{ is a factor
of }\sigma_{[0,n)}(i)\hbox{ for some
}i\in\mathcal{A},\,n\in\mathbb{N}\big{\\}},$

with

$\sigma_{[k,n)}=\sigma_{k}\circ\sigma_{k+1}\circ\cdots\circ\sigma_{n-1}\qquad(0\leq
k\leq n).$

The _$S$ -adic dynamical system_ $(X_{\boldsymbol{\sigma}},\Sigma)$ is then
defined by setting

$X_{\boldsymbol{\sigma}}=\\{\omega\in\mathcal{A}^{\mathbb{N}}\,:\,\hbox{each
factor of }\omega\hbox{ is contained in }L_{\boldsymbol{\sigma}}\\}.$

Note that the $S$-adic dynamical system of a periodic sequence of
substitutions $(\sigma_{0},\sigma_{1},\dots,\sigma_{n-1})^{\infty}$ is equal
to the substitutive dynamical system of $\sigma_{[0,n)}$.

We say that a sequence $\boldsymbol{\sigma}\in\mathcal{S}_{d}^{\mathbb{N}}$
has _purely discrete spectrum_ if the system
$(X_{\boldsymbol{\sigma}},\Sigma)$ is uniquely ergodic and has purely discrete
measure-theoretic spectrum.

We already mentioned that there is a link between $S$-adic dynamical systems
and continued fraction algorithms. For the classical continued fraction
algorithm, this is worked out in detail in [AF01, AF05]. This is also well
developed for multidimensional continued fractions. Indeed, for each given
vector, a continued fraction algorithm creates a sequence of “partial quotient
matrices”. If these matrices are nonnegative and integral, they can be
regarded as incidence matrices of a directive sequence of substitutions of an
$S$-adic dynamical system. However, a continued fraction algorithm produces a
whole shift of sequences of matrices, depending on the vector that has to be
approximated. The matrices are taken from a (finite or infinite) set
$\mathcal{M}$. While for some algorithms, all sequences in
$\mathcal{M}^{\mathbb{N}}$ occur as sequences of partial quotient matrices (as
is the case for instance for the Brun and Selmer algorithms), other algorithms
(like the Jacobi–Perron algorithm) impose some restrictions on these
admissible sequences, which are usually given a finite type condition. For
instance, in the formalization of multidimensional continued fraction
algorithms as Rauzy induction type algorithms developed in [CN13, Fou20],
inspired by interval exchanges, finite graphs allow to formalize admissibility
conditions. Here, we do not need to restrict ourselves to such Markovian type
admissibility conditions.

Let $D\subset\mathcal{S}_{d}^{\mathbb{N}}$ be a _shift-invariant set of
directive sequences_ , i.e., a shift-invariant set of sequences of
substitutions (which is not to be confused with the $S$-adic shift
$(X_{\boldsymbol{\sigma}},\Sigma)$ of a sequence $\boldsymbol{\sigma}\in D$).
We define the _linear cocycle_ $Z$ over $(D,\Sigma)$ by

$Z:\,D\to\mathcal{M}_{d},\qquad(\sigma_{n})_{n\in\mathbb{N}}\mapsto\vphantom{\\!M}^{t}\\!M_{\sigma_{0}}.$

(Recall that $M_{\sigma}$ is the incidence matrix of $\sigma$.) Analogously to
the linear cocycle $A$, we define

(2.2) $Z^{(n)}(\boldsymbol{\sigma})=Z(\Sigma^{n-1}\boldsymbol{\sigma})\cdots
Z(\Sigma\boldsymbol{\sigma})Z(\boldsymbol{\sigma}),$

so that
$Z^{(n)}(\boldsymbol{\sigma})=\vphantom{\\!}^{t}\\!M_{\sigma_{n-1}}\cdots\vphantom{\\!}^{t}\\!M_{\sigma_{1}}\vphantom{\\!}^{t}\\!M_{\sigma_{0}}=\vphantom{\\!}^{t}\\!M_{\sigma_{[0,n)}}$.

Like in the substitutive case, also in the $S$-adic case properties of the
incidence matrices of the substitutions $\sigma_{n}$ will be decisive for the
behavior of the $S$-adic dynamical system $(X_{\boldsymbol{\sigma}},\Sigma)$.
We have under mild conditions (see Section 4.1) that, for
$M_{n}=M_{\sigma_{n}}$,

(2.3) $\bigcap_{n\in\mathbb{N}}M_{0}M_{1}\cdots
M_{n}\mathbb{R}^{d}_{+}=\mathbb{R}_{+}\mathbf{u}$

for some vector $\mathbf{u}\in\mathbb{R}_{\geq 0}^{d}$, which is called the
_generalized right eigenvector_ of $\boldsymbol{\sigma}$ (or of
$(M_{n})_{n\in\mathbb{N}}$) and can be seen as the generalization of the
Perron–Frobenius eigenvector of a primitive matrix. Moreover, we wish to carry
over the Pisot property of the substitutive case to this more general setting.
This will be done by imposing the Pisot condition in Definition 2.1 to the
Lyapunov exponents of the cocycle $(D,\Sigma,Z,\nu)$ for a convenient
invariant measure $\nu$. Thus we do not consider a single sequence
$\boldsymbol{\sigma}$ but the behavior of $\nu$-almost all sequences in $D$.

###  2.3. $S$-adic shifts given by continued fraction algorithms

Our goal is to set up symbolic realizations of continued fraction algorithms
which in turn will provide symbolic models of toral translations, such as
described in Section 2.4. We want to associate with each $\mathbf{x}\in\Delta$
a sequence of substitutions
$\boldsymbol{\sigma}=(\sigma_{n})_{n\in\mathbb{N}}\in\mathcal{S}_{d}^{\mathbb{N}}$
with generalized right eigenvector $\mathbf{x}$. To achieve this, we choose
$\sigma_{n}$ with incidence matrix
$\vphantom{\\!}{}^{t}\\!A(T^{n}\mathbf{x})$, so that
$M_{\sigma_{[0,n)}}=\vphantom{\\!}^{t}\\!A^{(n)}(\mathbf{x})$.

######  Definition 2.2 ($S$-adic realizations).

We call a mapping $\varphi:\,\Delta\to\mathcal{S}_{d}$ a _substitution
selection_ for a positive $(d{-}1)$-dimensional continued fraction algorithm
$(\Delta,T,A)$ if the incidence matrix of $\varphi(\mathbf{x})$ is equal to
$\vphantom{\\!A}{}^{t}\\!A(\mathbf{x})$ for all $\mathbf{x}\in\Delta$. The
corresponding _substitutive realization_ of $(\Delta,T,A)$ is the map

$\boldsymbol{\varphi}:\,\Delta\to\mathcal{S}_{d}^{\mathbb{N}},\quad\mathbf{x}\mapsto(\varphi(T^{n}\mathbf{x}))_{n\in\mathbb{N}},$

together with the shift $(\boldsymbol{\varphi}(\Delta),\Sigma)$. For any
$\mathbf{x}\in\Delta$, $\boldsymbol{\varphi}(\mathbf{x})$ is called an _$S$
-adic expansion_ of $\mathbf{x}$, and
$(X_{\boldsymbol{\varphi}(\mathbf{x})},\Sigma)$ is called the _$S$ -adic
dynamical system of $\mathbf{x}$ w.r.t. $(\Delta,T,A,\varphi)$_.

If $\varphi(\mathbf{x})=\varphi(\mathbf{y})$ for all
$\mathbf{x},\mathbf{y}\in\Delta$ with $A(\mathbf{x})=A(\mathbf{y})$, then
$\varphi$ is called a _faithful substitution selection_ and
$\boldsymbol{\varphi}$ is a _faithful substitutive realization_.

Note that the diagram

(2.4)
${\Delta}$${\Delta}$${\boldsymbol{\varphi}(\Delta)}$${\boldsymbol{\varphi}(\Delta)}$$\scriptstyle{T}$$\scriptstyle{\boldsymbol{\varphi}}$$\scriptstyle{\boldsymbol{\varphi}}$$\scriptstyle{\Sigma}$

commutes. If $T$ converges weakly at $\mathbf{x}$ for almost all
$\mathbf{x}\in\Delta$, then $(\Delta,T,\nu)$ is measure-theoretically
conjugate to its substitutive realization, which we write as

(2.5)
$(\Delta,T,\nu)\overset{\boldsymbol{\varphi}}{\cong}(\boldsymbol{\varphi}(\Delta),\Sigma,\nu\circ\boldsymbol{\varphi}^{-1}).$

###  2.4. Natural codings, bounded remainder sets and Rauzy fractals

In this section we recall several definitions related to the notion of
symbolic codings of toral translations with respect to finite partitions.

For $\mathbf{t}\in\mathbb{R}^{d}$, we consider the minimal translation

$R_{\mathbf{t}}:\mathbb{T}^{d}\to\mathbb{T}^{d},\quad\mathbf{x}\mapsto\mathbf{x}+\mathbf{t}\pmod{\mathbb{Z}^{d}}$

on $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$. A measurable fundamental
domain of $\mathbb{T}^{d}$ is a set $\mathcal{F}\subset\mathbb{R}^{d}$ with
Lebesgue measure $1$ that satisfies
$\mathcal{F}+\mathbb{Z}^{d}=\mathbb{R}^{d}$. A collection
$\\{\mathcal{F}_{1},\ldots,\mathcal{F}_{h}\\}$ is said to be a natural
measurable partition of $\mathcal{F}$ with respect to $R_{\mathbf{t}}$ if the
sets $\mathcal{F}_{i}$ are measurable, they are the closure of their interior
and zero measure boundaries, $\bigcup_{i=1}^{h}\mathcal{F}_{i}=\mathcal{F}$,
the (Lebesgue) measure of $\mathcal{F}_{i}\cap\mathcal{F}_{j}$ is $0$ for all
$i\neq j$, and moreover there exist vectors
$\mathbf{t}_{1},\cdots,\mathbf{t}_{h}$ in $\mathbb{R}^{d}$ such that
$\mathbf{t}_{i}+\mathcal{F}_{i}\subset\mathcal{F}$ with
$\mathbf{t}_{i}\equiv\mathbf{t}\pmod{\mathbb{Z}^{d}}$, $1\leq i\leq h$. This
allows to define a map $\tilde{R}_{\mathbf{t}}$ (which depends on the
partition) as an exchange of domains defined a.e. on $\mathcal{F}$ as
$\tilde{R}_{\mathbf{t}}(\mathbf{x})=\mathbf{x}+\mathbf{t}_{i}$ whenever
$\mathbf{x}\in\mathring{\mathcal{F}_{i}}$. One has for a.e. $\mathbf{x}$ in
$\mathcal{F}$,
$\tilde{R}_{\mathbf{t}}(\mathbf{x})\equiv{R}_{\mathbf{t}}(\mathbf{x})\pmod{\mathbb{Z}^{d}}$.
The collection
$\\{\mathcal{F}^{\prime}_{1},\ldots,\mathcal{F}^{\prime}_{h}\\}$, with
$\mathcal{F}^{\prime}_{i}=\mathcal{F}_{i}+\mathbf{t}_{i}$, $1\leq i\leq h$,
forms also a natural measurable partition of $\mathcal{F}$, hence the
terminology exchange of domains. As an example, consider the translation
$R_{\alpha}$ on $\mathbb{T}$ with $\alpha$ being an irrational number: the
partition of the fundamental domain $[0,1)$ by the intervals $[0,1/2)$ and
$[1/2,1)$ is not a natural partition, whereas the partition by the intervals
$[0,\alpha)$ and $[\alpha,1)$ is a natural partition. The language associated
with the partition $\\{\mathcal{F}_{1},\dots,\mathcal{F}_{h}\\}$ is then
defined as the set of finite words $w_{0}\cdots w_{n}\in\\{1,\dots,h\\}^{*}$
such that
$\bigcap_{k=0}^{n}\tilde{R}_{\mathbf{t}}^{-k}\mathring{\mathcal{F}}_{w_{k}}\neq\emptyset$.

A  _subshift_ is a closed and shift-invariant set of infinite words over a
finite alphabet and its language is the set of factors of its elements.

######  Definition 2.3 (Natural coding).

A subshift $(X,\Sigma)$ is a _natural coding_ of
$(\mathbb{T}^{d},R_{\mathbf{t}})$ if its language is the language of a natural
measurable partition $\\{\mathcal{F}_{1},\ldots,\mathcal{F}_{h}\\}$ and
$\bigcap_{n\in\mathbb{N}}\overline{\bigcap_{k=0}^{n}\tilde{R}_{\mathbf{t}}^{-k}\mathring{\mathcal{F}}_{i_{k}}}$
is reduced to one point for any $(i_{n})_{n\in\mathbb{N}}\in X$, where
$\tilde{R}_{\mathbf{t}}$ stands for the associated exchange of domains.

A sequence $(i_{n})_{n\in\mathbb{N}}\in\\{1,\ldots,h\\}^{\mathbb{N}}$ is said
to be a _natural coding_ of $(\mathbb{T}^{d},R_{\mathbf{t}})$ w.r.t. the
natural measurable partition $\\{\mathcal{F}_{1},\ldots,\mathcal{F}_{h}\\}$ if
there exists $\mathbf{x}\in\mathcal{F}$ such that
$(i_{n})_{n\in\mathbb{N}}\in\\{1,\ldots,h\\}^{\mathbb{N}}$ codes the orbit of
$\mathbf{x}$ under the action of $\tilde{R}_{\mathbf{t}}$, i.e.,
$\tilde{R}_{\mathbf{t}}^{n}(\mathbf{x})=\mathbf{x}+\sum_{k=0}^{n-1}\mathbf{t}_{i_{k}}\equiv{R}_{\mathbf{t}}^{n}(\mathbf{x})\in\mathcal{F}_{i_{n}}$
for all $n\in\mathbb{N}$.

If $(X,\Sigma)$ is a natural coding of $(\mathbb{T}^{d},R_{\mathbf{t}})$ with
respect to the natural measurable partition
$\\{\mathcal{F}_{1},\ldots,\mathcal{F}_{h}\\}$, then one can define a
continuous and onto map $\chi:X\rightarrow\mathcal{F}$. Moreover there exists
a a one-to-one coding map $\Phi$ defined a.e. on $\mathcal{F}$ that satisfies
$\chi\circ\Phi(\mathbf{x})=\mathbf{x}$ for a.e. $\mathbf{x}$, and that
associates with $\mathbf{x}$ the natural coding of its orbit under the action
$\tilde{R}_{\mathbf{t}}$ w.r.t. the partition
$\\{\mathcal{F}_{1},\dots,\mathcal{F}_{h}\\}$. Moreover, the subshift
$(X_{\mathcal{F}},\Sigma)$ is minimal and uniquely ergodic,
$(\mathbb{T}^{d},R_{\mathbf{t}})$ is a topological factor of its symbolic
realization$(X_{\mathcal{F}},\Sigma)$, and one has a measure-theoretic
isomorphism between $(\mathbb{T}^{d},R_{\mathbf{t}})$ and $(X,\Sigma)$.

Natural codings with respect to bounded partitions and bounded remainder sets
are closely related. A  _bounded remainder set_ of a dynamical system
$(X,T,\mu)$ with invariant probability measure $\mu$ is a measurable set
$Y\subseteq X$ such that there exists $C>0$ with the property

$\big{|}\\#\\{0\leq n<N\,:\,T^{n}(x)\in Y\\}-N\mu(Y)\big{|}\leq
C\quad\mbox{for all}\ N\in\mathbb{N},\ \mbox{for a.e.}\ x\in X.$

If $(X,\Sigma)$ is a natural coding of $(\mathbb{T}^{d-1},R_{\mathbf{t}})$
w.r.t. a partition $\\{\mathcal{F}_{1},\dots,\mathcal{F}_{d}\\}$ of a bounded
fundamental domain, then, according to Theorem 3.8, each $\mathcal{F}_{i}$ is
a bounded remainder set of $R_{\mathbf{t}}$.

Moreover, we also prove that the elements of the partition $\mathcal{F}$ are
proved to be related to Rauzy fractals. Rauzy fractals are aimed at providing
fundamental domains and associated natural measurable partitions for toral
translations. In this setting, it is convenient to consider fundamental
domains leaving in the hyperplane $\mathbf{1}^{\perp}$ orthogonal to the
vector $\mathbf{1}=(1,1,\dots,1)$. More precisely, for an $S$-adic dynamical
system $(X_{\boldsymbol{\sigma}},\Sigma)$ with
$\boldsymbol{\sigma}\in\mathcal{S}_{d}^{\mathbb{N}}$ having the generalized
right eigenvector $\mathbf{u}$, we consider translation vectors
$\pi^{\prime}_{\mathbf{u}}\mathbf{e}_{i}$, $i\in\mathcal{A}=\\{1,\dots,d\\}$,
where

$\pi^{\prime}_{\mathbf{u}}$ denotes the projection along $\mathbf{u}$ on
$\mathbf{1}^{\perp}$.

Then the _Rauzy fractal_ associated with
$\boldsymbol{\sigma}=(\sigma_{n})_{n\in\mathbb{N}}$ is

$\mathcal{R}_{\boldsymbol{\sigma}}=\overline{\\{\pi^{\prime}_{\mathbf{u}}\,\mathbf{l}(p)\,:\,\mbox{$p$
is a prefix of $\sigma_{[0,n)}(j)$ for infinitely many $n\in\mathbb{N}$,
$j\in\mathcal{A}$}\\}}.$

It has subtiles

(2.6)
$\mathcal{R}_{\boldsymbol{\sigma}}(w)=\overline{\\{\pi^{\prime}_{\mathbf{u}}\,\mathbf{l}(p)\,:\,p\,w\
\mbox{is a prefix of $\sigma_{[0,n)}(j)$ for infinitely many}\
n\in\mathbb{N},j\in\mathcal{A}\\}}\ (w\in\mathcal{A}^{*}).$

We clearly have
$\mathcal{R}_{\boldsymbol{\sigma}}=\bigcup_{w\in\mathcal{A}^{n}}\mathcal{R}_{\boldsymbol{\sigma}}(w)$
for all $n\in\mathbb{N}$, and in particular
$\mathcal{R}_{\boldsymbol{\sigma}}=\bigcup_{i\in\mathcal{A}}\mathcal{R}_{\boldsymbol{\sigma}}(i)$.
We will show, for an $S$-adic dynamical system that is a natural coding with a
bounded fundamental domain $\mathcal{F}$, that the Rauzy fractal is an affine
image of $\mathcal{F}$ and that the collection
$\\{\mathcal{R}_{\boldsymbol{\sigma}}(1),\dots,\mathcal{R}_{\boldsymbol{\sigma}}(d)\\}$
of Rauzy fractals associated with letters forms a natural measurable partition
with respect to a minimal translation; see Theorem 3.8. We even prove (also in
Theorem 3.8), under the extra assumption of (left) properness for
$\boldsymbol{\sigma}$, that each collection of subtiles
$\\{\mathcal{R}_{\boldsymbol{\sigma}}(w)\,:\,w\in\mathcal{A}^{n}\\}$,
$n\in\mathbb{N}$, forms a natural measurable partition consisting of bounded
remainder sets, providing a sequence of refined natural measurable partitions.

###  2.5. Further definitions

To state our theorems, we need a few further definitions.

######  Definition 2.4 (Pisot sequences and points).

A periodic sequence
$(M_{0},M_{1},\dots,M_{n-1})^{\infty}\in\mathcal{M}_{d}^{\mathbb{N}}$ or
$(\sigma_{0},\sigma_{1},\dots,\sigma_{n-1})^{\infty}\in\mathcal{S}_{d}^{\mathbb{N}}$
is called a _periodic Pisot sequence_ if $M_{0}M_{1}\cdots M_{n-1}$ is a Pisot
matrix or $\sigma_{0}\circ\sigma_{1}\circ\cdots\circ\sigma_{n-1}$ is a Pisot
substitution.

For a multidimensional continued fraction algorithm $(\Delta,T,A,\nu)$, we say
that $\mathbf{x}\in\Delta$ is a _periodic Pisot point_ if there is an $n\geq
1$ such that $T^{n}(\mathbf{x})=\mathbf{x}$ and $A^{(n)}(\mathbf{x})$ is a
Pisot matrix.

######  Definition 2.5 (Cylinder and follower sets, positive range).

For a (symbolic) dynamical system $(D,\Sigma,\nu)$, we say that
$(\omega_{n})_{n\in\mathbb{N}}$ has _positive range_ if

$\inf_{n\in\mathbb{N}}\nu(\Sigma^{n}[\omega_{0},\dots,\omega_{n-1}])>0,$

where

$[\omega_{0},\dots,\omega_{n-1}]=\big{\\{}(\upsilon_{k})_{k\in\mathbb{N}}\in
D\,:\,(\upsilon_{0},\dots,\upsilon_{n-1})=(\omega_{0},\dots,\omega_{n-1})\big{\\}}$

is the _cylinder set_ of $(\omega_{0},\dots,\omega_{n-1})$ (and
$\Sigma^{n}[\omega_{0},\dots,\omega_{n-1}]$ is the _follower set_ of
$(\omega_{0},\dots,\omega_{n-1})$).

The cylinder sets of a multidimensional continued fraction algorithm
$(\Delta,T,A,\nu)$ are

(2.7)
$\Delta^{(n)}(\mathbf{x})=\\{\mathbf{y}\in\Delta\,:\,A(\mathbf{y})=A(\mathbf{x}),A(T\mathbf{y})=A(T\mathbf{x}),\dots,A(T^{n-1}\mathbf{y})=A(T^{n-1}\mathbf{x})\\},$

with $\Delta^{(0)}(\mathbf{x})=\Delta$, and the follower sets are
$T^{n}\Delta^{(n)}(\mathbf{x})$; for convenience, we set
$\Delta(\mathbf{x})=\Delta^{(1)}(\mathbf{x})$. Then $\mathbf{x}\in\Delta$ has
_positive range_ if

$\inf_{n\in\mathbb{N}}\nu(T^{n}\Delta^{(n)}(\mathbf{x}))>0.$

We note that all the classical algorithms we are aware of satisfy even the
_finite range property_ (cf. [IY87]) stating that the collection of sets

$\mathcal{D}=\\{T^{n}\Delta^{(n)}(\mathbf{x})\,:\,\mathbf{x}\in\Delta,\,n\in\mathbb{N}\\}$

is finite, where sets differing only on a set of $\nu$-measure zero are
identified. For instance, although the Jacobi-Perron algorithm is
_multiplicative_ in the sense that the range of its cocycle is infinite,
$\mathcal{D}$ consists of only two elements. The finite range property
obviously implies positive range for all $\mathbf{x}\in\Delta$ if we suppose
that all cylinders $\Delta(\mathbf{x})$ have positive measure.

If $(\Delta,T,A,\nu)$ has the finite range property and
$\bigcap_{n\in\mathbb{N}}\Delta^{(n)}(\mathbf{x})=\\{\mathbf{x}\\}$ for almost
all $\mathbf{x}\in\Delta$, i.e., the collection of cylinders
$\\{\Delta(\mathbf{x}):\,\mathbf{x}\in\Delta\\}$ generates $\Delta$, then
$\\{U\cap\Delta(\mathbf{x})\,:\,U\in\mathcal{D},\,\mathbf{x}\in\Delta\\}$
forms a (measurable countable) _Markov partition_ of $(\Delta,T)$ in the sense
of [Yur95, Theorem 10.1]. This is _not_ to be confused with the algorithm to
be Markovian in the sense of [Lag93] mentioned above. Most usual continued
fraction algorithms (like Brun, Selmer, Jacobi–Perron) are designed in a way
that the Markov partition property holds.

We also want that the intersection of a preimage $T^{-n}B$ of a set $B$ with
$\nu(B)>0$ to a cylinder $\Delta^{(n)}(\mathbf{x})$ has again positive
measure. To this end, we always assume that $\nu\circ T$ is absolutely
continuous w.r.t. $\nu$, or $\nu\circ T\ll\nu$ for short.

Finally, we define

(2.8)
$\pi\,:\,\mathbb{R}^{d}\to\mathbb{R}^{d-1},\quad(x_{1},\ldots,x_{d})\mapsto(x_{1},\dots,x_{d-1}),$

i.e., we omit the last coordinate of a vector. (In doing so, we make an
arbitrary choice; it would also be possible to omit any other coordinate.)

##  3\. Main results

###  3.1. Main results on multidimensional continued fraction algorithms

In this section we provide our main results for multidimensional continued
fraction algorithms.

######  Theorem 3.1.

Let $(\Delta,T,A,\nu)$ be a positive $(d{-}1)$-dimensional continued fraction
algorithm satisfying the Pisot condition and $\nu\circ T\ll\nu$, let
$\boldsymbol{\varphi}$ be a faithful substitutive realization of
$(\Delta,T,A,\nu)$, and assume that there is a periodic Pisot point
$\mathbf{x}$ with positive range such that $\boldsymbol{\varphi}(\mathbf{x})$
has purely discrete spectrum. Then for $\nu$-almost all $\mathbf{x}\in\Delta$
the $S$-adic dynamical system $(X_{\boldsymbol{\varphi}(\mathbf{x})},\Sigma)$
is a natural coding of the (minimal) translation by $\pi(\mathbf{x})$ on
$\mathbb{T}^{d-1}$ with respect to a partition of a bounded fundamental
domain; in particular, its measure-theoretic spectrum is purely discrete.

######  Remark 3.2.

We note that $(X_{\boldsymbol{\varphi}(\mathbf{x})},\Sigma)$ is a substitutive
dynamical system since $\boldsymbol{\varphi}(\mathbf{x})$ is a periodic
sequence of substitutions. For such systems, some combinatorial _coincidence
conditions_ (as for instance the ones used in [ABB+15, BK06, BST10, IR06])
ensure purely discrete measure-theoretic spectrum. We could therefore replace
the pure discrete spectrum condition in Definition 2.4 by
“$\varphi(\mathbf{x})\circ\varphi(T\mathbf{x})\circ\dots\circ\varphi(T^{n-1}\mathbf{x})$
satisfies the _super coincidence condition_ from [IR06, Definition 4.2]”.
However, since coincidence conditions require quite some notation we decided
to introduce them later in this paper in order to make our main results easier
to read. The Pisot substitution conjecture implies that all Pisot
substitutions satisfy the super coincidence condition.

######  Remark 3.3.

Note that we can omit in Theorem 3.1 the requirement that
$\boldsymbol{\varphi}$ is faithful, if we replace $A$ by $\varphi$ in the
definition of the cylinder sets $\Delta^{(n)}(\mathbf{x})$ in (2.7) and assume
that $\varphi$ is measurable.

Since the Pisot substitution conjecture is not proved, we cannot omit the
requirement of a periodic Pisot point with purely discrete spectrum in Theorem
3.1, and we do not even know whether there always exists a substitutive
realization $\boldsymbol{\varphi}$ admitting such a point. We cannot ensure
purely discrete spectrum of $(X_{\boldsymbol{\sigma}},\Sigma)$ for almost all
$\boldsymbol{\sigma}\in\boldsymbol{\varphi}(\Delta)$ when we have no
$\boldsymbol{\sigma}\in\boldsymbol{\varphi}(\Delta)$ with purely discrete
spectrum. However, we have the following unconditional theorem for
accelerations $(\Delta,T^{k})$.

######  Theorem 3.4.

Let $(\Delta,T,A,\nu)$ be a positive $({d{-}1})$-dimensional continued
fraction algorithm satisfying the Pisot condition and $\nu\circ T\ll\nu$, and
assume that there exists a periodic Pisot point with positive range. Then,
there exist a positive integer $k$ and a (faithful) substitutive realization
$\boldsymbol{\varphi}$ of $(\Delta,T^{k},A,\nu)$ such that for $\nu$-almost
all $\mathbf{x}\in\Delta$ the $S$-adic dynamical system
$(X_{\boldsymbol{\varphi}(\mathbf{x})},\Sigma)$ is a natural coding of the
(minimal) translation by $\pi(\mathbf{x})$ on $\mathbb{T}^{d-1}$ with respect
to a partition of a bounded fundamental domain; in particular, its measure-
theoretic spectrum is purely discrete. Moreover, we have
$(\Delta,T^{k},\nu)\overset{\boldsymbol{\varphi}}{\cong}(\boldsymbol{\varphi}(\Delta),\Sigma,\nu\circ\boldsymbol{\varphi}^{-1})$.

######  Remark 3.5.

Note that the set of translations in Theorems 3.1 and 3.4 does not cover
$\mathbb{T}^{d-1}$ since the translations are of the form
$\mathbf{x}\in[0,1]^{d-1}$ with $\|\mathbf{x}\|_{1}\leq 1$. However, the
translation by $\mathbf{x}$ on $\mathbb{T}^{d-1}$ is conjugate to all
translations by $\mathbf{y}\in\mathbf{x}\,\mathrm{GL}(d{-}1,\mathbb{Z})$, and
$\\{\mathbf{x}\in[0,1]^{d-1}\,:\,\|\mathbf{x}\|_{1}\leq 1\\}$ is mapped by

$(x_{1},\ldots,x_{d-1})\mapsto(x_{1},x_{1}+x_{2},\dots,x_{1}+x_{2}+\cdots+x_{d-1})$

to $\\{\mathbf{x}\in[0,1]^{d-1}\,:\,0\leq x_{1}\leq x_{2}\leq\cdots\leq
x_{d-1}\leq 1\\}$. Taking permutations of the coordinates of the latter set
gives $\mathbb{T}^{d-1}$.

Verifying purely discrete spectrum for some concrete substitutive dynamical
systems will allow us to use Theorem 3.1 in order to prove a.e. purely
discrete spectrum for many continued fraction algorithms like for instance the
Jacobi–Perron, Brun, Cassaigne–Selmer and Arnoux–Rauzy–Poincaré algorithms.
Indeed, it is well known that these algorithms have the finite range property,
and the Pisot condition holds for all these algorithms when $d=3$. In the case
of Brun, the Pisot property also holds for $d=4$. Applying Theorem 3.1 to
these algorithms, according to Remark 3.5 we are able to realize almost all
translations in $\mathbb{T}^{2}$ and $\mathbb{T}^{3}$ by systems of the form
$(X_{\boldsymbol{\varphi}(\mathbf{x})},\Sigma)$, $\mathbf{x}\in\Delta$. Since
Cassaigne–Selmer (for $d=3$) gives rise to languages
$L_{\boldsymbol{\varphi}(\mathbf{x})}$ of complexity $2n+1$, this entails that
there exist natural codings for almost all translations of $\mathbb{T}^{2}$
with complexity $2n+1$. Looking at [BCBD+19, BST19a], we also see many other
consequences for these algorithms and their associated shifts of directive
sequences. Since we will require more notation to formulate all these
consequences, we will come back to them in Proposition 5.11 and Section 6.

###  3.2. Main results for shifts of directive sequences

We now give variants of the results of the previous section in terms of
directive sequences.

######  Theorem 3.6.

Let $D\subset\mathcal{S}_{d}^{\mathbb{N}}$ be a shift-invariant set of
directive sequences equipped with an ergodic $\Sigma$-invariant probability
measure $\nu$ satisfying $\nu\circ\Sigma\ll\nu$. Assume that the linear
cocycle $(D,\Sigma,Z,\nu)$ defined by
$Z((\sigma_{n})_{n\in\mathbb{N}})=\vphantom{\\!}^{t}\\!M_{\sigma_{0}}$
satisfies the Pisot condition, and that there is a periodic Pisot sequence
with positive range and purely discrete spectrum in $(D,\Sigma,\nu)$. Then for
$\nu$-almost all $\boldsymbol{\sigma}\in D$ the $S$-adic dynamical system
$(X_{\boldsymbol{\sigma}},\Sigma)$ is a natural coding of the minimal
translation by $\pi(\mathbf{u})$ on $\mathbb{T}^{d-1}$ with respect to a
partition of a bounded fundamental domain, where $\mathbf{u}$ is the
generalized right eigenvector of $\boldsymbol{\sigma}$ with
$\|\mathbf{u}\|_{1}=1$; in particular, the measure-theoretic spectrum of
$(X_{\boldsymbol{\sigma}},\Sigma)$ is purely discrete.

We refer to Theorem 3.8 where the bounded fundamental domains for the coding
$(X_{\boldsymbol{\sigma}},\Sigma)$ of the translation are given explicitly in
terms of the $S$-adic Rauzy fractals associated with $\boldsymbol{\sigma}$.
Moreover, if the substitutions $\sigma_{n}$ enjoy some _properness condition_
, this proposition shows that we can refine the bounded remainder sets to
factors of $X_{\boldsymbol{\sigma}}$.

To get an analogue of Theorem 3.4 for directive sequences, we do not start
with a shift of directive sequences but rather with its abelianization, i.e.,
a shift of sequences of matrices $(\mathfrak{D},\Sigma)$, for which we would
like to find a map $s:\,\mathcal{M}_{d}\to\mathcal{S}_{d}$ such that almost
all $\boldsymbol{\sigma}\in s(\mathfrak{D})$, with
$s((M_{n})_{n\in\mathbb{N}})=(s(M_{n}))_{n\in\mathbb{N}}$, have purely
discrete spectrum. Again, we have to consider the accelerated shift
$(\mathfrak{D},\Sigma^{k})$ for a suitable power $\Sigma^{k}$ to gain such a
result.

######  Theorem 3.7.

Let $\mathfrak{D}\subset\mathcal{M}_{d}^{\mathbb{N}}$ be a shift-invariant set
of sequences of unimodular matrices equipped with an ergodic
$\Sigma$-invariant probability measure $\nu$ satisfying
$\nu\circ\Sigma\ll\nu$. Assume that the linear cocycle
$(\mathfrak{D},\Sigma,Z,\nu)$ defined by
$Z((M_{n})_{n\in\mathbb{N}})=\vphantom{\\!}^{t}\\!M_{0}$ satisfies the Pisot
condition, and that there is a periodic Pisot sequence with positive range in
$(\mathfrak{D},\Sigma,\nu)$. Then there exists a positive integer $k$ and a
map $\boldsymbol{\psi}:\,\mathfrak{D}\to\mathcal{S}_{d}^{\mathbb{N}}$
satisfying $\boldsymbol{\psi}\circ\Sigma^{k}=\Sigma\circ\boldsymbol{\psi}$
such that for $\nu$-almost all $\mathbf{M}\in\mathfrak{D}$ the $S$-adic
dynamical system $(X_{\boldsymbol{\psi}(\mathbf{M})},\Sigma)$ is a natural
coding of the minimal translation by $\pi(\mathbf{u})$ on $\mathbb{T}^{d-1}$
with respect to a partition of a bounded fundamental domain, where
$\mathbf{u}$ is the generalized right eigenvector of $\mathbf{M}$ with
$\|\mathbf{u}\|_{1}=1$. In particular, the measure-theoretic spectrum of
$(X_{\boldsymbol{\psi}(\mathbf{M})},\Sigma)$ is purely discrete.

The main difference between the results in Section 3.1 and the ones in Section
3.2 is that there can be several directive sequences in $D$ with the same
generalized right eigenvector.

###  3.3. Results on natural codings, bounded remainder sets and Rauzy
fractals

We prove that natural codings with respect to bounded partitions are closely
related to bounded remainder sets. Moreover, Rauzy fractals are canonical
bounded remainder sets, up to some affine map.

######  Theorem 3.8.

Assume that $(X,\Sigma)$ is the natural coding of a minimal translation
$R_{\mathbf{t}}$ on $\mathbb{T}^{d-1}$ w.r.t. a measurable partition
$\\{\mathcal{F}_{1},\ldots,\mathcal{F}_{d}\\}$ of a bounded fundamental domain
$\mathcal{F}$. Then the sets $\mathcal{F}_{i}$ are bounded remainder sets of
$R_{\mathbf{t}}$, and $(X,\Sigma)$ is uniquely ergodic.

If $X=X_{\boldsymbol{\sigma}}$ for some
$\boldsymbol{\sigma}\in\mathcal{S}_{d}^{\mathbb{N}}$, then there is an affine
map $A:\,\mathbb{R}^{d}\to\mathbb{R}^{d-1}$ such that
$\mathcal{F}_{i}=A(\mathcal{R}_{\boldsymbol{\sigma}}(i))$ for $1\leq i\leq d$.
In particular, $(X_{\boldsymbol{\sigma}},\Sigma)$ is a natural coding of
$R_{\pi(\mathbf{u})}$, where $\mathbf{u}$ is the generalized right eigenvector
of $\boldsymbol{\sigma}$ with $\|\mathbf{u}\|_{1}=1$; the domains of the
natural coding are $\pi(-\mathcal{R}_{\boldsymbol{\sigma}}(i))$, $1\leq i\leq
d$. If moreover $\boldsymbol{\sigma}$ is left proper, then for each
$w\in\\{1,\dots,d\\}^{*}$, the “cylinder set”
$A(\mathcal{R}_{\boldsymbol{\sigma}}(w))$ is a bounded remainder set of
$R_{\mathbf{t}}$ (and $\pi(-\mathcal{R}_{\boldsymbol{\sigma}}(w))$ is a
bounded remainder set of $R_{\pi(\mathbf{u})}$).

Note that we have $A(\mathcal{R}_{\boldsymbol{\sigma}}(i_{0}i_{1}\cdots
i_{n}))=\mathcal{F}_{i_{0}}\cap
R_{\mathbf{t}}^{-1}\mathcal{F}_{i_{1}}\cap\cdots\cap
R_{\mathbf{t}}^{-n}\mathcal{F}_{i_{n}}$ modulo $\mathbb{Z}^{d-1}$.

##  4\. Preparations for the proofs of the main theorems

Throughout the proofs of our main results we will need notation, definitions,
and results that are recalled in this section.

###  4.1. Properties of sequences of substitutions

In our main theorems, we put certain assumptions, like the Pisot condition, on
$S$-adic graphs. We will now discuss combinatorial properties that will be
satisfied by almost all directive sequences $\boldsymbol{\sigma}$ under these
assumptions. We need these combinatorial properties because they occur in some
results from [BST19a] that will be important for us. All the following
definitions are taken from [BST19a, Section 2]

Let $\boldsymbol{\sigma}=(\sigma_{n})\in\mathcal{S}_{d}^{\mathbb{N}}$ be a
sequence of substitutions over a given alphabet $\mathcal{A}=\\{1,\dots,d\\}$.
We say that $\boldsymbol{\sigma}$ is _primitive_ , if for each
$k\in\mathbb{N}$ there exists $n>k$ such that $M_{[k,n)}$ is a positive
matrix. If each factor $(\sigma_{0},\ldots,\sigma_{m})$, $m\in\mathbb{N}$,
occurs infinitely often in $\boldsymbol{\sigma}$, then $\boldsymbol{\sigma}$
is _recurrent_. As observed in [Fur60, p. 91–95], primitivity and recurrence
of $\boldsymbol{\sigma}$ allow for an analog of the Perron–Frobenius theorem
for the associated sequence $(M_{n})$ of incidence matrices. In particular, if
$\boldsymbol{\sigma}$ is primitive and recurrent, then the generalized right
eigenvector $\mathbf{u}$ defined in (2.3) exists and is positive.

A substitution $\sigma$ over $\mathcal{A}$ is _left [right] proper_ if there
exists $j\in\mathcal{A}$ such that $\sigma(i)$ starts [ends] with $j$ for all
$i\in\mathcal{A}$. A sequence of substitutions
$\boldsymbol{\sigma}=(\sigma_{n})$ is _left [right] proper_ if for each
$k\in\mathbb{N}$ there exists $n>k$ such that $\sigma_{[k,n)}$ is left [right]
proper.

Another important property is _algebraic irreducibility_. A sequence of
substitutions $\boldsymbol{\sigma}=(\sigma_{n})$ over the alphabet
$\mathcal{A}$ is called algebraically irreducible if for each $k\in\mathbb{N}$
the matrix $M_{[k,n)}$ has irreducible characteristic polynomial provided that
$n\in\mathbb{N}$ is large enough. For $S$-adic dynamical systems that arise
from multidimensional continued fraction algorithms which are almost
everywhere exponentially convergent we can even prove that for each
$k\in\mathbb{N}$ the characteristic polynomial of $M_{[k,n)}$ is the minimal
polynomial of a Pisot unit for $n$ large enough. This is true in particular if
we assume the _Pisot condition_.

Finally, we need a balance property for the language related to a sequence of
substitutions. Let $L$ be a language over a finite alphabet
$\mathcal{A}=\\{1,\ldots,d\\}$. We say that $L$ is _$C$ -balanced_ if for each
two words $w,w^{\prime}\in L$ with $|w|=|w^{\prime}|$ and for each
$i\in\mathcal{A}$, we have $\big{|}|w|_{i}-|w^{\prime}|_{i}\big{|}\leq C$. We
define

(4.1)
$B_{C}=\\{\boldsymbol{\sigma}\in\mathcal{S}_{d}^{\mathbb{N}}\,:\,\mbox{$L_{\boldsymbol{\sigma}}$
is $C$-balanced}\\}.$

Balance can be generalized to factors. We say that $L$ is _balanced on
factors_ if for each $v\in\mathcal{A}^{*}$ there exists some $C_{v}\geq 1$
such that, for any two words $w,w^{\prime}\in L$ with $|w|=|w^{\prime}|$, we
have $\big{|}|w|_{v}-|w^{\prime}|_{v}\big{|}\leq C_{v}$. Here, $|w|_{v}$
denotes the number of occurrences of the factor $v$ in $w$. Without further
precision, balanced refers a priori to letters hereafter.

In the sequel we need various results from [BST19a], some of which require the
so-called _PRICE_ property: A directive sequence
$\boldsymbol{\sigma}=(\sigma_{n})\in\mathcal{S}_{d}^{\mathbb{N}}$ has Property
PRICE if the following conditions hold for some strictly increasing sequences
$(n_{k})_{k\in\mathbb{N}}$ and $(\ell_{k})_{k\in\mathbb{N}}$ and a vector
$\mathbf{v}\in\mathbb{R}_{\geq 0}^{d}\setminus\\{\mathbf{0}\\}$.

  * •

There exists $h\in\mathbb{N}$ and a positive matrix $B$ such that
$M_{\sigma_{[\ell_{k}-h,\ell_{k})}}=B$ for all $k\in\mathbb{N}$.

  * •

We have
$(\sigma_{n_{k}},\sigma_{n_{k}+1},\ldots,\sigma_{n_{k}+\ell_{k}-1})=(\sigma_{0},\sigma_{1},\ldots,\sigma_{\ell_{k}-1})$
for all $k\in\mathbb{N}$.

  * •

The directive sequence $\boldsymbol{\sigma}$ is algebraically irreducible.

  * •

There is $C>0$ such that
$\mathcal{L}_{\boldsymbol{\sigma}}^{(n_{k}+\ell_{k})}$ is $C$-balanced for all
$k\in\mathbb{N}$.

  * •

We have
$\lim_{k\to\infty}\mathbf{v}^{(n_{k})}/\|\mathbf{v}^{(n_{k})}\|=\mathbf{v}$.

We note that if $\boldsymbol{\sigma}$ satisfies PRICE, then the same is true
for $\Sigma\boldsymbol{\sigma}$ by [BST19a, Lemma 5.10]. Moreover, the
condition (E) is required only in the proofs of [BST19a] and can be omitted by
[BST19a, Lemma 5.7].

###  4.2. Tilings by Rauzy fractals and coincidence conditions

As mentioned before, the Rauzy fractals defined in Section 2.4 play a crucial
role in proving that the $S$-adic dynamical system
$(X_{\boldsymbol{\sigma}},\Sigma)$ has purely discrete spectrum. Our
definition of $\mathcal{R}_{\boldsymbol{\sigma}}$ is equivalent to that one in
[BST19a, Section 2.9] which uses limit words of $\boldsymbol{\sigma}$, i.e.,
infinite words that are images of $\sigma_{[0,n)}$ for all $n\in\mathbb{N}$.
The importance of Rauzy fractals is due to the fact that one can “see” on them
the torus translation, to which we want to conjugate an $S$-adic dynamical
systems $(X_{\boldsymbol{\sigma}},\Sigma)$; this is worked out in [BST19a,
Section 8]. Rauzy fractals associated with periodic sequences
$\boldsymbol{\sigma}$ (and therefore related to substitutive dynamical systems
go back to [Rau82] and have been studied extensively; see for instance [AI01,
BS05, BST10, CS01, Fog02, IR06, ST09, Thu19]. In Figure 2, we illustrate the
definition of Rauzy fractals for the periodic directive sequence
$\boldsymbol{\sigma}=(\gamma_{1},\gamma_{2})^{\infty}$, with
$\gamma_{1},\gamma_{2}$ being the Cassaigne–Selmer Substitutions defined in
(6.1) below. Since $\gamma_{1}\gamma_{2}$ is a unit Pisot substitution, this
directive sequence satisfies the necessary properties.

Figure 2. Illustration of the definition of the Rauzy fractal
$\mathcal{R}_{\boldsymbol{\sigma}}$ corresponding to the periodic directive
sequence $\boldsymbol{\sigma}=(\gamma_{1},\gamma_{2})^{\infty}$, where
$\gamma_{1},\gamma_{2}$ are the Cassaigne-Selmer Substitutions defined in
(6.1). The abelianizations $\mathbf{l}(p)$ of the prefixes of the limit word
define a broken line. Its vertices are projected along $\mathbf{u}$ to
$\mathbf{1}^{\perp}$ in order to define the Rauzy fractal
$\mathcal{R}_{\boldsymbol{\sigma}}$, where $\mathbf{u}$ is the generalized
right eigenvector of $\boldsymbol{\sigma}$. Its subtiles
$\mathcal{R}_{\boldsymbol{\sigma}}(1)$,
$\mathcal{R}_{\boldsymbol{\sigma}}(2)$, and
$\mathcal{R}_{\boldsymbol{\sigma}}(3)$ are indicated by different shades of
grey.

We will need the collection of tiles

$\mathcal{C}_{\boldsymbol{\sigma}}=\\{\mathbf{x}+\mathcal{R}_{\boldsymbol{\sigma}}(i)\,:\,\mathbf{x}\in\mathbb{Z}^{d}\cap\mathbf{1}^{\perp},\,i\in\mathcal{A}\\}$

consisting of the translations of (the subtiles of) the Rauzy fractal by
vectors in the lattice $\mathbb{Z}^{d}\cap\mathbf{1}^{\perp}$. As shown e.g.
in [BST19a], the fact that $\mathcal{C}_{\boldsymbol{\sigma}}$ forms a tiling
of $\mathbf{1}^{\perp}$ implies that $(X_{\boldsymbol{\sigma}},\Sigma)$ has
purely discrete spectrum.

It is a priori not clear how to decide for a given directive sequence
$\boldsymbol{\sigma}$ whether $\mathcal{C}_{\boldsymbol{\sigma}}$ forms a
tiling of $\mathbb{R}^{d}$ or not. Here, a _tiling_ of $\mathbb{R}^{d}$ is a
collection of sets that covers $\mathbb{R}^{d}$ and where the intersection of
any two distinct sets has Lebesgue measure $0$. However, as shown in [BST19a,
Section 7] the following coincidence conditions can be used to get checkable
criteria for this tiling property. We say that
$\boldsymbol{\sigma}=(\sigma_{n})_{n\in\mathbb{N}}$ satisfies the _geometric
coincidence condition_ , if for each $R>0$ there is $k\in\mathbb{N}$, such
that, for all $n\geq k$, there exist $\mathbf{z}_{n}\in\mathbf{1}^{\perp}$,
$i_{n}\in\mathcal{A}$, such that

(4.2)
$\begin{split}\\{(\mathbf{y},j)\in\mathbb{Z}^{d}\times\mathcal{A}\,:\,&\|M_{\sigma_{[0,n)}}^{-1}(\mathbf{y}-\mathbf{z}_{n})\|\leq
R,\,0\leq\langle\mathbf{1},\mathbf{y}\rangle<|\sigma_{[0,n)}(j)|\\}\\\
&\subset\\{(\mathbf{l}(p),j)\,:\,p\in\mathcal{A}^{*},j\in\mathcal{A},\,p\,i_{n}\preceq\sigma_{[0,n)}(j)\\}.\end{split}$

Here, $v\preceq w$ means that $v$ is a prefix of $w$. The geometric
coincidence condition is a rephrasing of the one defined in [BST19a, Section
2.11]; since we do not want to define discrete hyperplanes and dual
substitutions here, we use equivalent statements with usual substitutions and
abelianizations of words. It turns out that the following _effective version_
from [BST19a, Proposition 7.9 (iv)], which states that there are
$n\in\mathbb{N}$, $\mathbf{z}\in\mathbf{1}^{\perp}$, $i\in\mathcal{A}$, $C>0$,
such that $L_{\Sigma^{n}\boldsymbol{\sigma}}$ is $C$-balanced and

(4.3)
$\begin{split}\big{\\{}(\mathbf{y},j)\in\mathbb{Z}^{d}\times\mathcal{A}\,:\,\|&\pi^{\prime}_{\mathbf{u}^{(n)}}M_{\sigma_{[0,n)}}^{-1}\mathbf{y}-\mathbf{z}\|_{\infty}\leq
C,\,0\leq\langle\mathbf{1},\mathbf{y}\rangle<|\sigma_{[0,n)}(j)|\big{\\}}\\\
&\subset\big{\\{}(\mathbf{l}(p),j)\,:\,p\in\mathcal{A}^{*},j\in\mathcal{A},\,p\,i\preceq\sigma_{[0,n)}(j)\big{\\}},\end{split}$

with $\mathbf{u}^{(n)}=M_{\sigma_{[0,n)}}^{-1}\mathbf{u}$, is more useful for
our purposes. These conditions guarantee that
$\mathcal{C}_{\boldsymbol{\sigma}}$ contains an exclusive point, i.e., a point
contained in only one tile of $\mathcal{C}_{\boldsymbol{\sigma}}$. The fact
that $\mathcal{C}_{\boldsymbol{\sigma}}$ is a multiple tiling then leads to
the conclusion that $\mathcal{C}_{\boldsymbol{\sigma}}$ is actually a tiling.
This is illustrated in Figure 3; see also Proposition 4.1 below.

The geometric coincidence condition can be seen as an $S$-adic analogue to the
geometric coincidence condition (or super-coincidence condition) in [BK06,
IR06, BST10], which provides a tiling criterion. Recall that the periodic
tiling yields the isomorphism with a minimal toral translation and thus purely
discrete spectrum (see [BST19a, Proposition 8.5]; related results for the
periodic case is contained in [AI01, Theorem 2] and [CS01, Theorem 3.8]; for
the classical case that that initiated the whole theory we refer to [Rau82]).
This criterion is a coincidence type condition in the same vein as the various
coincidence conditions introduced in the usual Pisot framework; see e.g.
[Sol97, AL11].

Results from [BST19a] that are central for our proofs are contained in the
following proposition.

######  Proposition 4.1.

Let $\boldsymbol{\sigma}\in\mathcal{S}_{d}^{\mathbb{N}}$ be a directive
sequence satisfying PRICE. Then the following assertions are equivalent.

  1. (i)

The collection $\mathcal{C}_{\boldsymbol{\sigma}}$ forms a tiling.

  2. (ii)

The collection $\mathcal{C}_{\Sigma^{n}\boldsymbol{\sigma}}$ forms a tiling
for some $n\in\mathbb{N}$.

  3. (iii)

The collection $\mathcal{C}_{\Sigma^{n}\boldsymbol{\sigma}}$ forms a tiling
for all $n\in\mathbb{N}$.

  4. (iv)

The sequence $\boldsymbol{\sigma}$ satisfies the geometric coincidence
condition.

  5. (v)

The sequence $\boldsymbol{\sigma}$ satisfies the effective version of the
geometric coincidence condition.

Figure 3. Illustration of the effective version of the geometric coincidence
condition and of the proof of Proposition 4.1. The large tiles are the tiles
of $\mathcal{C}_{\boldsymbol{\sigma}}$, the points are the translation points
of their level $n$ subtiles (drawn in light grey). Each of these subtiles is
contained in a projected parallelepiped
$\pi^{\prime}_{\mathbf{u}}M_{[0,n)}[-C,C]^{3}$ centered at its translation
point. This is indicated for one point
$\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{z}$; note that up to three subtiles
can share the same translation point in this three letter example. The
constant $C$ is chosen in a way that $\Sigma^{n}\boldsymbol{\sigma}$ has
$C$-balanced language. All translation points inside the dark grey
parallelepiped belong to the same tile of $\mathcal{C}_{\boldsymbol{\sigma}}$,
namely $\mathcal{R}_{\boldsymbol{\sigma}}(i)$. Therefore,
$\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{z}$ is an exclusive point of
$\mathcal{C}_{\boldsymbol{\sigma}}$, hence $\mathcal{C}_{\boldsymbol{\sigma}}$
is a tiling.

###### Proof.

This is proved in [BST19a, Lemma 7.2 and Proposition 7.9]. However, the proof
of the implication (v)$\Rightarrow$(i) in [BST19a, Proposition 7.9] is very
concise. Since this assertion will be important in the sequel and in order to
explain the (effective version of the) geometric coincidence condition, we
give a more detailed proof here, and we illustrate it in Figure 3.

Assume that there are $n\in\mathbb{N}$, $\mathbf{z}\in\mathbf{1}^{\perp}$,
$i\in\mathcal{A}$, $C>0$, such that $\Sigma^{n}\boldsymbol{\sigma}\in B_{C}$
and (4.3) holds. We show that
$\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}\mathbf{z}$ is an exclusive point
of $\mathcal{C}_{\boldsymbol{\sigma}}$, where $\mathbf{u}$ is the generalized
right eigenvector of $\boldsymbol{\sigma}$ (which exists since
$\boldsymbol{\sigma}$ is primitive and recurrent). First note that each
$\mathcal{R}_{\boldsymbol{\sigma}}(i^{\prime})$, $i^{\prime}\in\mathcal{A}$,
can be written as

(4.4)
$\mathcal{R}_{\boldsymbol{\sigma}}(i^{\prime})=\bigcup_{p\in\mathcal{A}^{*},j\in\mathcal{A}\,:\,pi^{\prime}\preceq\sigma_{[0,n)}(j)}\pi^{\prime}_{\mathbf{u}}\big{(}\mathbf{l}(p)+M_{\sigma_{[0,n)}}\mathcal{R}_{\Sigma^{n}\boldsymbol{\sigma}}(j)\big{)},$

cf. [BST19a, Proposition 5.6]. Indeed, we have
$p^{\prime}i^{\prime}\preceq\sigma_{[0,k)}(j^{\prime})$ with $k>n$ if and only
if

$p^{\prime}i^{\prime}=\sigma_{[0,n)}(\tilde{p})\,pi^{\prime}\ \mbox{with}\
pi^{\prime}\preceq\sigma_{[0,n)}(j),\
\tilde{p}j\preceq\sigma_{[n,k)}(j^{\prime})\ \mbox{for some}\
p,\tilde{p}\in\mathcal{A}^{*},\,j\in\mathcal{A};$

since the projections satisfy
$\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}\pi^{\prime}_{\mathbf{u}^{(n)}}=\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}$,
where $\mathbf{u}^{(n)}=M_{\sigma_{[0,n)}}^{-1}\mathbf{u}$ is the generalized
right eigenvector of $\Sigma^{n}\boldsymbol{\sigma}$, this shows that (4.4)
holds. We remark here that $\mathcal{R}_{\boldsymbol{\sigma}}$ has two
different kinds of subtiles: the sets $\mathcal{R}_{\boldsymbol{\sigma}}(w)$
defined in (2.6) and the sets
$\pi^{\prime}_{\mathbf{u}}\big{(}\mathbf{l}(p)+M_{\sigma_{[0,n)}}\mathcal{R}_{\Sigma^{n}\boldsymbol{\sigma}}(j)\big{)}$
as in (4.4).

Let now $\mathbf{x}\in\mathbb{Z}^{d}\cap\mathbf{1}^{\perp}$,
$i^{\prime}\in\mathcal{A}$ be such that
$\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}\mathbf{z}\in\mathbf{x}+\mathcal{R}_{\boldsymbol{\sigma}}(i^{\prime})$.
Then by (4.4) there exist $p\in\mathcal{A}^{*}$, $j\in\mathcal{A}$ such that
$pi^{\prime}\preceq\sigma_{[0,n)}(j)$ and

$\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}\mathbf{z}\in\pi^{\prime}_{\mathbf{u}}\big{(}\mathbf{x}+\mathbf{l}(p)+M_{\sigma_{[0,n)}}\mathcal{R}_{\Sigma^{n}\boldsymbol{\sigma}}(j)\big{)}.$

Using
$\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}=\pi^{\prime}_{\mathbf{u}}\,M_{\sigma_{[0,n)}}\pi^{\prime}_{\mathbf{u}^{(n)}}$,
we obtain that

$\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}\mathbf{z}\in\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}\Big{(}\pi^{\prime}_{\mathbf{u}^{(n)}}M_{\sigma_{[0,n)}}^{-1}\big{(}\mathbf{x}+\mathbf{l}(p)\big{)}+\mathcal{R}_{\Sigma^{n}\boldsymbol{\sigma}}(j)\Big{)}.$

Since $\mathbf{z}\in\mathbf{1}^{\perp}$,
$\pi^{\prime}_{\mathbf{u}^{(n)}}\mathbf{y}\in\mathbf{1}^{\perp}$ for all
$\mathbf{y}\in\mathbb{R}^{d}$, and
$\mathcal{R}_{\Sigma^{n}\boldsymbol{\sigma}}(j)\subset\mathbf{1}^{\perp}$,
this yields

$\mathbf{z}\in\pi^{\prime}_{\mathbf{u}^{(n)}}M_{\sigma_{[0,n)}}^{-1}\big{(}\mathbf{x}+\mathbf{l}(p)\big{)}+\mathcal{R}_{\Sigma^{n}\boldsymbol{\sigma}}(j).$

Since $\Sigma^{n}\boldsymbol{\sigma}\in B_{C}$, we have
$\|\mathbf{y}\|_{\infty}\leq C$ for all
$\mathbf{y}\in\mathcal{R}_{\Sigma^{n}\boldsymbol{\sigma}}$, thus (4.3) implies
that $(\mathbf{x}+\mathbf{l}(p),j)=(\mathbf{l}(p^{\prime}),j^{\prime})$ for
some $p^{\prime}\in\mathcal{A}^{*}$, $j^{\prime}\in\mathcal{A}$ with
$p^{\prime}i\preceq\sigma_{[0,n)}(j^{\prime})$. Since $j=j^{\prime}$ and
$\mathbf{x}\in\mathbf{1}^{\perp}$, we obtain that $\mathbf{x}=\mathbf{0}$ and
$p=p^{\prime}$, thus $i^{\prime}=i$. (The set
$\\{\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}\mathbf{y}\,:\,\|\mathbf{z}-\mathbf{y}\|_{\infty}\leq
C\\}$ is the shaded cube in Figure 3; in view of (4.4), (4.3) says that this
cube contains only translation points of subtiles that correspond to subtiles
of $\mathcal{R}_{\boldsymbol{\sigma}}(i)$.) Therefore,
$\mathcal{R}_{\boldsymbol{\sigma}}(i)$ is the only tile of
$\mathcal{C}_{\boldsymbol{\sigma}}$ containing
$\pi^{\prime}_{\mathbf{u}}M_{\sigma_{[0,n)}}\mathbf{z}$. Since
$\mathcal{C}_{\boldsymbol{\sigma}}$ is a multiple tiling by [BST19a,
Proposition 7.5], it follows that it is a tiling. ∎

###  4.3. Purely discrete spectrum implies geometric coincidence

In our main theorems, substitutive dynamical systems with purely discrete
spectrum play a role. The following lemma links this property to the geometric
coincidence condition.

######  Lemma 4.2.

Let $\sigma$ be a Pisot substitution. If $(X_{\sigma},\Sigma)$ has purely
discrete spectrum, then $\sigma$ satisfies the geometric coincidence
condition.

###### Proof.

As mentioned in the introduction of [BK06], it follows from [CS03, Theorem
3.1] that $(X_{\sigma},\Sigma)$ has purely discrete spectrum if and only if
the associated tiling flow $T$ has purely discrete spectrum (as e.g. in
[Sad16], just note that if all the tiles in the self-similar tiling space
$\mathcal{T}$ have length $1$, the spectrum of $T$ on $\mathcal{T}$ is (up to
a multiplicative constant) the logarithm of the spectrum of the shift operator
$\Sigma$ on $X_{\sigma}$). According to [BK06, Corollary 9.4 and Remark 18.5],
the flow $T$ has purely discrete spectrum if and only if the collection
$\mathcal{C}_{\sigma}$ of Rauzy fractals associated with $\sigma$ forms a
tiling. Thus, Proposition 4.1 implies that the substitution $\sigma$ has
geometric coincidence. ∎

##  5\. Proofs of the main results

We first prove the results of Section 3.2. Later we will use these results in
the proofs of the results contained in Section 3.1.

###  5.1. Proof of Theorem 3.6

Let $D\subset\mathcal{S}_{d}^{\mathbb{N}}$ be a shift-invariant set of
directive sequences equipped with an ergodic $\Sigma$-invariant measure $\nu$
satisfying $\nu\circ\Sigma\ll\nu$. Assume that

  * •

the linear cocycle $(D,\Sigma,Z,\nu)$ defined by
$Z((\sigma_{n})_{n\in\mathbb{N}})=\vphantom{\\!}^{t}\\!M_{\sigma_{0}}$
satisfies the Pisot condition,

  * •

there is a periodic Pisot sequence with purely discrete spectrum and positive
range in $(D,\Sigma,\nu)$.

We first show that $\nu$-almost all $\boldsymbol{\sigma}\in D$ satisfy the
property PRICE; recall that $B_{C}$ denotes the set of sequences in
$\mathcal{S}_{d}^{\mathbb{N}}$ with $C$-balanced language.

######  Lemma 5.1.

Under the assumptions of Theorem 3.6, we have $\lim_{C\to\infty}\nu(D\cap
B_{C})=1$, in particular $D\cap B_{C}$ is $\nu$-measurable for all $C>0$.

###### Proof.

The Pisot condition yields that $\nu\big{(}\bigcup_{C\in\mathbb{N}}(D\cap
B_{C})\big{)}=1$ by [BD14, Theorem 6.4]. Since $B_{C}\subseteq B_{C^{\prime}}$
for all $C<C^{\prime}$, it only remains to show that $D\cap B_{C}$ is
$\nu$-measurable for all $C>0$. Let

$B^{\prime}_{C}=\bigcap_{n\in\mathbb{N}}\bigcup_{(\sigma_{0},\dots,\sigma_{n-1})\in\mathcal{S}_{d}^{n}:\,[\sigma_{0},\dots,\sigma_{n-1}]\cap
B_{C}\neq\emptyset}[\sigma_{0},\dots,\sigma_{n-1}].$

Then we clearly have $D\cap B_{C}\subseteq B^{\prime}_{C}$. If
$\boldsymbol{\sigma}\in B^{\prime}_{C}$, then we have $\boldsymbol{\sigma}\in
D$ and the finite languages

$L_{\boldsymbol{\sigma}}^{(n)}=\big{\\{}w\in\mathcal{A}^{*}\,:\,w\hbox{ is a
factor of }\sigma_{[0,n)}(i)\hbox{ for some }i\in\mathcal{A}\big{\\}}$

are $C$-balanced for all $n\in\mathbb{N}$. Since
$L_{\boldsymbol{\sigma}}^{(0)}\subseteq
L_{\boldsymbol{\sigma}}^{(1)}\subseteq\cdots$, also
$L_{\boldsymbol{\sigma}}=\bigcup_{n\in\mathbb{N}}L_{\boldsymbol{\sigma}}^{(n)}$
is $C$-balanced, i.e., $\boldsymbol{\sigma}\in B_{C}$. Hence, we have $D\cap
B_{C}=B^{\prime}_{C}$. Since countable unions and intersections of measurable
sets are measurable, we obtain that $B_{C}$ is $\nu$-measurable. ∎

######  Lemma 5.2.

Under the assumptions of Theorem 3.6, $\nu$-almost every
$\boldsymbol{\sigma}\in D$ satisfies PRICE.

###### Proof.

By the assumptions of Theorem 3.6, $D$ contains a periodic Pisot sequence with
positive range. In particular, there exists a sequence
$\boldsymbol{\tau}=(\tau_{n})\in D$ with
$\Sigma^{h}\boldsymbol{\tau}=\boldsymbol{\tau}$ such that $\tau_{[0,h)}$ is a
Pisot substitution and $\nu([\tau_{0},\dots,\tau_{n-1}])>0$ for all
$n\in\mathbb{N}$. From [CS01, Proposition 1.3], we get that $\tau_{[0,n)}$ is
primitive and, hence, there is $k\in\mathbb{N}$ such that $\tau_{[0,kn)}$ has
positive incidence matrix. Set $h=kn$. By Lemma 5.1, we can choose $C$ such
that $\nu(\Sigma^{-h}(D\cap B_{C}))=\nu(D\cap
B_{C})>1-\nu([\tau_{0},\dots,\tau_{h-1}])$, thus
$\nu\big{(}[\tau_{0},\dots,\tau_{h-1}]\cap\Sigma^{-h}B_{C}\big{)}>0$.

By Poincaré’s Recurrence Theorem, we have for almost all
$\boldsymbol{\sigma}=(\sigma_{n})_{n\in\mathbb{N}}\in D$ some
$\ell_{0}(\boldsymbol{\sigma})\geq h$ such that
$\Sigma^{\ell_{0}(\boldsymbol{\sigma})-h}\boldsymbol{\sigma}\in[\tau_{0},\dots,\tau_{h-1}]\cap\Sigma^{-h}B_{C}$,
i.e., $(\sigma_{0},\dots,\sigma_{\ell_{0}(\boldsymbol{\sigma})-1})$ ends with
$(\tau_{0},\dots,\tau_{h-1})$ and
$\Sigma^{\ell_{0}(\boldsymbol{\sigma})}\boldsymbol{\sigma}\in B_{C}$. We
extend $\ell_{0}(\boldsymbol{\sigma})$ for almost all $\boldsymbol{\sigma}\in
D$ to a sequence $(\ell_{k}(\boldsymbol{\sigma}))_{k\in\mathbb{N}}$ such that

  * •

$(\sigma_{0},\dots,\sigma_{\ell_{k+1}(\boldsymbol{\sigma})-1})$ ends with
$(\sigma_{0},\dots,\sigma_{\ell_{k}(\boldsymbol{\sigma})-1})$ (and, a
fortiori, with $(\tau_{0},\ldots,\tau_{h-1})$),

  * •

$\Sigma^{\ell_{k+1}(\boldsymbol{\sigma})}\boldsymbol{\sigma}\in B_{C}$,

  * •

$\ell_{k+1}(\boldsymbol{\sigma})\geq 2\ell_{k}(\boldsymbol{\sigma})$,

for all $k\in\mathbb{N}$. To this end, assume that
$\ell_{0}(\boldsymbol{\sigma}),\ldots,\ell_{k}(\boldsymbol{\sigma})$ are
already defined for almost all $\boldsymbol{\sigma}\in D$. Consider the set of
all $\boldsymbol{\sigma}$ having a given value
$\ell_{k}=\ell_{k}(\boldsymbol{\sigma})$ and a given prefix
$(\sigma_{0},\dots,\sigma_{\ell_{k}-1})$. Assume that this set has positive
measure, which implies that
$\nu\big{(}[\sigma_{0},\dots,\sigma_{\ell_{k}-1}]\cap\Sigma^{-\ell_{k}}B_{C}\big{)}>0$.
Then, for almost all $\boldsymbol{\sigma}$ in this set, we obtain (by
Poincaré’s Recurrence Theorem) some $\ell_{k+1}(\boldsymbol{\sigma})$ with the
required properties. Applying this for all choices of $\ell_{k}$ and
$(\sigma_{0},\dots,\sigma_{\ell_{k}-1})$, we get some
$\ell_{k+1}(\boldsymbol{\sigma})$ for almost all $\boldsymbol{\sigma}\in D$.
Therefore, such a sequence $(\ell_{k}(\boldsymbol{\sigma}))_{k\in\mathbb{N}}$
exists for almost all $\boldsymbol{\sigma}\in D$.

Setting
$n_{k}(\boldsymbol{\sigma})=\ell_{k+1}(\boldsymbol{\sigma})-\ell_{k}(\boldsymbol{\sigma})$,
we obtain that conditions (P), (R) and (C) of Property PRICE hold for almost
all $\boldsymbol{\sigma}\in E_{G}$. By [BST19a, Lemma 5.7], we can replace
$(n_{k})$ and $(\ell_{k})$ by subsequences such that condition (E) holds.
These subsequences also satisfy (P), (R) and (C). From the Pisot condition and
[BST19a, Lemma 8.7], we obtain that almost all $\boldsymbol{\sigma}\in D$ are
algebraically irreducible, i.e., (I) holds. ∎

So far we could use a slight variation of [BST19a, Theorem 3.1] to show that
for almost all $\boldsymbol{\sigma}\in D$ the dynamical system
$(X_{\boldsymbol{\sigma}},\Sigma)$ has a $m$-to-$1$ factor which is a minimal
translation on $\mathbb{T}^{d-1}$ for some $m\in\mathbb{N}$. The most
difficult part of the proof is to show that $m=1$, i.e., that
$(X_{\boldsymbol{\sigma}},\Sigma)$ actually is measurably conjugate to a
minimal translation on $\mathbb{T}^{d-1}$. According to [BST19a, Theorem 3.1],
in order to achieve this we have to prove that $\boldsymbol{\sigma}$ admits
geometric coincidence almost always.

Geometric coincidence is defined in [BST19a, Section 2.11] as the property
that certain subsets of discrete hyperplanes defined by dual substitutions
contain arbitrarily large balls in these hyperplanes. However, by the
effective version, it is sufficient to have a ball of a certain radius. In the
following lemmas, we use that sufficiently large balls given by geometric
coincidence for the substitutive system of a Pisot substitution $\tau$ provide
geometric coincidence for $S$-adic systems that contains a sufficiently long
block $(\sigma_{n},\ldots,\sigma_{n+\ell-1})$ satisfying
$\sigma_{[n,n+\ell)}=\tau^{m}$. Theorem 3.6 will then follow by Poincaré
recurrence.

######  Lemma 5.3.

Let $\tau$ be a Pisot substitution with geometric coincidence. Then for each
$C>0$ there are $m=m_{\tau}(C)\in\mathbb{N}$,
$\mathbf{z}\in\mathbf{1}^{\perp}$, and $i\in\mathcal{A}$ such that for each
$\mathbf{t}\in\mathbb{R}^{d}_{\geq 0}\setminus\\{\mathbf{0}\\}$ we have

(5.1)
$\begin{split}\big{\\{}(\mathbf{y},j)\in\mathbb{Z}^{d}\times\mathcal{A}\,:\,\|\pi^{\prime}_{\mathbf{t}}M_{\tau}^{-m}\mathbf{y}-\mathbf{z}\|_{\infty}\leq
C,&\,0\leq\langle\mathbf{1},\mathbf{y}\rangle<|\tau^{m}(j)|\big{\\}}\\\
&\subset\\{(\mathbf{l}(p),j)\,:\,p\in\mathcal{A}^{*},j\in\mathcal{A},\,p\,i\preceq\tau^{m}(j)\\}.\end{split}$

######  Remark 5.4.

If we look at the definition of geometric coincidence in (4.2) the lemma
states that the inclusion in the definition of geometric coincidence still
holds if we add the projection $\pi^{\prime}_{\mathbf{t}}$ for some
nonnegative vector $\mathbf{t}$. Indeed, because the elements
$M_{\tau}^{-m}\mathbf{y}$ that are projected are close to a hyperplane that is
“sufficiently orthogonal” to $\mathbf{t}$ and $\mathbf{1}$, this projection
does not change these vectors too much.

###### Proof.

Since $\tau$ satisfies the geometric coincidence condition, there exist, for
each $R>0$ and sufficiently large $m\in\mathbb{N}$, some $i\in\mathcal{A}$ and
$\mathbf{z}^{\prime}\in
M_{\tau}^{-m}\mathbf{1}^{\perp}=(\vphantom{\\!M}^{t}\\!M_{\tau}^{m}\mathbf{1})^{\perp}$,
such that

(5.2)
$\begin{split}\\{(\mathbf{y},j)\in\mathbb{Z}^{d}\times\mathcal{A}\,:\,\|M_{\tau}^{-m}\mathbf{y}-\mathbf{z}^{\prime}\|_{\infty}&\leq
R,\,0\leq\langle\mathbf{1},\mathbf{y}\rangle<|\tau^{m}(j)|\\}\\\
&\subset\\{(\mathbf{l}(p),j)\,:\,p\in\mathcal{A}^{*},\,j\in\mathcal{A},\,p\,i\preceq\tau^{m}(j)\\}.\end{split}$

Since
$\vphantom{\\!M}{}^{t}\\!M_{\tau}^{m}\mathbf{1}/\|\vphantom{\\!M}^{t}\\!M_{\tau}^{m}\mathbf{1}\|$
converges to a dominant eigenvector of $\vphantom{\\!M}{}^{t}\\!M_{\tau}$,
which is positive, there exists a constant $c_{1}>0$ such that
$\|\mathbf{x}\|_{\infty}\leq
c_{1}\|\pi^{\prime}_{\mathbf{t}}\mathbf{x}\|_{\infty}$ for all
$\mathbf{t}\in\mathbb{R}^{d}_{\geq 0}\setminus\\{\mathbf{0}\\}$,
$\mathbf{x}\in(\vphantom{\\!M}^{t}\\!M_{\tau}^{m}\mathbf{1})^{\perp}$,
$m\in\mathbb{N}$. Let $\tilde{\pi}_{\mathbf{t},m}$ denote the projection along
$\mathbf{t}$ on $(\vphantom{\\!M}^{t}\\!M_{\tau}^{m}\mathbf{1})^{\perp}$.
There is another constant $c_{2}>0$ such that
$\|\mathbf{x}-\tilde{\pi}_{\mathbf{t},m}\mathbf{x}\|_{\infty}\leq c_{2}$ for
all $\mathbf{x}\in\mathbb{R}^{d}$ with
$0\leq\langle\vphantom{\\!M}^{t}\\!M_{\tau}^{m}\mathbf{1},\mathbf{x}\rangle<\max_{j\in\mathcal{A}}\langle\vphantom{\\!M}^{t}\\!M_{\tau}^{m}\mathbf{1},\mathbf{e}_{j}\rangle=\max_{j\in\mathcal{A}}|\tau^{m}(j)|$.
Therefore, we have

$\|M_{\tau}^{-m}\mathbf{y}-\mathbf{z}^{\prime}\|_{\infty}\leq\|\tilde{\pi}_{\mathbf{t},m}M_{\tau}^{-m}\mathbf{y}-\mathbf{z}^{\prime}\|_{\infty}+c_{2}\leq
c_{1}\|\pi^{\prime}_{\mathbf{t}}(M_{\tau}^{-m}\mathbf{y}-\mathbf{z}^{\prime})\|_{\infty}+c_{2}$

for all $\mathbf{y}\in\mathbb{Z}^{d}$,
$\mathbf{z}^{\prime}\in(\vphantom{\\!M}^{t}\\!M_{\tau}^{m}\mathbf{1})^{\perp}$
with
$0\leq\langle\mathbf{1},\mathbf{y}\rangle<\max_{j\in\mathcal{A}}|\tau^{m}(j)|$.
Choosing $m=m_{\tau}(C)$ such that (5.2) holds for $R=c_{1}C+c_{2}$ and some
$\mathbf{z}^{\prime}\in\mathbf{1}^{\perp}$, $i\in\mathcal{A}$, we obtain that
(5.1) holds with $\mathbf{z}=\pi^{\prime}_{\mathbf{t}}\mathbf{z}^{\prime}$. ∎

This lemma is now used in order to prove geometric coincidence for directive
sequences $\boldsymbol{\sigma}=(\sigma_{n})$ containing a long block
$(\sigma_{n},\ldots,\sigma_{n+\ell-1})$ satisfying
$\sigma_{[n,n+\ell)}=\tau^{m}$ followed by a tail
$\Sigma^{n+\ell}\boldsymbol{\sigma}\in B_{C}$. Indeed, this constellation will
allow us to apply Lemma 5.3 in order to fulfill the effective version of the
geometric coincidence condition for $\Sigma^{n+\ell}\boldsymbol{\sigma}$. Thus
$\Sigma^{n+\ell}\boldsymbol{\sigma}$ gives rise to tilings which will lead to
the desired conclusion.

######  Lemma 5.5.

Let $\tau$ be a Pisot substitution that satisfies geometric coincidence. Let
$\boldsymbol{\sigma}=(\sigma_{n})$ be a sequence satisfying PRICE and $C>0$
such that there are $\ell,n\in\mathbb{N}$ such that for $m=m_{\tau}(C)$ as in
Lemma 5.3 we have $\sigma_{[n,n+\ell)}=\tau^{m}$ and
$\Sigma^{n+\ell}\boldsymbol{\sigma}\in B_{C}$. Then
$\mathcal{C}_{\boldsymbol{\sigma}}$ forms a tiling of $\mathbf{1}^{\perp}$.

###### Proof.

Let $\tau$, $\boldsymbol{\sigma}$ be as in the statement of the lemma, with
incidence matrices $M_{\tau}$ and $M_{n}=M_{\sigma_{n}}$, respectively. Since
$\boldsymbol{\sigma}$ satisfies PRICE, $\Sigma^{n}\boldsymbol{\sigma}$ also
satisfies PRICE by [BST19a, Lemma 5.10]. Now we apply Lemma 5.3 to $\tau$ and
$\mathbf{t}=M_{\sigma_{[n,n+\ell)}}^{-1}M_{\sigma_{[0,n)}}^{-1}\mathbf{u}$;
recall that $M_{\sigma_{[0,n)}}^{-1}\mathbf{u}$ is the generalized right
eigenvalue of $\Sigma^{n}\boldsymbol{\sigma}$. Since
$\sigma_{[n,n+\ell)}=\tau^{m}$ this yields that

$\displaystyle\\{(\mathbf{y},j)\in\mathbb{Z}^{d}\times\mathcal{A}\,:\,$
$\displaystyle\|\pi^{\prime}_{\mathbf{t}}M_{\sigma_{[n,n+\ell)}}^{-1}\mathbf{y}-\mathbf{z}\|\leq
C,\,0\leq\langle\mathbf{1},\mathbf{y}\rangle<|\sigma_{[n,n+\ell)}(j)|\\}$
$\displaystyle=\\{(\mathbf{y},j)\in\mathbb{Z}^{d}\times\mathcal{A}\,:\,\|\pi^{\prime}_{\mathbf{t}}M_{\tau}^{-m}\mathbf{y}-\mathbf{z}\|\leq
C,\,0\leq\langle\mathbf{1},\mathbf{y}\rangle<|\tau^{m}(j)|\\}$
$\displaystyle\subset\\{(\mathbf{l}(p),j)\,:\,p\in\mathcal{A}^{*},j\in\mathcal{A},\,p\,i\preceq\sigma_{[n,n+\ell)}(j)\\}.$

Thus all conditions of Proposition 4.1 (v) are satisfied by
$\Sigma^{n}\boldsymbol{\sigma}$, hence
$\mathcal{C}_{\Sigma^{n}\boldsymbol{\sigma}}$ forms a tiling, hence
$\mathcal{C}_{\boldsymbol{\sigma}}$ forms a tiling by Proposition 4.1. ∎

We are now in a position to prove Theorem 3.6. Indeed, we use the Poincaré
Recurrence Theorem in order to show that under the conditions of Theorem 3.6,
Lemma 5.5 can be applied to almost all directive sequences
$\boldsymbol{\sigma}\in D$.

###### Conclusion of the proof of Theorem 3.6.

According to the assumptions of Theorem 3.6, there is a periodic sequence
$(\tau_{0},\dots,\tau_{k-1})^{\infty}\in D$ such that
$\tau=\tau_{0}\circ\cdots\circ\tau_{k-1}$ is a Pisot substitution and the
substitutive system $(X_{\tau},\Sigma)$ has purely discrete spectrum. By Lemma
4.2, $\tau$ satisfies the geometric coincidence condition. By Lemma 5.1, there
is a $C\in\mathbb{N}$ such that

$\nu(D\cap
B_{C})>1-\inf_{n\in\mathbb{N}}\nu(\Sigma^{n}[\tau_{0},\dots,\tau_{n-1}]).$

Then we have $\nu(\Sigma^{n}[\tau_{0},\dots,\tau_{n-1}]\cap B_{C})>0$ and,
since $\nu\circ\Sigma\ll\nu$, $\nu([\tau_{0},\dots,\tau_{n-1}]\Sigma^{-n}\cap
B_{C})>0$ for all $n\in\mathbb{N}$. Choose $m=m_{\tau}(C)$ as in Lemma 5.3. By
Poincaré’s Recurrence Theorem, for almost all sequences
$\boldsymbol{\sigma}\in D$, there exists $n$ such that
$\Sigma^{n}\boldsymbol{\sigma}\in[\tau_{0},\dots,\tau_{km-1}]\cap\Sigma^{-km}B_{C}$.
Thus Lemma 5.5 yields geometric coincidence for almost all
$\boldsymbol{\sigma}\in D$. This implies that
$\mathcal{C}_{\boldsymbol{\sigma}}$ forms a tiling of $\mathbf{1}^{\perp}$. We
may thus apply [BST19a, Proposition 8.5] to conclude that
$(X_{\boldsymbol{\sigma}},\Sigma,\mu)$ is conjugate to the translation by
$\mathbf{e}_{i}-\mathbf{u}$ on $\mathbf{1}^{\perp}/\mathbb{Z}^{d}$ for all
$i\in\\{1,\ldots,d\\}$, where $\mathbf{u}$ is the generalized right
eigenvector of $\boldsymbol{\sigma}$ with $\|\mathbf{u}\|_{1}=1$. Taking $i=d$
and omitting the $d$-th coordinate, we obtain that
$(X_{\boldsymbol{\sigma}},\Sigma,\mu)$ is conjugate to the translation by
$-\pi(\mathbf{u})$ on $\mathbb{T}^{d-1}$, thus also to the translation by
$\pi(\mathbf{u})$. ∎

###  5.2. Proof of Theorem 3.7

The proof of Theorem 3.7 consists in getting rid of the pure discrete spectrum
condition for the periodic Pisot sequence in Theorem 3.6. In other words,
under the conditions of Theorem 3.7 we have to provide a substitution with
purely discrete spectrum, i.e., satisfying the geometric coincidence
condition. We need two technical lemmas.

######  Lemma 5.6.

Let $M$ be a Pisot matrix with dominant right eigenvector $\mathbf{u}$. There
exists a constant $C>0$ such that $L_{\sigma}$ is $C$-balanced for all
substitutions $\sigma$ satisfying $M_{\sigma}=M^{k}$ for some $k\in\mathbb{N}$
and

$\max_{p\in\mathcal{A}^{*}\,:\,p\preceq\sigma(i),\,i\in\mathcal{A}}\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(p)\|<2.$

###### Proof.

Let $\sigma$ be a substitution satisfying the conditions indicated in the
statement of the lemma. Let $n\in\mathbb{N}$ be arbitrary but fixed and choose
a prefix $p$ of $\sigma^{n}(i_{n})$ for some $i\in\mathcal{A}$. Then we have
$p=\sigma^{n-1}(p_{n-1})\cdots\sigma(p_{1})p_{0}$ for some prefixes $p_{j}$ of
$\sigma(i_{j})$, $i_{j}\in\mathcal{A}$, (with $\sigma(i_{j})\in
p_{j}i_{j-1}\mathcal{A}^{*}$), thus

$\mathbf{l}(p)=M^{k(n-1)}\mathbf{l}(p_{n-1})+\cdots+M^{k}\mathbf{l}(p_{1})+\mathbf{l}(p_{0}).$

Let $\mathbf{v}$ be a dominant left eigenvector of $M$, $\varrho<1$ the
maximal absolute value of the non-dominant eigenvalues of $M$ and
$\tilde{\pi}_{\mathbf{u}}$ the projection along $\mathbf{u}$ on
$\mathbf{v}^{\perp}$. Then we have a constant $c_{1}>0$ such that
$\|M^{\ell}\mathbf{x}\|\leq c_{1}\varrho^{\ell}\|\mathbf{x}\|$ for all
$\ell\in\mathbb{N}$, $\mathbf{x}\in\mathbf{v}^{\perp}$. Thus we have
$\|\tilde{\pi}_{\mathbf{u}}M^{\ell}\mathbf{x}\|=\|M^{\ell}\tilde{\pi}_{\mathbf{u}}\mathbf{x}\|\leq
c_{1}\varrho^{\ell}\|\tilde{\pi}_{\mathbf{u}}\mathbf{x}\|$ for all
$\mathbf{x}\in\mathbb{R}^{d}$, hence

$\|\tilde{\pi}_{\mathbf{u}}\mathbf{l}(p)\|<\frac{c_{1}}{1-\varrho^{k}}\max_{q\in\mathcal{A}^{*}:\,q\preceq\sigma(i),\,i\in\mathcal{A}}\|\tilde{\pi}_{\mathbf{u}}\mathbf{l}(q)\|.$

There is a constant $c_{2}>0$ such that
$\|\pi^{\prime}_{\mathbf{u}}\mathbf{x}\|\leq c_{2}\|\mathbf{x}\|$ for all
$\mathbf{x}\in\mathbf{v}^{\perp}$ and
$\|\tilde{\pi}_{\mathbf{u}}\mathbf{x}\|\leq c_{2}\|\mathbf{x}\|$ for all
$\mathbf{x}\in\mathbf{1}^{\perp}$, thus
$\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(p)\|<\frac{2c_{1}c_{2}^{2}}{1-\varrho^{k}}$.
If $v\in L_{\sigma}$, then $v$ is a factor of $\sigma^{n}(i)$ for some
$n\in\mathbb{N}$, $i\in\mathcal{A}$. Thus there are two prefixes $p_{1},p_{2}$
of $\sigma^{n}(i)$ such that $p_{1}v=p_{2}$ and, hence,
$\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(v)\|\leq\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(p_{1})\|+\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(p_{2})\|<\frac{4c_{1}c_{2}^{2}}{1-\varrho^{k}}$.
Moreover, for two factors $v_{1},v_{2}$ with
$\mathbf{l}(v_{1})=\mathbf{l}(v_{2})$ we have
$\|\mathbf{l}(v_{1})-\mathbf{l}(v_{2})\|=\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(v_{1})-\pi^{\prime}_{\mathbf{u}}\mathbf{l}(v_{2})\|\leq\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(v_{1})\|+\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(v_{2})\|\leq\frac{8c_{1}c_{2}^{2}}{1-\varrho^{k}}$
and thus $L_{\sigma}$ is $C$-balanced with
$C=\frac{8c_{1}c_{2}^{2}}{1-\varrho^{k}}$. ∎

######  Lemma 5.7.

Let $\mathbf{x}\in\mathbb{N}^{d}$. Then there exists a word
$w\in\mathcal{A}^{*}$ such that $\mathbf{l}(w)=\mathbf{x}$ and
$\|\pi^{\prime}_{\mathbf{x}}\mathbf{l}(p)\|\leq 1-\frac{1}{2d-2}$ for all
prefixes $p$ of $w$. Moreover, $w$ starts with the letter corresponding to the
largest coordinate of $\mathbf{x}$.

###### Proof.

This is proved in [Mei73, Tij80]. ∎

The construction of the desired substitution is contained in the following
proposition.

######  Proposition 5.8.

Let $M$ be a nonnegative Pisot matrix. Then there exists a substitution
$\sigma$ with $M_{\sigma}=M^{n}$ for some $n\in\mathbb{N}$ such that the
geometric coincidence condition holds. We can choose $\sigma$ in a way that
$\sigma(i)$ is a prefix of $\sigma(j)$ or $\sigma(j)$ is a prefix of
$\sigma(i)$ ($i,j\in\mathcal{A}$).

###### Proof.

Let $\mathbf{u}$ be a dominant right eigenvector of $M$ and

$P=\\{\mathbf{y}\in\mathbb{Z}^{d}:\,0\leq\langle\mathbf{1},M^{n}\mathbf{y}\rangle\leq\max_{i\in\mathcal{A}}\langle\mathbf{1},M^{n}\mathbf{e}_{i}\rangle\
\mbox{for some}\ n\in\mathbb{N},\,\|\pi^{\prime}_{\mathbf{u}}\mathbf{y}\|\leq
C\\},$

with $C$ as in Lemma 5.6. Note that $P$ is a finite set since
$\langle\mathbf{1},M^{n}\mathbf{y}\rangle=\langle\vphantom{\\!M}^{t}\\!M^{n}\mathbf{1},\mathbf{y}\rangle$
and $\mathbf{u}\in\mathbb{R}_{+}^{d}$. Write
$P=\\{\mathbf{y}_{\ell}:\,0\leq\ell\leq L\\}$ such that
$0=\langle\mathbf{u},\mathbf{y}_{0}\rangle<\langle\mathbf{u},\mathbf{y}_{1}\rangle<\cdots<\langle\mathbf{u},\mathbf{y}_{L}\rangle$;
this is possible since $\mathbf{u}$ has rationally independent coordinates.
Then for $n\in\mathbb{N}$ large enough we have
$\|\pi^{\prime}_{\mathbf{u}}M^{n}\mathbf{y}\|\leq 1/2$ for all $\mathbf{y}\in
P$ and $M^{n}(\mathbf{y}_{\ell+1}-\mathbf{y}_{\ell})\in\mathbb{N}^{d}$ for all
$0\leq\ell<L$. Let the words $w_{\ell}$ be given by Lemma 5.7 with
$\mathbf{x}=\mathbf{x}_{\ell}=M^{n}(\mathbf{y}_{\ell+1}-\mathbf{y}_{\ell})$
for $0\leq\ell<L$, and set $\sigma(j)=w_{0}w_{1}\cdots w_{L_{j}-1}$ for all
$j\in\mathcal{A}$, with $L_{j}$ such that $\mathbf{y}_{L_{j}}=\mathbf{e}_{j}$.
By Lemma 5.7, we have
$\|\pi^{\prime}_{\mathbf{u}}\mathbf{l}(p)\|\leq\|\pi_{M^{n}(\mathbf{y}_{\ell+1}-\mathbf{y}_{\ell}),\mathbf{1}}\mathbf{l}(p)\|+\|\pi^{\prime}_{\mathbf{u}}M^{n}(\mathbf{y}_{\ell+1}-\mathbf{y}_{\ell})\|<2$
for all prefixes $p$ of $\sigma(j)$, $j\in\mathcal{A}$, thus $L_{\sigma}$ is
$C$-balanced by Lemma 5.6. By the construction of $\sigma$, we have
$M_{\sigma}=M^{n}$ and

(5.3)
$\begin{split}\\{(\mathbf{y},j)\in\mathbb{Z}^{d}\times\mathcal{A}:\,\|\pi^{\prime}_{\mathbf{u}}M_{\sigma}^{-1}\mathbf{y}\|\leq
C,&\,0\leq\langle\mathbf{1},\mathbf{y}\rangle<|\sigma(j)|\\}\\\
&=\bigcup_{j\in\mathcal{A}}\\{(\mathbf{l}(w_{0}\cdots
w_{\ell}),j)\;:\;0\leq\ell<L_{j}-1\\}\\\
&\subset\bigcup_{i\in\mathcal{A}}\\{(\mathbf{l}(p),j)\,:\,p\in\mathcal{A}^{*},j\in\mathcal{A},\,p\,i\preceq\sigma(j)\\}.\end{split}$

Moreover, $\sigma(i)$ is a prefix of $\sigma(j)$ if and only if
$\langle\mathbf{u},\mathbf{e}_{i}\rangle<\langle\mathbf{u},\mathbf{e}_{j}\rangle$.
Let $i_{0}\in\mathcal{A}$ be chosen in a way that
$\langle\mathbf{u},\mathbf{e}_{i_{0}}\rangle=\max_{j\in\mathcal{A}}\langle\mathbf{u},\mathbf{e}_{j}\rangle$.
Then the $i_{0}$-th coordinate of $\mathbf{x}_{\ell}$ is the largest one for
each $1\leq\ell<L$ if $n$ is chosen large enough. In the construction of
[Tij80], the word $w=w_{\ell}$ in Lemma 5.7 starts with the letter $i_{0}$ for
each $\mathbf{x}=\mathbf{x}_{\ell}$ ($1\leq\ell<L$). Therefore, we can choose
$n$ large enough such that all words $w_{\ell}$, $0\leq\ell<L$, start with
$i_{0}$. This means that we can sharpen the inclusion in (5.3) to

$\bigcup_{j\in\mathcal{A}}\\{(\mathbf{l}(w_{0}\cdots
w_{\ell}),j)\,:\,0\leq\ell<L_{j}-1\\}\subset\\{(\mathbf{l}(p),j)\,:\,p\in\mathcal{A}^{*},j\in\mathcal{A},\,p\,i_{0}\preceq\sigma(j)\\}.$

Together with (5.3) this yields

$\begin{split}\\{(\mathbf{y},j)\in\mathbb{Z}^{d}\times\mathcal{A}:\,\|\pi^{\prime}_{\mathbf{u}}M_{\sigma}^{-1}\mathbf{y}\|\leq
C,&\,0\leq\langle\mathbf{1},\mathbf{y}\rangle<|\sigma(j)|\\}\\\
&\subset\\{(\mathbf{l}(p),j)\,:\,p\in\mathcal{A}^{*},j\in\mathcal{A},\,p\,i_{0}\preceq\sigma(j)\\},\end{split}$

hence $\sigma$ satisfies geometric coincidence by Proposition 4.1. ∎

######  Remark 5.9.

To prove Proposition 5.8, we could also have used the condition from [Bar16,
Corollary 2] to check geometric coincidence. This condition requires that the
last letter of $\sigma(i)$ is equal for all $i\in\mathcal{A}$ and the first
letter of $\sigma(i)$ is different from the first letter of $\sigma(j)$ if
$i\neq j$; if $M$ is a positive matrix with non-zero determinant, then there
is clearly a substitution $\sigma$ with incidence matrix $M$ having this
property. However, since [Bar16] deals with an $\mathbb{R}$-action which is a
suspension of the shift $\Sigma$, a some more detailed discussion (like the
one contained in [BK06]) would be needed to adapt the results of [Bar16] to
our setting.

We can now finish the proof of Theorem 3.7.

###### Proof of Theorem 3.7.

Let $(\mathfrak{D},\Sigma,Z,\nu)$ be as in the statement of Theorem 3.7. Then
there is a periodic sequence
$(\tilde{M}_{0},\dots,\tilde{M}_{k-1})^{\infty}\in\mathfrak{D}$ such that
$\tilde{M}_{[0,k)}$ is a Pisot matrix. Since $\tilde{M}_{[0,k)}$ and
$\tilde{M}_{[i,k)}\tilde{M}_{[0,i)}$ are similar matrices, also
$\tilde{M}_{[i,k)}\tilde{M}_{[0,i)}$ is a Pisot matrix for all $0\leq i<k$. By
Proposition 5.8, we can assume that there is a substitution $\tau_{i}$ with
$M_{\tau_{i}}=\tilde{M}_{[i,k)}\tilde{M}_{[0,i)}$ satisfying the geometric
coincidence condition (replace $k$ by $kn$ for some $n\in\mathbb{N}$ if
necessary). We choose $\tau_{i}$ in a way that $\tau_{i}=\tau_{j}$ if
$\tilde{M}_{[i,k)}\tilde{M}_{[0,i)}=\tilde{M}_{[j,k)}\tilde{M}_{[0,j)}$
($0\leq i,j<k)$.

Choose a map $s:\,\mathcal{M}_{d}^{k}\to\mathcal{S}_{d}$ with the properties
that

  * •

the incidence matrix of $s(M_{0},\dots,M_{k-1})$ is $M_{[0,k)}$ for all
$(M_{0},\dots,M_{k-1})\in\mathcal{M}_{d}^{k}$,

  * •

$s(M_{0},\dots,M_{k-1})=s(M^{\prime}_{0},\dots,M^{\prime}_{k-1})$ if
$M_{[0,k)}=M^{\prime}_{[0,k)}$,

  * •

$s(M_{0},\dots,M_{k-1})=\tau_{i}$ if
$M_{[0,k)}=\tilde{M}_{[i,k)}\tilde{M}_{[0,i)}$ for some $0\leq i<k$.

Then the map

$\boldsymbol{\psi}:\,\mathfrak{D}\to
D,\quad(M_{n})_{n\in\mathbb{N}}\mapsto\big{(}s(M_{kn},\dots,M_{kn+k-1})\big{)}_{n\in\mathbb{N}}$

is well defined, and we have the commutative diagram

${\mathfrak{D}}$${\mathfrak{D}}$${D}$${D}$$\scriptstyle{\Sigma^{k}}$$\scriptstyle{\boldsymbol{\psi}}$$\scriptstyle{\boldsymbol{\psi}}$$\scriptstyle{\Sigma}$

We now show that $(D,\Sigma,\nu^{\prime})$, with
$\nu^{\prime}=\nu\circ\boldsymbol{\psi}^{-1}$, can be partitioned into ergodic
systems that satisfy the conditions of Theorem 3.6. Suppose that
$(D,\Sigma,\nu^{\prime})$ is not ergodic. Then there exists a
$\Sigma$-invariant (up to measure zero) subset $\tilde{D}\subseteq D$ with
$0<\nu^{\prime}(\tilde{D})<1$. Then
$\boldsymbol{\psi}^{-1}(\tilde{D})\subset\mathfrak{D}$ is
$\Sigma^{k}$-invariant, hence
$\bigcup_{i=0}^{k-1}\Sigma^{-i}\boldsymbol{\psi}^{-1}(\tilde{D})$ is
$\Sigma$-invariant and, by ergodicity of $\nu$, equal to $\mathfrak{D}$ up to
measure zero. Therefore, we have
$\nu^{\prime}(\tilde{D})=\nu(\boldsymbol{\psi}^{-1}(\tilde{D}))\geq 1/k$.
Since $D\setminus\tilde{D}$ is also $\Sigma$-invariant, we also have
$\nu^{\prime}(\tilde{D})\leq 1-1/k$. We repeat the argument until we have a
measurable partitition $\\{D_{1},\dots,D_{\ell}\\}$ of $D$, with
$1\leq\ell\leq k$, such that $(D_{j},\Sigma,\nu^{\prime}|_{D_{j}})$ is ergodic
for all $1\leq j\leq\ell$. Let now $j$ be fixed. Since
$\inf_{n\in\mathbb{N}}\nu(\Sigma^{kn}[(\tilde{M}_{0},\ldots,\tilde{M}_{k-1})^{n}])>0$,
there is some $0\leq i<k$ such that

$\inf_{n\in\mathbb{N}}\nu\big{(}\Sigma^{kn}[(\tilde{M}_{0},\ldots,\tilde{M}_{k-1})^{n}]\cap\boldsymbol{\psi}(\Sigma^{-i}\boldsymbol{\psi}^{-1}(D_{j}))\big{)}>0.$

Therefore, we have

$\inf_{n\in\mathbb{N}}\nu\big{(}\Sigma^{kn}[(\tilde{M}_{i},\ldots,\tilde{M}_{k-1},\tilde{M}_{0},\ldots,\tilde{M}_{i-1})^{n}]\cap\boldsymbol{\psi}(\Sigma^{-i}\boldsymbol{\psi}^{-1}(D_{j}))\big{)}>0.$

Since $Z$ satisfies the Pisot condition, the same holds for
$\mathcal{Z}_{j}:\,D_{j}\to\mathcal{M}_{d}$,
$(\sigma_{n})\mapsto\vphantom{\\!}^{t}\\!M_{\sigma_{0}}$. Therefore, we can
apply Theorem 3.6 to $(D_{j},\Sigma,\mathcal{Z}_{j},\nu^{\prime}|_{D_{j}})$.
This proves the result. ∎

###  5.3. Proofs of Theorems 3.1 and 3.4

We now prove Theorems 3.1 and 3.4 by reducing them to Theorems 3.6 and 3.7.

###### Proof of Theorem 3.1.

Let $(\Delta,T,A,\nu)$ be a positive $(d{-}1)$-dimensional continued fraction
algorithm satisfying the Pisot condition and $\nu\circ T\ll\nu$, let
$\boldsymbol{\varphi}$ be a faithful substitutive realization of
$(\Delta,T,A,\nu)$, and assume that there is a periodic Pisot point such that
$\boldsymbol{\varphi}(\mathbf{x})$ has purely discrete spectrum and the local
postive range property in $(\Delta,T,A,\nu)$. Then we have
$(\Delta,T,\nu)\overset{\boldsymbol{\varphi}}{\cong}(\boldsymbol{\varphi}(\Delta),\Sigma,\nu\circ\boldsymbol{\varphi}^{-1})$,
hence $\nu\circ\boldsymbol{\varphi}^{-1}$ is an ergodic $\Sigma$-invariant
measure satisfying
$\nu\circ\boldsymbol{\varphi}^{-1}\circ\Sigma\ll\nu\circ\boldsymbol{\varphi}^{-1}$,
the linear cocycle
$(\boldsymbol{\varphi}(\Delta),\Sigma,Z,\nu\circ\boldsymbol{\varphi}^{-1})$
defined by
$Z((\sigma_{n})_{n\in\mathbb{N}})=\vphantom{\\!}^{t}\\!M_{\sigma_{0}}$
satisfies the Pisot condition, and $\boldsymbol{\varphi}(\mathbf{x})$ is a
periodic Pisot sequence with purely discrete spectrum and the local positive
range property in
$(\boldsymbol{\varphi}(\Delta),\Sigma,\nu\circ\boldsymbol{\varphi}^{-1})$.
Therefore, by Theorem 3.6, for $\nu$-almost all $\mathbf{x}\in\Delta$ the
$S$-adic dynamical system $(X_{\boldsymbol{\sigma}},\Sigma)$ is a natural
coding of the minimal translation by $\pi(\mathbf{u})$ on $\mathbb{T}^{d-1}$
with respect to a partition of a bounded fundamental domain, where
$\mathbf{u}$ is the generalized right eigenvector of
$\boldsymbol{\varphi}(\mathbf{x})$ with $\|\mathbf{u}\|_{1}=1$. Since
$\mathbf{x}$ is a generalized right eigenvector of
$\boldsymbol{\varphi}(\mathbf{x})$, this proves Theorem 3.1. ∎

Theorem 3.4 follows from Theorem 3.7 in the following way.

###### Proof of Theorem 3.4.

Let $(\Delta,T,A,\nu)$ be a positive $({d{-}1})$-dimensional continued
fraction algorithm satisfying the Pisot condition and $\nu\circ T\ll\nu$, and
assume that the local positive range property holds for some periodic Pisot
point $\mathbf{y}\in\Delta$. Define
$\boldsymbol{\eta}:\,\Delta\to\mathcal{M}_{d}^{\mathbb{N}}$ by
$\mathbf{y}\to(\vphantom{\\!}^{t}\\!A(T^{n}\mathbf{y}))_{n\in\mathbb{N}}$.
Then we have
$(\Delta,T,\nu)\overset{\boldsymbol{\eta}}{\cong}(\boldsymbol{\eta}(\Delta),\Sigma,\nu\circ\boldsymbol{\eta}^{-1})$,
hence $\nu\circ\boldsymbol{\eta}^{-1}$ is an ergodic $\Sigma$-invariant
measure satisfying
$\nu\circ\boldsymbol{\eta}^{-1}\circ\Sigma\ll\nu\circ\boldsymbol{\eta}^{-1}$,
the linear cocycle
$(\boldsymbol{\eta}(\Delta),\Sigma,Z,\nu\circ\boldsymbol{\eta}^{-1})$ defined
by $Z((M_{n})_{n\in\mathbb{N}})=\vphantom{\\!}^{t}\\!M_{0}$ satisfies the
Pisot condition, and the local positive range property in
$(\boldsymbol{\eta}(\Delta),\Sigma,\nu\circ\boldsymbol{\eta}^{-1})$ holds at
$\boldsymbol{\eta}(\mathbf{y})$. Therefore, by Theorem 3.7, there exists a
positive integer $k$ and a map
$\boldsymbol{\psi}:\,\boldsymbol{\eta}(\Delta)\to\mathcal{S}_{d}^{\mathbb{N}}$
satisfying $\boldsymbol{\psi}\circ\Sigma^{k}=\Sigma\circ\boldsymbol{\psi}$
such that for $\nu$-almost all $\mathbf{x}\in\Delta$ the $S$-adic dynamical
system $(X_{\boldsymbol{\psi}\circ\boldsymbol{\eta}(\mathbf{x})},\Sigma)$ is a
natural coding of the minimal translation by $\pi(\mathbf{x})$ on
$\mathbb{T}^{d-1}$ with respect to a partition of a bounded fundamental
domain. Setting
$\boldsymbol{\varphi}=\boldsymbol{\psi}\circ\boldsymbol{\eta}$, we obtain that
the diagram

${\Delta}$${\Delta}$${\boldsymbol{\eta}(\Delta)}$${\boldsymbol{\eta}(\Delta)}$${\boldsymbol{\varphi}(\Delta)}$${\boldsymbol{\varphi}(\Delta)}$$\scriptstyle{T^{k}}$$\scriptstyle{\boldsymbol{\eta}}$$\scriptstyle{\boldsymbol{\varphi}}$$\scriptstyle{\boldsymbol{\eta}}$$\scriptstyle{\boldsymbol{\varphi}}$$\scriptstyle{\Sigma^{k}}$$\scriptstyle{\boldsymbol{\psi}}$$\scriptstyle{\boldsymbol{\psi}}$$\scriptstyle{\Sigma}$

commutes. Therefore, $\boldsymbol{\varphi}$ is a substitutive realization of
$(\Delta,T^{k},A,\nu)$ such that for $\nu$-almost all $\mathbf{x}\in\Delta$
the $S$-adic dynamical system $(X_{\boldsymbol{\varphi}(\mathbf{x})},\Sigma)$
is a natural coding of the (minimal) translation by $\pi(\mathbf{x})$ on
$\mathbb{T}^{d-1}$ with respect to a partition of a bounded fundamental
domain. By the construction of $\boldsymbol{\psi}$ in the proof of Theorem
3.7, we can choose $\boldsymbol{\varphi}$ to be a faithful substitutive
realization. Since $\mathbf{x}$ is a generalized right eigenvector of
$\boldsymbol{\varphi}(\mathbf{x})$, the map $\boldsymbol{\varphi}$ is
injective, thus
$(\Delta,T^{k},\nu)\overset{\boldsymbol{\varphi}}{\cong}(\boldsymbol{\varphi}(\Delta),\Sigma,\nu\circ\boldsymbol{\varphi}^{-1})$.
∎

###  5.4. Results on natural codings and bounded remainder sets

We give now a relation between a natural coding with $d$ atoms and Rauzy
fractals. To this end, we need the following result on strong convergence.
This was proved in [BST19a, Proposition 4.3] with the additional assumption
that $\boldsymbol{\sigma}$ is recurrent. We give a slightly simpler proof that
does not require recurrence.

######  Lemma 5.10.

Let $\boldsymbol{\sigma}\in\mathcal{S}_{d}^{\mathbb{N}}$. If
$L_{\boldsymbol{\sigma}}$ is balanced and the generalized right eigenvector
$\mathbf{u}$ of $\boldsymbol{\sigma}$ has rationally independent coordinates,
then
$\lim_{n\to\infty}\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{e}_{i}=\mathbf{0}$
for all $i\in\mathcal{A}$ and

(5.4)
$\lim_{n\to\infty}\sup\big{\\{}\|\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\,\mathbf{l}(w)\|\,:\,w\in
L_{\Sigma^{n}\boldsymbol{\sigma}}\big{\\}}=0.$

###### Proof.

The balancedness of $L_{\boldsymbol{\sigma}}$ implies that
$\boldsymbol{\sigma}$ has a generalized right eigenvector $\mathbf{u}$, and
the irrationality of $\mathbf{u}$ implies that $\boldsymbol{\sigma}$ is
primitive.

Let $(i_{n})_{n\in\mathbb{N}}\in\mathcal{A}^{\mathbb{N}}$ be such that
$i_{n}\preceq\sigma_{n}(i_{n+1})$ for all $n\in\mathbb{N}$, and let
$\omega^{(n)}$ be such that $\sigma_{[n,\ell)}(i_{\ell})\prec\omega^{(n)}$ for
all $\ell>n$, i.e., $\omega^{(n)}$ is a limit word of
$\Sigma^{n}\boldsymbol{\sigma}$. Let

$P=\\{w\in\mathcal{A}^{*}\,:\,w\prec\omega^{(0)}\\}\quad\mbox{and}\quad
P_{n}^{(j)}=\\{w\in\mathcal{A}^{*}\,:\,w\prec\sigma_{[0,n)}(j)\\}\quad(j\in\mathcal{A},n\in\mathbb{N}).$

Since $\boldsymbol{\sigma}$ is balanced, the set
$\pi^{\prime}_{\mathbf{u}}\mathbf{l}(P)$ is bounded. From
$P_{0}^{(i_{0})}\subseteq
P_{1}^{(i_{1})}\subseteq\cdots\subseteq\bigcup_{n\in\mathbb{N}}P_{n}^{(i_{n})}=P$,
we obtain that there is a sequence of positive numbers
$(\varepsilon_{n})_{n\in\mathbb{N}}$ with $\lim_{n\to\infty}\varepsilon_{n}=0$
such that

$\|\mathbf{x}\|\leq\varepsilon_{n}\quad\mbox{for all}\
\mathbf{x}\in\mathbf{1}^{\perp}\ \mbox{satisfying}\
\mathbf{x}+\pi^{\prime}_{\mathbf{u}}\mathbf{l}\big{(}P_{n}^{(i_{n})}\big{)}\subseteq\pi^{\prime}_{\mathbf{u}}\mathbf{l}(P).$

We can now show that $\pi^{\prime}_{\mathbf{u}}M_{[0,n)}Q_{n}$ is small, where

$Q_{n}=\\{w\in\mathcal{A}^{*}\,:\,\mbox{$pj\prec\omega^{(n)}$ and
$pwj\prec\omega^{(n)}$ for some $p\in\mathcal{A}^{*}$, $j\in\mathcal{A}$}\\}$

is the set of return words in $\omega^{(n)}$ to some letter. More precisely,
we have $\|\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{l}(w)\|\leq
2\varepsilon_{k}$ for all $w\in Q_{n}$, provided that $M_{[k,n)}$ is a
positive matrix. Indeed, if $M_{[k,n)}$ is a positive matrix and
$j\in\mathcal{A}$, then $v\,i_{k}\preceq\sigma_{[k,n)}(j)$ for some
$v\in\mathcal{A}^{*}$, hence we have some $p,v\in\mathcal{A}^{*}$ such that

$\pi^{\prime}_{\mathbf{u}}\mathbf{l}\big{(}\sigma_{[0,n)}(p)\sigma_{[0,k)}(v)\big{)}+\pi^{\prime}_{\mathbf{u}}\mathbf{l}\big{(}P_{k}^{(i_{k})}\big{)}\subseteq\pi^{\prime}_{\mathbf{u}}\mathbf{l}(P)\
\mbox{and}\
\pi^{\prime}_{\mathbf{u}}\mathbf{l}\big{(}\sigma_{[0,n)}(pw)\sigma_{[0,k)}(v))+\pi^{\prime}_{\mathbf{u}}\mathbf{l}\big{(}P_{k}^{(i_{k})}\big{)}\subseteq\pi^{\prime}_{\mathbf{u}}\mathbf{l}(P),$

which implies that

$\|\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{l}(w)\|\leq\big{\|}\pi^{\prime}_{\mathbf{u}}\mathbf{l}\big{(}\sigma_{[0,n)}(p)\sigma_{[0,k)}(v)\big{)}\big{\|}+\big{\|}\pi^{\prime}_{\mathbf{u}}\mathbf{l}\big{(}\sigma_{[0,n)}(pw)\sigma_{[0,k)}(v))\big{\|}\leq
2\varepsilon_{k}.$

Since $\boldsymbol{\sigma}$ is primitive, $M_{[k,n)}$ is positive for all
$k\in\mathbb{N}$ and sufficiently large $n$ (depending on $k$).

Next we show that, for each $n\in\mathbb{N}$, the Minkowski sum
$\mathbf{l}(Q_{n})-\sum_{j=1}^{d}\mathbf{l}(Q_{n})$ contains a basis of
$\mathbb{R}^{d}$ with vectors in $\\{0,1\\}^{d}$. First note that
$\mathbf{l}(Q_{n})$ contains a basis of $\mathbb{R}^{d}$ by the rational
indepence of $\mathbf{u}$ and the balancedness of $L_{\boldsymbol{\sigma}}$.
Indeed, we cannot have $\mathbf{l}(Q_{n})\subset\mathbf{v}^{\perp}$ for some
$\mathbf{v}\in\mathbb{Z}^{d}$ because $Q_{n}$ contains arbitarily long
prefixes of $\omega^{(n)}$, hence $M_{[0,n)}\mathbf{l}(Q_{n})$ contains
arbitrarily large vectors with bounded distance from $\mathbb{R}\mathbf{u}$
(by the balancedness of $L_{\boldsymbol{\sigma}}$), which implies that
$\mathbf{u}\in M_{[0,n)}\mathbf{v}^{\perp}$, contradicting that $\mathbf{u}$
is rationally independent. Now choose words $w_{i}\in Q_{n}$ such that
$\\{\mathbf{l}(w_{i}):1\leq i\leq d\\}$ forms a basis of $\mathbb{R}^{d}$. If
$\mathbf{l}(w_{i})\in\\{0,1\\}^{d}$ for all $i$, then we have found a basis of
the wanted form because $\mathbf{0}\in\mathbf{l}(Q_{n})$. Otherwise note that
each non-empty factor $w$ of $\omega^{(n)}$ can be written as

(5.5) $w=v_{1}a_{1}v_{2}a_{2}\cdots v_{\ell}a_{\ell}\quad\mbox{with}\
1\leq\ell\leq d,\,v_{j}\in Q_{n},\,a_{j}\in\mathcal{A}\ \mbox{for all}\ 1\leq
j\leq\ell,\,a_{j}\neq a_{k}\ \mbox{if}\ j\neq k.$

Indeed, let $a_{1}$ be the first letter of $w$ and $v_{1}$ the longest
(possibly empty) word such that $v_{1}a_{1}\preceq w$; then $v_{1}\in Q_{n}$
and $(v_{1}a_{1})^{-1}w$ has no occurrence of $a_{1}$; if $w\neq v_{1}a_{1}$,
then let $a_{2}\in\mathcal{A}$ be the first letter of $(v_{1}a_{1})^{-1}w$ and
$v_{2}$ the longest word such that $v_{2}a_{2}\preceq(v_{1}a_{1})^{-1}w$;
repeat this procedure until $(v_{1}a_{1}\dots v_{\ell}a_{\ell})^{-1}w$ (which
has no occurrences of $a_{1},\dots,a_{\ell}$) is the empty word. Now, if
$w_{i}\notin\\{0,1\\}^{d}$ and $w_{i}=v_{1}a_{1}v_{2}a_{2}\cdots
v_{\ell}a_{\ell}$, then we can replace $w_{i}$ by the shorter word $v_{j}$ for
some $j$ or, when all $\mathbf{l}(v_{j})$ are in the span of the other basis
vectors, we replace $\mathbf{l}(w_{i})$ by
$\mathbf{l}(w_{i})-\sum_{j=1}^{\ell}\mathbf{l}(v_{j})$ without losing the
basis property. Since
$\mathbf{l}(w_{i})-\sum_{j=1}^{\ell}\mathbf{l}(v_{j})=\mathbf{l}(a_{1}\cdots
a_{\ell})\in\\{0,1\\}^{d}$ and the replacement by a shorter word can happen
only finitely many times, this proves the claim.

The previous paragraphs provide, for each $n\in\mathbb{N}$, a basis of
$\mathbb{R}^{d}$ with vectors $\mathbf{x}\in\\{0,1\\}^{d}$ satisfying
$\|\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{x}\|\leq 2(d{+}1)\varepsilon_{k}$
for all $k<n$ such that $M_{[k,n)}$ is positive. In particular, we have the
same basis for infinitely many $n$, and obtain that
$\lim_{n\to\infty}\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{e}_{i}=\mathbf{0}$
for all $i\in\mathcal{A}$.

Finally, let $w\in L_{\Sigma^{n}\boldsymbol{\sigma}}$. By primitivity, $w$ is
a factor of $\omega^{(n)}$. Writing $w$ as in (5.5), we obtain that
$\|\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{l}(w)\|\leq
2d\varepsilon_{k}+\sum_{i=1}^{d}\|\pi^{\prime}_{\mathbf{u}}M_{[0,n)}\mathbf{e}_{i}\|$
for all $k<n$ such that $M_{[k,n)}$ is positive. This proves the lemma. ∎

###### Proof of Theorem 3.8.

Let $\mathbf{t}_{i}$ be such that
$\tilde{R}_{\mathbf{t}}(\mathbf{x})=\mathbf{x}+\mathbf{t}_{i}$ on
$\mathcal{F}_{i}$, and let $\mathbf{u}=(u_{1},\dots,u_{d})$ with
$u_{i}=\lambda(\mathcal{F}_{i})$. Since $\mathcal{F}$ is bounded and
$(\mathcal{F},\tilde{R}_{\mathbf{t}},\lambda|_{\mathcal{F}})$ is ergodic, we
have by Birkhoff’s theorem, for almost all $\mathbf{x}\in\mathcal{F}$,

$\sum_{i=1}^{d}u_{i}\mathbf{t}_{i}=\sum_{i=1}^{d}\mathbf{t}_{i}\int_{\mathcal{F}_{i}}\mathrm{d}\lambda=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(\tilde{R}_{\mathbf{t}}^{k+1}(\mathbf{x})-\tilde{R}_{\mathbf{t}}^{k}(\mathbf{x}))=\lim_{n\to\infty}\frac{1}{n}(\tilde{R}_{\mathbf{t}}^{n}(\mathbf{x})-\mathbf{x})=\mathbf{0}.$

Define the matrix $N\in\mathbb{R}^{(d-1)\times d}$ by
$N\mathbf{e}_{i}=\mathbf{t}_{i}$, i.e., $N$ is such that its columns are given
by the vectors $\mathbf{t}_{i}$. Then we have $N\mathbf{u}=\mathbf{0}$ and, by
the minimality of $\tilde{R}_{\mathbf{t}}$, the vectors $\mathbf{t}_{i}$ span
$\mathbb{R}^{d-1}$, thus the kernel of $N$ is $\mathbb{R}\mathbf{u}$. Hence
there exists some $c>0$ such that $\|\mathbf{x}\|_{\infty}\leq
c\,\|N\mathbf{x}\|_{\infty}$ for all $\mathbf{x}\in\mathbf{1}^{\perp}$. If $w$
is in the language $L_{X}$ of $X$, then
$N\mathbf{l}(w)=\sum_{i=1}^{d}|w|_{i}\mathbf{t}_{i}=\tilde{R}_{\mathbf{t}}^{|w|}(\mathbf{x})-\mathbf{x}$
for some $\mathbf{x}\in\mathcal{F}$, thus
$\|N\mathbf{l}(w)\|_{\infty}\leq\mathrm{diam}(\mathcal{F})$, where
$\mathrm{diam}(\mathcal{F})$ stands for the diameter of $\mathcal{F}$. For
$v,w\in L_{X}$ with $|v|=|w|$, we have
$\mathbf{l}(v)-\mathbf{l}(w)\in\mathbf{1}^{\perp}$, hence
$\|\mathbf{l}(v)-\mathbf{l}(w)\|_{\infty}\leq
c\,\|N\mathbf{l}(v)-N\mathbf{l}(w)\|_{\infty}\leq
2\,c\,\mathrm{diam}(\mathcal{F})$. This means that $L_{X}$ is
$(2\,c\,\mathrm{diam}(\mathcal{F}))$-balanced.

ssume now that $X=X_{\boldsymbol{\sigma}}$ for some sequence of substitutions
$\boldsymbol{\sigma}\in\mathcal{S}_{d}^{\mathbb{N}}$. The minimality of
$R_{\mathbf{t}}$ implies that $\mathbf{u}$ is rationally independent. Indeed,
suppose that $\langle\mathbf{z},\mathbf{u}\rangle=0$ for some
$\mathbf{z}\in\mathbb{Z}^{d}\setminus\\{\mathbf{0}\\}$. Consider the matrix
$\tilde{N}\in\mathbb{Z}^{d\times d}$ that is obtained from $N$ by subtracting
$\mathbf{t}$ from each column and adding the row $(1,\dots,1)$ at the bottom.
Since $N\mathbf{u}=0$, we have $\tilde{N}\mathbf{u}=\binom{-\mathbf{t}}{1}$.
If $\det\tilde{N}\neq 0$, then we have
$\vphantom{\\!}{}^{t}\\!\tilde{N}\mathbf{y}=\mathbf{z}$ for some
$\mathbf{y}\in\mathbb{Q}^{d}\setminus\\{\mathbf{0}\\}$; if $\det\tilde{N}=0$,
then we have $\vphantom{\\!}{}^{t}\\!\tilde{N}\mathbf{y}=\mathbf{0}$ for some
$\mathbf{y}\in\mathbb{Z}^{d}\setminus\\{\mathbf{0}\\}$. In both cases, we have
$0=\langle\vphantom{\\!}^{t}\\!\tilde{N}\mathbf{y},\mathbf{u}\rangle=\langle\mathbf{y},\binom{-\mathbf{t}}{1}\rangle$,
contradicting that $\mathbf{t}$ is totally irrational.

Rational independence of $\mathbf{u}$ and balancedness of
$L_{\boldsymbol{\sigma}}$ entail that $\boldsymbol{\sigma}$ is primitive and
$\mathbf{u}$ is a generalized right eigenvector of $\boldsymbol{\sigma}$; see
[BD14, Theorem 5.7]. Let $\omega^{(0)}\in X_{\boldsymbol{\sigma}}$ be as in
the proof of Lemma 5.10, and write $\omega^{(0)}=\omega_{0}\omega_{1}\cdots$
with $\omega_{n}\in\mathcal{A}$. Then there is some $\mathbf{z}\in\mathcal{F}$
such that $R_{\mathbf{t}}^{n}(\mathbf{z})\in\mathcal{F}_{\omega_{n}}$ for all
$n\in\mathbb{N}$. Define the affine map
$A:\,\mathbb{R}^{d}\to\mathbb{R}^{d-1}$ by
$A(\mathbf{x})=\mathbf{z}+N\mathbf{x}$. Then we have
$A(\mathbf{x})=A(\pi^{\prime}_{\mathbf{u}}\mathbf{x})$. By minimality, we have

(5.6)
$\mathcal{F}_{i}=\overline{\big{\\{}\mathbf{z}+N\mathbf{l}(p)\,:\,p\in\mathcal{A}^{*},\,p\,i\prec\omega^{(0)}\big{\\}}}\subseteq
A(\mathcal{R}_{\boldsymbol{\sigma}}(i))\quad\mbox{for all}\ i\in\mathcal{A}.$

On the other hand, if $p\,i\preceq\sigma_{[0,n)}(j)$ for infinitely many
$n\in\mathbb{N}$, $j\in\mathcal{A}$, then there are words $w_{n}\in
L_{\Sigma^{n}\boldsymbol{\sigma}}$ such that
$\sigma_{[0,n)}(w_{n})\,p\,i\prec\omega^{(0)}$ for all $n\in\mathbb{N}$, hence
$A\big{(}M_{[0,n)}\mathbf{l}(w_{n})+\mathbf{l}(p)\big{)}\in\mathcal{F}_{i}$,
which implies that $A(\mathbf{l}(p))\in\mathcal{F}_{i}$ by Lemma 5.10.
Therefore, we have
$A(\mathcal{R}_{\boldsymbol{\sigma}}(i))\subseteq\mathcal{F}_{i}$, thus
$A(\mathcal{R}_{\boldsymbol{\sigma}}(i))=\mathcal{F}_{i}$.

Assume that a directive sequence is left proper (infinitely often). We can
transform the directive sequence to make it proper. From [BCBD+19, Corollary
5.5], if $(X,S)$ is a primitive unimodular proper $S$-adic subshift, then if
it is balanced on letters, then it is also balanced on all its factors. Hence
cylinders associated to factors are also bounded remainder sets. ∎

The following proposition makes the assertions of Theorem 3.6 more concrete
and extends the results in case of proper substitutions.

######  Proposition 5.11.

Let $D\subset\mathcal{S}_{d}^{\mathbb{N}}$ be a shift-invariant set of
directive sequences equipped with an ergodic $\Sigma$-invariant probability
measure $\nu$ satisfying $\nu\circ\Sigma\ll\nu$. Assume that the linear
cocycle $(D,\Sigma,Z,\nu)$ defined by
$Z((\sigma_{n})_{n\in\mathbb{N}})=\vphantom{\\!}^{t}\\!M_{\sigma_{0}}$
satisfies the Pisot condition, and that there is a periodic Pisot sequence
$(\tau_{0},\dots,\tau_{k-1})^{\infty}$ with purely discrete spectrum and the
local positive range property in $(D,\Sigma,\nu)$. Then for $\nu$-almost all
$\boldsymbol{\sigma}\in D$ the following assertions hold:

  1. (i)

the $S$-adic dynamical system $(X_{\boldsymbol{\sigma}},\Sigma)$ is a natural
coding of the (minimal) translation by $\pi(\mathbf{u})$ on
$\mathbb{T}^{d-1}$, where $\mathbf{u}$ is the generalized right eigenvector of
$\boldsymbol{\sigma}$ with $\|\mathbf{u}\|=1$;

  2. (ii)

the domains of the natural coding are the embeddings of the (bounded) Rauzy
fractals $\pi(-\mathcal{R}_{\boldsymbol{\sigma}}(i))$, $i\in\mathcal{A}$; in
particular, the sets $\pi(-\mathcal{R}_{\boldsymbol{\sigma}}(i))$ are bounded
remainder sets and $\pi(-\mathcal{R})$ is a fundamental domain of
$R_{\pi(\mathbf{u})}$;

Furthermore, if $\tau_{[0,k)}$ is left or right proper, then for $\nu$-almost
all $\boldsymbol{\sigma}\in D$ also the following holds:

  1. (iii)

for each $w\in L_{\boldsymbol{\sigma}}$,
$\pi(-\mathcal{R}_{\boldsymbol{\sigma}}(w))$ is a bounded remainder set of the
translation by $\pi(\mathbf{u})$ on $\mathbb{T}^{d-1}$.

###### Proof.

Assertion (i) follows immediately from the proof of Theorem 3.6 by the
definition of natural coding (see Definition 2.3). According to [Ada03,
Proposition 7], the $C$-balancedness of $L_{\boldsymbol{\sigma}}$ implies that
$\pi(-\mathcal{R}_{\boldsymbol{\sigma}}(i))$ is a bounded remainder set for
each $i\in\mathcal{A}$, which proves assertion (ii). Assertion (iii) follows
from [BCBD+19, Corollary 5.5]. ∎

##  6\. Examples

In this section we show that our theory can easily be applied to many well-
known multidimensional continued fraction algorithms. According results for
the case of the Brun and the Arnoux–Rauzy algorithm for $d=3$ are treated in
[BST19a], and for the Cassaigne–Selmer algorithm ($d=3$) in [FN20]. However,
using our new theory the conditions we need to check are easier to verify than
the ones in [BST19a, FN20]. This even allows us to treat the Arnoux–Rauzy
algorithm in arbitrary dimensions $d\geq 3$ as well as the multiplicative
Jacobi–Perron algorithm ($d=3$) and the Brun algorithm for $d=4$.

###  6.1. The Cassaigne–Selmer algorithm

Cassaigne announced in 2015 a $2$-dimensional continued fraction algorithm
that was first studied in [CLL17]. This Markovian algorithm
$(\Delta,T_{\scriptscriptstyle\mathrm{C}},A_{\scriptscriptstyle\mathrm{C}})$,
with $\Delta=\Delta_{3}$ is called _Cassaigne–Selmer_ algorithm because it is
measurably conjugate to the $2$-dimensional Selmer algorithm (see [CLL17];
Selmer’s algorithm goes back to [Sel61]). Cassaigne’s representation of this
algorithm is remarkable because it admits a set of substitutions that is
particularly relevant from a symbolic point of view. As shown in [CLL17], the
$S$-adic symbolic dynamical systems defined in terms of these substitutions
have factor complexity $2n+1$ and, as underlined in [BCBD+19], belong to the
family of so-called dendric subshifts.

Recall that
$\Delta=\Delta_{3}=\\{(x_{1},x_{2},x_{3})\in[0,1]^{3}\,:\,x_{1}+x_{2}+x_{3}=1\\}$.
Using the matrices

$C_{1}=\begin{pmatrix}1&1&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}\qquad\hbox{and}\qquad C_{2}=\begin{pmatrix}0&1&0\\\
1&0&0\\\ 0&1&1\end{pmatrix},$

we define the matrix valued function

$A_{\scriptscriptstyle\mathrm{C}}:\,\Delta\to\mathrm{GL}(3,\mathbb{Z}),\quad\mathbf{x}\mapsto\begin{cases}\vphantom{C}{}^{t}C_{1}&\mbox{if}\
\mathbf{x}\in\Delta(1)=\\{(x_{1},x_{2},x_{3})\in\Delta:\,x_{1}\geq
x_{3}\\},\\\ \vphantom{C}{}^{t}C_{2}&\mbox{if}\
\mathbf{x}\in\Delta(2)=\\{(x_{1},x_{2},x_{3})\in\Delta:\,x_{1}<x_{3}\\}\end{cases}$

(here, we slightly abuse notation by writing $\Delta(j)$ for
$\Delta(\mathbf{x})$ with
$A_{\scriptscriptstyle\mathrm{C}}(\mathbf{x})=\vphantom{C}^{t}C_{j}$). Then
$T_{\scriptscriptstyle\mathrm{C}}$ is given by

$T_{\scriptscriptstyle\mathrm{C}}:\,\Delta\to\Delta,\quad\mathbf{x}\mapsto\begin{cases}(\frac{x_{1}-x_{3}}{x_{1}+x_{2}},\frac{x_{3}}{x_{1}+x_{2}},\frac{x_{2}}{x_{1}+x_{2}})&\mbox{if}\
x_{1}\geq x_{3},\\\
(\frac{x_{2}}{x_{2}+x_{3}},\frac{x_{1}}{x_{2}+x_{3}},\frac{x_{3}-x_{1}}{x_{2}+x_{3}},)&\mbox{if}\
x_{1}<x_{3}.\end{cases}$

In [AL18, Proposition 22] it is proved that the density of the invariant
probability measure $\nu_{\scriptscriptstyle\mathrm{C}}$ of
$T_{\scriptscriptstyle\mathrm{C}}$ is given by
$\frac{12}{\pi^{2}(1-x_{1})(1-x_{3})}$. Following [CLL17] we define the
_Cassaigne–Selmer substitutions_

(6.1) $\gamma_{1}:\begin{cases}1\mapsto 1\\\ 2\mapsto 13\\\ 3\mapsto
2\end{cases}\qquad\qquad\gamma_{2}:\begin{cases}1\mapsto 2\\\ 2\mapsto 13\\\
3\mapsto 3\end{cases}$

The corresponding faithful substitution selection is defined by
$\varphi(\mathbf{x})=\gamma_{j}$ if $\mathbf{x}\in\Delta(j)$. By Definition
2.2 the map

(6.2)
$\boldsymbol{\varphi}:\,\Delta\to\\{\gamma_{1},\gamma_{2}\\}^{\mathbb{N}}\quad\hbox{with}\quad\boldsymbol{\varphi}(\mathbf{x})=(\varphi(T^{n}\mathbf{x}))_{n\in\mathbb{N}}$

is a faithful substitutive realization of
$(\Delta,T_{\scriptscriptstyle\mathrm{C}},A_{\scriptscriptstyle\mathrm{C}})$.
We have
$T_{\scriptscriptstyle\mathrm{C}}(\Delta(1))=T_{\scriptscriptstyle\mathrm{C}}(\Delta(2))=\Delta$,
thus the algorithm satisfies the finite range property and, because
$\nu(\Delta)=1$, each $\mathbf{x}\in\Delta$ has the local positive range
property (see Definition 2.5). Moreover,
$\boldsymbol{\varphi}(\Delta)=\\{\gamma_{1},\gamma_{2}\\}^{\mathbb{N}}$ (up to
a set of measure zero). According to [Lag93, Section 6] and [Sch00, Chapter
7], $\\{\Delta(1),\Delta(2)\\}$ is a generating (Markov) partition for
$T_{\scriptscriptstyle\mathrm{C}}$, hence,
$(\Delta,T_{\scriptscriptstyle\mathrm{C}},\nu_{\scriptscriptstyle\mathrm{C}})\overset{\boldsymbol{\varphi}}{\cong}(\\{\gamma_{1},\gamma_{2}\\}^{\mathbb{N}},\Sigma,\nu_{\scriptscriptstyle\mathrm{C}}\circ\boldsymbol{\varphi}^{-1}).$
The linear cocycle $A_{\scriptscriptstyle\mathrm{C}}$ is $\log$-integrable
since $A_{\scriptscriptstyle\mathrm{C}}$ takes only 2 values. By [Sch04,
BST19b], we know that
$(\Delta,T_{\scriptscriptstyle\mathrm{C}},A_{\scriptscriptstyle\mathrm{C}},\nu_{\scriptscriptstyle\mathrm{C}})$
satisfies the Pisot condition. Moreover, since
$\nu_{\scriptscriptstyle\mathrm{C}}$ is a Borel probability measure equivalent
to the Lebesgue measure and $T_{\scriptscriptstyle\mathrm{C}}$ maps open sets
to open sets we have $\nu_{\scriptscriptstyle\mathrm{C}}\circ
T\ll\nu_{\scriptscriptstyle\mathrm{C}}$.

To apply Theorem 3.1 we have to find a periodic Pisot point
$\mathbf{x}\in\Delta$ (see Definition 2.4) such that
$\boldsymbol{\varphi}(\mathbf{x})$ has purely discrete spectrum. To this end
consider

(6.3) $\tau=\gamma_{1}\circ\gamma_{2}:\,\begin{cases}1\mapsto 13\\\ 2\mapsto
12\\\ 3\mapsto 2\end{cases}$

and let $\mathbf{x}\in\Delta$ be the dominant right eigenvector of $M_{\tau}$.
Then we have
$\boldsymbol{\varphi}(\mathbf{x})=(\gamma_{1},\gamma_{2})^{\infty}$. Since
$M_{\tau}$ is a Pisot matrix we conclude that $\mathbf{x}$ is a periodic Pisot
point which certainly has the local positive range property by the above
considerations. It only remains to prove the following lemma to be able to
apply Theorem 3.1.

######  Lemma 6.1.

Let $\tau=\gamma_{1}\circ\gamma_{2}$. Then $\tau$ is a unit Pisot substitution
and the substitutive dynamical system $(X_{\tau},\Sigma)$ has purely discrete
spectrum.

###### Proof.

The characteristic polynomial $X^{3}-2X^{2}+X-1$ of
$M_{\tau}=\vphantom{C}^{t}C_{1}\vphantom{C}^{t}C_{2}$ is the minimal
polynomial of a Pisot unit. We have to check if $(X_{\tau},\Sigma)$ has purely
discrete spectrum.

Let $\sigma$ be a unit Pisot substitution over the alphabet
$\mathcal{A}=\\{1,2,3\\}$. To check if the substitutive dynamical system
$(X_{\sigma},\Sigma)$ is measurably conjugate to a minimal translation on
$\mathbb{T}^{2}$ one can prove by the _balanced pair algorithm_ (see e.g.
[SS02, Section 3] or [BST10, Section 5.8]). A  _balanced pair_ is a pair
$(v_{1},v_{2})\in\mathcal{A}^{*}\times\mathcal{A}^{*}$ with
$\mathbf{l}(v_{1})=\mathbf{l}(v_{2})$. It is called irreducible, if no proper
prefixes of $v_{1}$ and $v_{2}$ give rise to a balanced pair. Each balanced
pair can be decomposed in irreducible balanced pairs in an obvious way. We
recall the balanced pair algorithm for $\sigma$. It starts with
$I_{0}=\\{(12,21),(13,31),(23,32)\\}$ (see also [BK06, Section 17]). Given
$I_{k}$ for some $k\in\mathbb{N}$ the set $I_{k+1}$ is defined recursively by
the set of all irreducible balanced pairs occurring in a decomposition of a
balanced pair $(\sigma(v_{1}),\sigma(v_{2}))$ with $(v_{1},v_{2})\in I_{k}$.
We say that the balanced pair algorithm _terminates_ if for some
$k\in\mathbb{N}$ the set $I_{k}\setminus(I_{0}\cup\dots\cup
I_{k-1})=\emptyset$ and if each $(v_{1},v_{2})\in\bigcup_{j=0}^{k}I_{j}$
eventually contains a _coincidence_ , i.e., a pair of the form
$(i,i)\in\mathcal{A}\times\mathcal{A}$ occurs in
$(\sigma^{j}(v_{1}),\sigma^{j}(v_{2}))$ for some $j\in\mathbb{N}$. According
to [SS02, Section 3] the balanced pair algorithm terminates if and only if
$(X_{\sigma},\Sigma)$ has purely discrete spectrum.

In our case we get $(12,21)\xrightarrow{\tau}(1312,1213)$ which splits into
the irreducible pairs $(1,1)$, a coincidence, and $(312,213)$. Moreover,
$(13,31)\xrightarrow{\tau}(132,213)$ does not split and
$(23,32)\xrightarrow{\tau}(122,212)$ splits into $(12,21)$, and the
coincidence $(2,2)$. Thus

$I_{1}=\\{(1,1),(2,2),(12,21),(312,213),(132,213)\\}.$

We have to go on with the new pairs $(1,1),(2,2),(312,213),(132,213)$
occurring in $I_{1}$. While coincidences yield only coincidences again, we get
the pairs $(312,213)\xrightarrow{\tau}(21312,12132)$ and
$(132,213)\xrightarrow{\tau}(13212,12132)$. Splitting these yields the new
pair $(321,213)$. Summing up the set $I_{2}$ contains the new pairs $(3,3)$
and $(321,213)$. We only have to check the one which is not a coincidence
getting $(321,213)\xrightarrow{\tau}(21213,12132)$. This gives (up to
switching the order of the pair) no new pairs in $I_{3}$. Since all occurring
pairs eventually end up in coincidences, the balanced pair algorithm
terminates for $\tau$ and, hence, $(X_{\tau},\Sigma)$ has purely discrete
spectrum. ∎

Moreover, the periodic directive sequence $(\gamma_{1},\gamma_{2})^{\infty}$
is proper, thus we can also apply Proposition 5.11 (iii). Combining Theorem
3.1 with Proposition 5.11 we thus obtain the following result.

######  Theorem 6.2.

Let
$(\Delta,T_{\scriptscriptstyle\mathrm{C}},A_{\scriptscriptstyle\mathrm{C}},\nu_{\scriptscriptstyle\mathrm{C}})$
be the Cassaigne–Selmer algorithm, and let $\boldsymbol{\varphi}$ be the
substitutive realization defined in (6.2). Then
$(\Delta,T_{\scriptscriptstyle\mathrm{C}},\nu_{\scriptscriptstyle\mathrm{C}})\overset{\boldsymbol{\varphi}}{\cong}(\\{\gamma_{1},\gamma_{2}\\}^{\mathbb{N}},\Sigma,\nu_{\scriptscriptstyle\mathrm{C}}\circ\boldsymbol{\varphi}^{-1})$
and for $\nu_{\scriptscriptstyle\mathrm{C}}$-almost all $\mathbf{x}\in\Delta$
the following assertions hold.

  * (i)

The $S$-adic dynamical system
$(X_{\boldsymbol{\varphi}(\mathbf{x})},\Sigma)\cong(\mathbb{T}^{2},R_{\pi(\mathbf{x})})$
has purely discrete spectrum.

  * (ii)

The shift $X_{\boldsymbol{\varphi}(\mathbf{x})}$ is a natural coding of
$R_{\pi(\mathbf{x})}$ w.r.t. the partition
$\\{-\pi\,\mathcal{R}_{\boldsymbol{\sigma}}(i)\,:\,i\in\mathcal{A}\\}$.

  * (iii)

The set $-\pi\,\mathcal{R}_{\boldsymbol{\sigma}}(w)$ is a bounded remainder
set for $R_{\pi(\mathbf{x})}$ for each $w\in\mathcal{A}^{*}$.

######  Remark 6.3.

Let $\gamma_{1},\gamma_{2}$ be the Cassaigne–Selmer substitutions defined in
(6.1) and consider the shift
$(\\{\gamma_{1},\gamma_{2}\\}^{\mathbb{N}},\Sigma,\nu)$, where $\nu$ is an
ergodic invariant measure on $\\{\gamma_{1},\gamma_{2}\\}^{\mathbb{N}}$
satisfying $\nu\circ T\ll\nu$. Then the same conclusions as in Theorem 6.2
hold.

According to [CLL17], the sequences in $X_{\boldsymbol{\varphi}(\mathbf{x})}$
have factor complexity $2n+1$. Thus Theorem 6.2 has the following consequence
(observe also Remark 3.5).

######  Corollary 6.4.

For almost all $\mathbf{t}\in\mathbb{T}^{2}$, there exists a minimal subshift
$X\subset\\{1,2,3\\}^{\mathbb{N}}$ with factor complexity $2n+1$ and language
balanced on factors such that $(X,\Sigma)$ is a natural coding of
$R_{\mathbf{t}}$.

Note that balancedness on factors means that all $\mathcal{F}_{i_{0}}\cap
R_{\mathbf{t}}^{-1}\mathcal{F}_{i_{1}}\cap\cdots\cap
R_{\mathbf{t}}^{-n}\mathcal{F}_{i_{n}}$ are bounded remainder sets of
$R_{\mathbf{t}}$, with the notation of Theorem 3.8. This extends properties of
Sturmian words to words on $3$-letter alphabets. We mention that the dimension
group of $X$ can be completely described, in particular its group part is
$\mathbb{Z}^{3}$; see [BCBD+19] for more on this topic.

The Selmer algorithm exists in higher dimensions, but it does not lead to
sequences of complexity $(d{-}1)n+1$, and the second Lyapunov exponent seems
to be negative only for $d\leq 4$ [BST19b].

###  6.2. The Arnoux–Rauzy algorithm

In this section we apply our results to the Arnoux–Rauzy algorithm in
arbitrary dimension $d\geq 3$. As for the Cassaigne–Selmer algorithm (with
$d=3$), the Arnoux–Rauzy algorithm generates symbolic dynamical systems that
have factor complexity $(d{-}1)n+1$ and belong to the family of dendric
subshifts.

Now we have $\Delta=\Delta_{d}$, and the set of _Arnoux–Rauzy substitutions_
over $\mathcal{A}$ is defined by

(6.4) $\alpha_{i}:\ i\mapsto i,\ j\mapsto ij\ \mbox{for}\
j\in\mathcal{A}\setminus\\{i\\}\qquad(i\in\mathcal{A}).$

Let

$\Delta(i)=\bigg{\\{}(x_{1},\dots,x_{d})\,:\,x_{i}\geq\sum_{j\neq
i}x_{j}\bigg{\\}}.$

Using the transposed incidence matrices of $\alpha_{i}$, we define the matrix
valued function

$A_{\scriptscriptstyle\mathrm{AR}}:\,\Delta\to\mathrm{GL}(3,\mathbb{Z}),\quad\mathbf{x}\mapsto\vphantom{\\!M}^{t}\\!M_{\alpha_{i}}\quad\mbox{if}\
\mathbf{x}\in\Delta(i),$

which gives that

$T_{\scriptscriptstyle\mathrm{AR}}(x_{1},\dots,x_{d})=\bigg{(}\frac{x_{1}}{x_{i}},\dots,\frac{x_{i-1}}{x_{i}},\frac{x_{i}-\sum_{j\neq
i}x_{j}}{x_{i}},\frac{x_{i+1}}{x_{i}},\dots,\frac{x_{d}}{x_{i}}\bigg{)}\quad\mbox{if}\
\mathbf{x}\in\Delta(i).$

The algorithm
$(\Delta,T_{\scriptscriptstyle\mathrm{AR}},A_{\scriptscriptstyle\mathrm{AR}})$
is called _Arnoux–Rauzy algorithm_. We have
$T_{\scriptscriptstyle\mathrm{AR}}(\Delta(i))=\Delta$ for all
$i\in\mathcal{A}$, but the Lebesgue measure of
$\bigcup_{i\in\mathcal{A}}\Delta(i)$ is smaller than that of $\Delta$.
Generalizing the _Rauzy gasket_ , we call the set

$\Delta_{\scriptscriptstyle\mathrm{AR}}=\bigg{\\{}\mathbf{x}\in\Delta:\,T_{\scriptscriptstyle\mathrm{AR}}^{n}(\mathbf{x})\in\bigcup_{i\in\mathcal{A}}\Delta(i)\
\mbox{for all}\ n\in\mathbb{N}\bigg{\\}}$

the $d$-dimensional _Rauzy simplex_. It has zero Lebesgue measure by [AS13,
Section 7]. We consider invariant measures $\nu$ of
$(\Delta,T_{\scriptscriptstyle\mathrm{AR}})$ with support
$\Delta_{\scriptscriptstyle\mathrm{AR}}$ satisfying $\nu\circ T\ll\nu$ (the
latter condition is satisfied for instance for Borel probability measures
$\nu$ w.r.t. the subspace topology on
$\Delta_{\scriptscriptstyle\mathrm{AR}}$). Clearly, the map $\varphi$ defined
by $\varphi(\mathbf{x})=\alpha_{j}$ when $\mathbf{x}\in\Delta(j)$ is a
faithful substitution selection. We have
$T_{\scriptscriptstyle\mathrm{AR}}(\Delta_{\scriptscriptstyle\mathrm{AR}}(i))=\Delta_{\scriptscriptstyle\mathrm{AR}}$,
thus the algorithm satisfies the finite range property and, because
$\nu(\Delta_{\scriptscriptstyle\mathrm{AR}})=1$, each $\mathbf{x}\in\Delta$
has finite range (see Definition 2.5). The associated substitutive realization
$\boldsymbol{\varphi}$ thus satisfies
$\boldsymbol{\varphi}(\Delta_{\scriptscriptstyle\mathrm{AR}})=\\{\alpha_{1},\dots,\alpha_{d}\\}^{\mathbb{N}}$
(up to a set of measure zero). By [AD15], we know that the Lyapunov exponents
of the Arnoux–Rauzy algorithm satisfy
$\theta_{1}(A_{\scriptscriptstyle\mathrm{AR}})>0>\theta_{2}(A_{\scriptscriptstyle\mathrm{AR}})$
for any ergodic invariant measure $\nu$ with support
$\Delta_{\scriptscriptstyle\mathrm{AR}}$.

By induction on $d$, we can show that

$\alpha_{1}\circ\alpha_{2}\circ\cdots\circ\alpha_{d}=\tilde{\alpha}^{d},\quad\mbox{with}\
\tilde{\alpha}(i)=1(i\\!+\\!1)\ \mbox{for}\ 1\leq i<d,\ \tilde{\alpha}(d)=1.$

The substitution $\tilde{\alpha}$ is the $d$-bonacci substitution; the
characteristic polynomial of the incidence matrix $M_{\tilde{\alpha}}$ of