\section{Introduction}

A numeration system for representing nonnegative integers as well as real
numbers in a certain interval was proposed in \cite{MR1020484}. 
It is based on right-infinite fixed points of substitutions.
Dumont-Thomas numeration system was later explained by
so-called prefix-suffix automata associated to a primitive substitution
\cite{MR1879663}.
It is related to the order of a Bratteli Vershik
diagram and its natural successor map \cite{zbMATH00219514,MR1322556,MR3791491}.
Their main motivation was
to prove that every dynamical system
generated by a substitution of Pisot type on $d$ letters admits as a
topological factor a minimal translation on the torus $\mathbb{T}^{d-1}$
\cite{MR1852097}. As a consequence, we obtain a numeration system representing the
elements of $\mathbb{T}^{d-1}$ by infinite paths in the prefix-suffix automaton,
see \cite{MR1970385}.

Note that the Dumont-Thomas numeration system for $\mathbb{N}$ is deduced by
considering the cylinders of finite length words \cite[Corollary
6.2]{MR1879663}.
It was also generalized to numeration systems for $\mathbb{N}$ \cite{MR1799066} 
and for real numbers in an interval \cite{zbMATH01748010} 
associated to arbitrary regular languages,
see also \cite[\S 7]{MR1905123}, \cite[\S 4]{MR2290784}, 
\cite{MR2282862}, \cite[\S 3]{MR2742574}, \cite{MR2799274}.
A recent article extending Dumont-Thomas also concerns $\beta$-numeration of
real numbers in an interval \cite{MR4143682}.

A very general framework was proposed recently to 
extend the Dumont-Thomas numeration system to all integers
based on the notion of coding prescription which allows the image of
letters to be scattered words of nonconsecutive letters \cite{MR3805464}.
The goal of this contribution is to propose a simple extension of Dumont-Thomas
numeration system to all integers in $\mathbb{Z}$.
It is based on two-sided periodic points of a substitution
as opposed to right-infinite fixed points.
Intuitively, it is inspired by the adic transformation of a Bratteli-Vershik
diagram in the sense that we want to represent the integer $-n$ by a finite
word which is the $n$-th predecessor of $0$ under the successor map. The proofs
provided here are simple and follow as much as possible the approach originally
proposed by Dumont-Thomas \cite{MR1020484}. 
The motivation for this work comes from the study of substitutive Wang tilings
of the plane in general (a preliminary work was done in \cite{MR4364231}) where
a theory of substitutive numeration systems for $\mathbb{Z}^2$ is needed.


\section{Preliminaries}

An \emph{alphabet} $A$ is a finite set and its elements $a\in A$
are called $\emph{letters}$. 
A \emph{finite word} $u = u_0 u_1 \cdots u_{n-1}$ is a concatenation of letters 
$u_i \in A$ for every $i\in \{0,1,\dots,n-1\}$ and $|u|$ denotes its \emph{length}.
The \emph{empty word} is denoted $\varepsilon$.
The set of all finite words over alphabet $A$ is denoted $A^*$ 
and the set of all nonempty words over alphabet $A$ is denoted 
$A^+ = A^*\setminus \{\varepsilon\}$.
The set $A^*$ with concatenation as the monoid operation
and $\varepsilon$ as the neutral element
forms a monoid.

A \emph{morphism} over $A$ is a map $\eta:A^* \to A^*$ 
such that $\eta(uv) = \eta(u)\eta(v)$  for all words $u, v \in A^*$.
A \emph{substitution} $\eta:A^* \to A^*$ is a morphism such that 
$\eta(a)\in A^+$ is nonempty
for every $a\in A$
and there exists $a\in A$ such that $\lim_{n\to +\infty} |\eta^n(a)| = +\infty$.
A morphism $\eta$ is said \emph{primitive} if 
there exists $k\in\mathbb{N}$ such that
for every $a,b\in A$ 
the letter $a$ appears in $\eta^k(b)$.

We call $u_0 u_1 u_2 \cdots \in A^{\mathbb{Z}_{\geq 0}}$ a \emph{right-infinite word}
and $ \cdots  u_{-3} u_{-2} u_{-1} \in A^{{\mathbb{Z}_{< 0}}}$ a \emph{left-infinite word}.
We call $u\in A^{\mathbb{Z}}$ a \emph{two-sided word} and
we separate by a dot its elements $u_{-1}$ and $u_0$ to indicate the origin, i.e.,
$u = \cdots u_{-3} u_{-2} u_{-1} . u_{0} u_1 u_2 \cdots$.

Substitutions can be applied naturally to two-sided words $u\in A^{\mathbb{Z}}$ by setting
\[
\eta(\dots u_{-3} u_{-2} u_{-1} . u_0 u_1 u_2 \cdots) 
= \cdots\eta(u_{-3}) \eta(u_{-2}) \eta(u_{-1}) .\eta(u_0) \eta(u_1) \eta(u_2) \cdots.
\]
Let $\mathbb{D}\in\{\mathbb{Z},{\mathbb{Z}_{\geq 0}},{\mathbb{Z}_{< 0}}\}$.
A word $u\in A^\mathbb{D}$ is called \emph{periodic point} of the substitution
$\eta$ if there exists an integer $p\geq 1$ such that $\eta^p(u)=u$.
Such an integer $p$ is called a \emph{period}.
A periodic point with a period $p=1$ is called a \emph{fixed point} of $\eta$.
The set of periodic points of $\eta$ is denoted
$\Per_\mathbb{D}(\eta)=\{u\in A^\mathbb{D} \mid \eta^p(u)=u\}$.
Since we are mostly interested in two-sided words in this contribution,
we omit the domain when $\mathbb{D}=\mathbb{Z}$ and we write
$\Per(\eta)=\Per_\mathbb{Z}(\eta)$.

A word $u\in \Per(\eta)$ of period $p$ is of the form 
$u = \lim_{n\to +\infty} \eta^{pn}(b).\eta^{pn}(a)$
where $a,b\in A$ are letters such that $\eta^p(a) =au$ and 
$\eta^p(b) =vb$ for some nonempty words $u,v\in A^+$.
In this case, we call $s = b.a$ 
the \emph{seed} of
the periodic point $u$.

Let $u=u_0 u_1 \cdots\in\Per_\mathbb{N}(\eta)$ with $u_0=a$.
The previous terminology is inspired by \cite{MR1879663}
where a prefix-suffix automaton is associated with $\eta$.
However, for our goal an automaton associated with $\eta$
as in \cite{MR2742574} is sufficient.
Let $\mathcal{D}=\left\{0,...,\max_{c\in A}|\eta(c)|-1\right\}$ be an alphabet. 
The deterministic finite automaton with output (DFAO)
associated to the substitution $\eta$ and letter $a$ is
the 5-tuple,
$\mathcal{A}_{\eta,a} = (A,\mathcal{D},\delta,a,A),$
where the transition function $\delta:A\times \mathcal{D} \rightarrow A$ 
is a partial function 
such that $\delta(b,i)=c$ 
if and only if 
$c=w_i$ and $\eta(b)=w_0\dots w_{|\eta(b)|-1}$. 
We denote by $\mathcal{L}(\mathcal{A}_{\eta,a})$ the words
accepted by the automaton $\mathcal{A}_{\eta,a}$
and for $q\in\mathbb{N}$ we denote $\mathcal{L}_q(\mathcal{A}_{\eta,a})$
the set of words $w\in\mathcal{L}(\mathcal{A}_{\eta,a})$ such that $|w|=q$.

\section{Dumont-Thomas numeration system for $\mathbb{N}$}

In this section, we recall Dumont-Thomas numeration system for $\mathbb{N}$ which was 
based on  substitutions having a right-infinite fixed point \cite{MR1020484}. 
It uses the definition of admissible sequences.

\begin{definition}[admissible sequence]
    {\rm\cite{MR1020484}}
Let $a\in A$ be a letter, $k$ an integer and, for each integer $i$, $0\leq i\leq k$,
$(m_i,a_i)$ be an element of $A^*\times A$. We say that the finite sequence 
$(m_i,a_i)_{i=0,\dots,k}$ is \emph{admissible} if and only if, for all $i$, 
$1\leq i\leq k$,
$m_{i-1}a_{i-1}$ is a prefix of $\eta(a_i)$. We say that this sequence is
\emph{$a$-admissible} if it is admissible and, moreover, $m_ka_k$ is a prefix of
$\eta(a)$.
\end{definition}



Dumont-Thomas proved the following result which we rewrite in our notation.

\begin{theorem}\label{thm:dumont-thomas}
    {\rm\cite[Theorem 1.5]{MR1020484}}
    Let $a\in A$ and let $\eta:A^*\to A^*$ be a substitution. 
    Let $u=\eta(u)$ be a right-infinite fixed point of $\eta$ with $u_0 = a$.
    For every integer $n\geq1$,
    there exists a unique integer $k=k(n)$ and a unique sequence
    $(m_i,a_i)_{i=0,\dots,k}$ such that
    \begin{itemize}
        \item this sequence is $a$-admissible and $m_k\neq\varepsilon$, 
        \item $u_0u_1\cdots u_{n-1} = \eta^k(m_k) \eta^{k-1}(m_{k-1}) \cdots \eta^0(m_0)$.
    \end{itemize}
\end{theorem}

The proof of the above theorem was based on the following lemmas
which we cite here as we need them in what follows.

\begin{lemma}\label{lem:admissible-sequence_sum}
    {\rm\cite[Lemma 1.1]{MR1020484}}
    Let $k$ be an integer. If 
    $(m_i,a_i)_{i=0,\dots,k}$ is an admissible sequence, then 
    $\sum_{j=0}^k |\eta^j(m_j)| < |\eta^k(m_k a_k)|$.
\end{lemma}


\begin{lemma}\label{lem:unique_admissible_sequence}
    {\rm\cite[Lemma 1.3]{MR1020484}}
    Let $k$ be an integer. Let $b\in A$,
    $(m_i,a_i)_{i=0,\dots,k}$ and
    $(m'_i,a'_i)_{i=0,\dots,k}$ be two $b$-admissible sequences
    and $n$ be an integer such that
    \[
        n = \sum_{j=0}^k |\eta^j(m_j)| = \sum_{j=0}^{k} |\eta^j(m'_j)|.
    \]
    Then for every $i$, $0\leq i \leq k$, we have $(m_i,a_i) = (m'_i, a'_i)$.
\end{lemma}

\begin{lemma}\label{lem:prefix_recursion}
    {\rm\cite[Lemma 1.4]{MR1020484}}
    Let $\eta\colon A^*\to A^*$ be a substitution.
    Let $\ell\geq 1$ be an integer, $a\in A$ a letter and $m\in A^*$
    a proper prefix of the word $\eta^\ell(a)$.
    Then there exist $(m',a')\in A^*\times A$ and $m''\in A^*$
    such that $m'a'$ is a prefix of $\eta(a)$,
    $m''$ is a proper prefix of $\eta^{\ell-1}(a')$
    and $m = \eta^{\ell-1}(m') m''$.
\end{lemma}

\subsection{Few extensions of Dumont-Thomas results}

In this subsection, we propose few small extensions of Dumont-Thomas lemmas.
Firstly, we observe that admissible sequences are related to automata as follows.

\begin{lemma}\label{lemma:admissible-feed-into-automaton}
    If $(m_i,a_i)_{i=0,\dots,k}$ is a $x$-admissible sequence,
    then 
    \[
        a_i=\mathcal{A}_{\eta,x}(|m_{k}|,|m_{k-1}|,\dots,|m_i|), \qquad \text{ for every } i=0,\dots,k.
    \]
\end{lemma}

\begin{proof}
    The proof is done by induction on $i$.
    If $i=k$, then
        $a_i=a_k=\eta(x)[|m_k|]=\mathcal{A}_{\eta,x}(|m_{k}|)$.
    If $i<k$, then
    \begin{align*}
        a_i 
        &= \eta(a_{i+1})[|m_i|]
        = \eta\left(\mathcal{A}_{\eta,x}(|m_{k}|,\dots,|m_{i+1}|)\right)[|m_i|]\\
        &= \mathcal{A}_{\eta,x}(|m_{k}|,\dots,|m_{i+1}|,|m_i|). \quad\qed
    \end{align*}
\end{proof}

The following lemma reformulates and strenghtens Lemma 1.2 from
\cite{MR1020484}. 
It allows
to conclude that
two admissible sequences are of the same length.

\begin{lemma}\label{lem:intervals-for-sums}
    Let $\eta:A^*\to A^*$ be a substitution and $x\in A$.
    Let $(m_i,a_i)_{i=0,\dots,k}$
    be an $x$-admissible sequence.
    \begin{enumerate}[(i)]
    \item If $m_k \neq \varepsilon$, then
        $  |\eta^{k}(x)| 
            \leq \textstyle\sum_{j=0}^k |\eta^j(m_j)| 
            <    |\eta^{k+1}(x)|$.
    \item If $m_km_{k-1}\cdots m_{k-p+1} \neq \varepsilon$, then
            $ |\eta^{k-p+1}(x)| 
            \leq \textstyle\sum_{j=0}^k |\eta^j(m_j)| 
            < |\eta^{k+1}(x)|$.
    \item If $m_k a_k\neq \eta(x)$ and $x$ is a suffix of $\eta(x)$, then
        \[
            -|\eta^{k+1}(x)| 
            \leq -|\eta^{k+1}(x)| + \textstyle\sum_{j=0}^k |\eta^j(m_j)| 
            <    -|\eta^{k}(x)|.
        \]
    \item If $\eta^{p-1}(m_{k})\eta^{p-2}(m_{k-1})\cdots\eta^0(m_{k-p+1})a_{k-p+1}
        \neq\eta^p(x)$ and $x$ is a suffix of $\eta^p(x)$, then
        \[
            -|\eta^{k+1}(x)| 
            \leq -|\eta^{k+1}(x)| + \textstyle\sum_{j=0}^k |\eta^j(m_j)| 
            <    -|\eta^{k-p+1}(x)|.
        \]
    \end{enumerate}
\end{lemma}

\begin{proof}
    (i) Since $m_k\neq\varepsilon$, the word $m_k$ starts with letter $x$, thus
    using Lemma~\ref{lem:admissible-sequence_sum}, we have
    \[
        |\eta^{k}(x)| 
        \leq |\eta^{k}(m_k)| 
            \leq \textstyle\sum_{j=0}^k |\eta^j(m_j)| 
            <    |\eta^{k}(m_ka_k)|
            \leq |\eta^{k+1}(x)|.
    \]

    (ii)
    If $m_km_{k-1}\cdots m_{k-p+1} \neq \varepsilon$, then
    the word $w=\eta^{p-1}(m_{k})\eta^{p-2}(m_{k-1})\cdots\eta^0(m_{k-p+1})$
    is nonempty and starts with letter $x$.
    Thus
    \[
        |\eta^{k-p+1}(x)| 
        \leq |\eta^{k-p+1}(w)| 
            \leq \textstyle\sum_{j=0}^k |\eta^j(m_j)| 
            <    |\eta^{k}(m_ka_k)|
            \leq |\eta^{k+1}(x)|.
    \]

    (iii)
    If $m_k a_k\neq \eta(x)$ and $x$ is a suffix of
    $\eta(x)$, and using Lemma~\ref{lem:admissible-sequence_sum}, we get
        \[
            -|\eta^{k+1}(x)| 
            \leq -|\eta^{k+1}(x)| + \textstyle\sum_{j=0}^k |\eta^j(m_j)| 
            <    -|\eta^{k+1}(x)| + |\eta^{k}(m_ka_k)|
            <    -|\eta^{k}(x)|.
        \]

    (iv)
    Assume $w=\eta^{p-1}(m_{k})\eta^{p-2}(m_{k-1})\cdots\eta^0(m_{k-p+1})a_{k-p+1}
        \neq\eta^p(x)$ and $x$ is a suffix of $\eta^p(x)$. 
    Observe that $w$ is a prefix of $\eta^p(x)$
    and $|\eta^{k+1}(x)| 
         =|\eta^{k-p+1}(\eta^p(x))| 
         \geq |\eta^{k-p+1}(w)| + |\eta^{k-p+1}(x)|$.
        Thus 
        \begin{align*}
            -|\eta^{k+1}(x)| 
            &\leq -|\eta^{k+1}(x)| + \textstyle\sum_{j=0}^k |\eta^j(m_j)| \\
            &= -|\eta^{k+1}(x)| 
            + \textstyle\sum_{j=0}^{k-p} |\eta^j(m_j)|
            + \textstyle\sum_{j=k-p+1}^k |\eta^j(m_j)| \\
            &< -|\eta^{k+1}(x)| 
            + |\eta^{k-p}(m_{k-p}a_{k-p})|
            + |\eta^{k-p+1}(w)| - |\eta^{k-p+1}(a_{k-p+1})|\\
            &\leq  -|\eta^{k+1}(x)| + |\eta^{k-p+1}(w)|
             \leq  -|\eta^{k-p+1}(x)|.\qquad\qed
        \end{align*}
\end{proof}


Lemma~\ref{lem:prefix_recursion} can be used to construct
an admissible sequence from a prefix of the image of a letter
under the $p$-th power of a substitution.

\begin{lemma}\label{lem:when-m-prefix-of-eta-p-a}
    Let $\eta:A^*\to A^*$ be a substitution and
    $p\geq 1$ be an integer.
    If $m\in A^*$ and $x\in A$ are such that
    $m$ is a prefix of $\eta^p(x)$, then
    there exists a unique $x$-admissible sequence
    $(m_i,a_i)_{i=0,\dots,p-1}$ such that
    \begin{equation}\label{eq:m-as-product}
        m = \eta^{p-1}(m_{p-1}) \eta^{p-2}(m_{p-2}) \cdots \eta^0(m_0).
    \end{equation}
\end{lemma}

\begin{proof}
    (Unicity)
    Let
    $(m_i,a_i)_{i=0,\dots,p}$ and
    $(m'_i,a'_i)_{i=0,\dots,p}$ be two $x$-admissible sequences satisfying the hypothesis.
    Let 
    $k=\max\{i=0,\dots,p\mid m_i \neq\varepsilon\}$
    and
    $k'=\max\{i=0,\dots,p\mid m'_i \neq\varepsilon\}$.
    Then
    \[
            \textstyle\sum_{j=0}^{k} |\eta^j(m_j)| 
          = |m|
          = \textstyle\sum_{j=0}^{k'} |\eta^j(m'_j)|.
    \]
    From Lemma~\ref{lem:intervals-for-sums},
    we have
    $ |\eta^{k}(a)| \leq |m| < |\eta^{k+1}(a)|$
    and
    $ |\eta^{k'}(a)| \leq |m| < |\eta^{k'+1}(a)|$,
    so that $k = k'$.
    The unicity of the sequence $(m_i,a_i)$
    follows from Lemma~\ref{lem:unique_admissible_sequence}.


    (Existence)
    We do the proof by induction on $p$.
    If $p=1$, then $m$ is a prefix of $\eta(x)$.
    Let $m_0=m$ and $a_0=x$.
    The length-1 sequence $(m_i,a_i)_{i=0}$ is $x$-admissible
    and satisfies $m=\eta^0(m_0)$.

    Now let $m\in A^*$ and $x\in A$ be such that
    $m$ is prefix of $\eta^{p+1}(x)$.
    From Lemma~\ref{lem:prefix_recursion}, 
    there exist $(m_p,a_p)\in A^*\times A$ and $m''\in A^*$
    such that $m_pa_p$ is a prefix of $\eta(x)$,
    $m''$ is a proper prefix of $\eta^{p}(a_p)$
    and $m = \eta^{p}(m_p) m''$.
    By the induction hypothesis,
    there exists a $a_p$-admissible sequence
    $(m_i,a_i)_{i=0,\dots,p-1}$ such that
    \[
        m'' = \eta^{p-1}(m_{p-1}) \eta^{p-2}(m_{p-2}) \cdots \eta^0(m_0).
    \]
    Therefore, 
    \[
        m = \eta^{p}(m_p) m''
          = \eta^{p}(m_p)\eta^{p-1}(m_{p-1}) \eta^{p-2}(m_{p-2}) \cdots \eta^0(m_0).
    \]
    The extended sequence $(m_i,a_i)_{i=0,\dots,p}$ is $x$-admissible
    since $m_{p-1}a_{p-1}$ is a prefix of $\eta(a_p)$
    and $m_pa_p$ is a prefix of $\eta(x)$.\qquad \qed
\end{proof}

Let $\eta:A^*\to A^*$ be a substitution and
$\mathcal{D}=\left\{0,...,\max_{c\in A}|\eta(c)|-1\right\}$.
Lemma~\ref{lem:when-m-prefix-of-eta-p-a} allows to define
a map for every integer $p\geq 1$ and $x\in A$ as follows
	\[
	\begin{array}{rccl}
        \tail_{\eta,p,x}:&\{0,1,\dots,|\eta^p(x)|-1\} & \to & \mathcal{D}^p\\
		&n & \mapsto & |m_{p-1}|,|m_{p-2}|,\dots,|m_0|.
	\end{array}
	\]
where 
$(m_i,a_i)_{i=0,\dots,p-1}$ is the unique $x$-admissible sequence
satisfying Equation~\eqref{eq:m-as-product}
with $m$ being the prefix of length $n$ of $\eta^p(x)$.
The map $\tail_{\eta,p,x}$ will be used in Section~\ref{sec:NS-for-periodic-points}.

\begin{lemma}\label{lem:automaton_q_r}
    Let $\eta:A^*\to A^*$ be a substitution and $p\geq 1$ be an integer.
    Let $x\in A$. Then for every $\ell\in\{0,1,\dots, |\eta^p(x)|-1\}$ we have
    \begin{equation*}
        \eta^p(x)[\ell] = \mathcal{A}_{\eta,x}(\tail_{\eta,p, x}(\ell)).
    \end{equation*}
   
   
\end{lemma}

\begin{proof}
    Let $m$ be the prefix of $\eta^p(x)$ of length $\ell$. 
    From Lemma~\ref{lem:when-m-prefix-of-eta-p-a},
    there exists a unique $x$-admissible sequence
    $(m_i,a_i)_{i=0,\dots,p-1}$ such that
    \[
        m = \eta^{p-1}(m_{p-1}) \eta^{p-2}(m_{p-2}) \cdots \eta^0(m_0).
    \]
    The word $ma_0$ is a prefix of $\eta^p(x)$, thus 
    $\eta^p(x)[\ell]=a_0$.
    From Lemma~\ref{lemma:admissible-feed-into-automaton},
    \[
        \eta^p(x)[\ell]=
        a_0=\mathcal{A}_{\eta,x}(|m_{p-1}|,|m_{p-2}|,\dots,|m_0|)
           =\mathcal{A}_{\eta,x}(\tail_{\eta,p, x}(\ell)).\quad\qed
    \]
\end{proof}


\section{Dumont-Thomas numeration system for $\mathbb{Z}$}

In this section, we prove extensions of
Theorem~\ref{thm:dumont-thomas} to right-infinite and left-infinite periodic
points of substitutions.

\begin{theorem}\label{thm:dumont-thomas-right-p}
    Let $a\in A$ and $\eta:A^*\to A^*$ be a substitution. 
    Let $u\in \Per_{{\mathbb{Z}_{\geq 0}}}(\eta)$ with a period $p\geq 1$ such that $u_0 = a$.
    For every integer $n\geq1$,
    there exists a unique integer $k=k(n)$
    such that $p$ divides $k+1$
    and a unique sequence $(m_i,a_i)_{i=0,\dots,k}$ such that
    \begin{enumerate}[(i)]
        \item this sequence is $a$-admissible and 
            $m_{k} m_{k-1} \cdots m_{k-p+1}\neq\varepsilon$,
        \item $u_0u_1\cdots u_{n-1} 
            = \eta^{k}(m_{k}) \eta^{k-1}(m_{k-1}) \cdots \eta^0(m_0)$.
    \end{enumerate}
\end{theorem}


\begin{proof}
    Let $n\geq 1$ be an integer. 
    There exists a unique integer $k\in\mathbb{N}$
    such that $p$ divides $k+1$
    and
    \[
        |\eta^{k-p+1}(a)| \leq n <
        |\eta^{k+1}(a)|.
    \]
    The word $m = u_0u_1\cdots u_{n-1}$ is thus a proper prefix
    of $\eta^{k+1}(a)$.
    From Lemma~\ref{lem:when-m-prefix-of-eta-p-a},
    there exists a unique $a$-admissible sequence
    $(m_i,a_i)_{i=0,\dots,k}$ such that
    \[
        m = \eta^{k}(m_{k}) \eta^{k-1}(m_{k-1}) \cdots \eta^0(m_0).
    \]
    Assume by contradiction that $m_{k} m_{k-1} \cdots m_{k-p+1}=\varepsilon$. 
    Then $a_{k-p+1}=a$ and
    from Lemma~\ref{lem:admissible-sequence_sum}, we have
    \[
        n = |m| 
        = \textstyle\sum_{j=0}^{k-p}|\eta^j(m_j)|
        < |\eta^{k-p}(m_{k-p})|
        \leq |\eta^{k-p+1}(a_{k-p+1})|,
    \]
    a contradiction.
    Thus $m_{k} m_{k-1} \cdots m_{k-p+1}\neq\varepsilon$.
    \qed
\end{proof}

We now adapt Dumont-Thomas's theorem to the left-infinite periodic points.

\begin{theorem}\label{thm:dumont-thomas-for-left-side-p}
    Let $b\in A$ and $\eta:A^*\to A^*$ be a substitution. 
    Let $u\in \Per_{{\mathbb{Z}_{< 0}}}(\eta)$ with a period $p\geq 1$
    such that $u_{-1} = b$.
    For every integer $n\leq -2$,
    there exists a unique integer $k=k(n)$ 
    such that $p$ divides $k+1$
    and a unique sequence
    $(m_i,a_i)_{i=0,\dots,k}$ such that
    \begin{enumerate}[(i)]
        \item this sequence is $b$-admissible and 
            \begin{equation}\label{eq:left-periodic-condition}
                \eta^{p-1}(m_{k})\eta^{p-2}(m_{k-1})\cdots\eta^0(m_{k-p+1})a_{k-p+1}
                \neq\eta^p(b),
            \end{equation}
        \item $u_{-|\eta^{k+1}(b)|}\cdots u_{n-2}u_{n-1} 
            = \eta^{k}(m_{k}) \eta^{k-1}(m_{k-1}) \cdots \eta^0(m_0)$.
    \end{enumerate}
\end{theorem}

\begin{proof}
    Let $n\leq -2$ be an integer. 
    There exists a unique integer $k\in\mathbb{N}$
    such that $p$ divides $k+1$
    and
    \[
        -|\eta^{k+1}(b)|\leq n  < -|\eta^{k-p+1}(b)|.
    \]
    Therefore the word 
    $m = u_{-|\eta^{k+1}(b)|}\cdots u_{n-2}u_{n-1}$ of length
    \[
    |m| = |\eta^{k+1}(b)| + n 
        < |\eta^{k+1}(b)| - |\eta^{k-p+1}(b)| 
        < |\eta^{k+1}(b)|
    \]
    is a proper prefix of the word $\eta^{k+1}(b)$.
    From Lemma~\ref{lem:when-m-prefix-of-eta-p-a},
    there exists a unique $b$-admissible sequence
    $(m_i,a_i)_{i=0,\dots,k}$ such that
    \[
        m = \eta^{k}(m_{k}) \eta^{k-1}(m_{k-1}) \cdots \eta^0(m_0).
    \]
    By contradiction, assume that \eqref{eq:left-periodic-condition}
    is an equality. Then $a_{k-p+1}=b$ and
    \begin{equation*}
        \begin{aligned}
        |m| 
       
       
       
        &= |\eta^{k-p+1}(\eta^p(b))| - |\eta^{k-p+1}(a_{k-p+1})|
          + \textstyle\sum_{j=0}^{k-p}|\eta^{j}(m_{j})|\\
        &\geq |\eta^{k+1}(b)| - |\eta^{k-p+1}(b)|,
        \end{aligned}
    \end{equation*}
    a contradiction. \qed
\end{proof}



\section{Numeration systems for $\mathbb{Z}$ based on periodic points}\label{sec:NS-for-periodic-points}

In this section, we define a numeration system for $\mathbb{Z}$
using the results proved in the previous section.

\begin{definition}[Numeration system]\label{def:rep-not-prolongable}
    Let $\eta:A^*\to A^*$ be a substitution and
    let $u\in\Per(\eta)$.
    Let $\mathcal{D}=\left\{0,...,\max_{c\in A}|\eta(c)|-1\right\}$.
    We define
	\[
	\begin{array}{rccl}
		\rep_{u}:&\mathbb{Z} & \to & \mathcal{D}^*\\
		&n & \mapsto &
		\begin{cases}
            \mathtt{0},|m_{k}|,|m_{k-1}|,\dots,|m_0|               & \text{ if } n \geq 1,\\
			\mathtt{0}                                          & \text{ if } n = 0,\\
			\mathtt{1}                                          & \text{ if } n =-1,\\
			\mathtt{1},|m_{k}|,|m_{k-1}|,\dots,|m_0|               & \text{ if } n \leq -2.
		\end{cases}
	\end{array}
	\]
    where $k=k(n)$ is the unique integer and
    $(m_i,a_i)_{i=0,\dots,k}$ is the unique sequence
    obtained from Theorem~\ref{thm:dumont-thomas-right-p}
    (Theorem~\ref{thm:dumont-thomas-for-left-side-p}) applied on $u|_{\mathbb{Z}_{\geq 0}}$
    ($u|_{\mathbb{Z}_{< 0}}$)
    if $n\geq 1$
    (if $n\leq -2$, respectively).
\end{definition}

\noindent
Note that if $p\in\mathbb{N}$ is a period of $u$,
then it divides $|\rep_{u}(n)|-1$ for every $n\in\mathbb{Z}$.

When $u=\eta^p(u)$ is a periodic point of a substitution $\eta$,
then it is also a fixed point of the substitution $\eta^p$.
Thus, Theorem~\ref{thm:dumont-thomas} may be used to define a
numeration system for $\mathbb{N}$, but it
leads to a much larger alphabet size $\#\mathcal{D}$.
One advantage of Definition~\ref{def:rep-not-prolongable} is
that the size of the alphabet $\mathcal{D}$ is independent of the period $p$.

\begin{definition}[quotient, remainder]\label{def:quotient-p}
    Let $\eta:A^*\to A^*$ be a substitution and
    let $u\in\Per(\eta)$.
    Let $n\in\mathbb{Z}\setminus\{-1,0\}$ be an integer
    and $k=k(n)$ be the unique integer and
    $(m_i,a_i)_{i=0,\dots,k}$ be the unique sequence
    obtained from Theorem~\ref{thm:dumont-thomas-right-p}
    (Theorem~\ref{thm:dumont-thomas-for-left-side-p}) applied on $u|_{\mathbb{Z}_{\geq 0}}$
    ($u|_{\mathbb{Z}_{< 0}}$)
    if $n\geq 1$
    (if $n\leq -2$, respectively).
    We define the \emph{$u$-quotient} of $n$ as
    \[
        q=
		\begin{cases}
            |\eta^{k-p}(m_{k}) \eta^{k-p-1}(m_{k-1}) \cdots \eta^0(m_p)|
			            & \text{ if } n \geq 1,\\
            |\eta^{k-p}(m_{k}) \eta^{k-p-1}(m_{k-1}) \cdots \eta^0(m_p)| -|\eta^{k-p+1}(b)|
			            & \text{ if } n \leq -2,
		\end{cases}
    \]
    and the \emph{$u$-remainder} of $n$  as 
    $r=|\eta^{p-1}(m_{p-1}) \eta^{p-2}(m_{p-2})\cdots\eta^0(m_0)|$.
\end{definition}

Notice that the $u$-quotient $q$ and $u$-remainder $r$ of an integer
$n\in\mathbb{Z}\setminus\{-1,0\}$ fulfil that if $n \geq1$ then $0\leq q < n$ and if $n \leq
-2$ then $-1 \geq q > n$. Consequently, $|q| < |n|$.  Also, if $\eta$ is $d$-uniform,
then the $u$-quotient and $u$-remainder of $n$ correspond to the quotient and
remainder of the division of $n$ by~$d^p$. 

\begin{remark}
Note that if we know the $u$-quotient $q$ and the $u$-remainder $r$, we
can recover the sequence $|m_{p-1}|,|m_{p-2}|,\cdots,|m_0|$.
Indeed, it is equal to $\tail_{\eta,p,u_q}(r)$.
\end{remark}

\begin{lemma}\label{lem:rep_recurrence-p}
    Let $\eta:A^*\to A^*$ be a substitution and
    $u\in \Per(\eta)$ with period $p\geq 1$.
    Let $n\in\mathbb{Z}\setminus\{-1,0\}$ be an integer.
    If $q\in\mathbb{Z}$ is the $u$-quotient 
    and $r\in\mathbb{N}$ is the $u$-remainder
    of $n$, then
    \begin{equation*}
        u_n  =  \eta^p(u_q)[r] 
        \quad
        \text{ and }
        \quad
        \rep_{u}(n) = \rep_{u}(q)\cdot \tail_{\eta,p,u_q}(r).
    \end{equation*}
\end{lemma}

\begin{proof}
    Let $n\in\mathbb{Z}\setminus\{-1,0\}$
    and let $q$ be the $u$-quotient and $r$ the $u$-remainder of~$n$.
    Let $u\in\Per(\eta)$ and let $a,b\in A$ denote letters such that $u_{-1}=b$ and $u_0=a$.

    Suppose $n\geq 1$.
    From Theorem~\ref{thm:dumont-thomas-right-p},
    there exists a unique $a$-admissible sequence
    $(m_i,a_i)_{i=0,\dots,k}$
    such that
    $u_0 \dots u_{n-1} = \eta^{k}(m_{k}) \dots \eta^0(m_0)$.
    Also $\eta^{k}(m_{k}) \dots \eta^0(m_0)a_0$
    is a prefix of $\eta^{k+1}(a)$ which is a prefix of $u$,
    thus $u_n=a_0$.
    Since $u$ has period $p$, the word
    \[
        \eta^{k-p}(m_{k}) \eta^{k-p-1}(m_{k-1}) \cdots \eta^0(m_p) a_p
    \]
    is a prefix of $\eta^{k+1-p}(a)$ which is a prefix of $u$.
    Thus $a_p=u_q$.
    Since $\eta^{p-1}(m_{p-1})\cdots \eta^{0}(m_{0})a_0$ is a prefix of
    $\eta^p(a_p)$, we deduce that 
    $u_n = a_0 = \eta^p(a_p)[r] = \eta^p(u_q)[r]$.

    Suppose $n\leq -2$.
    From Theorem~\ref{thm:dumont-thomas-for-left-side-p},
    there exists a unique $b$-admissible sequence
    $(m_i,a_i)_{i=0,\dots,k}$ such that
    $u_{-|\eta^{k+1}(b)|} \dots u_{n-1} 
    = \eta^{k}(m_{k}) \dots \eta^0(m_0)$.
    Also $\eta^{k}(m_{k}) \dots \eta^0(m_0)a_0$
    is a prefix of $\eta^{k+1}(b)$, 
    which is a prefix of 
    $u_{-|\eta^{k+1}(b)|} \dots u_{-1}$, thus $u_n=a_0$.
    Since $u$ has period $p$, the word
    \[
        \eta^{k-p}(m_{k}) \eta^{k-p-1}(m_{k-1}) \cdots \eta^0(m_p) a_p
    \]
    is a prefix of $\eta^{k+1-p}(a)$ which is a prefix of
    $u_{-|\eta^{k-p+1}(b)|} \dots u_{-1}$, thus $a_p=u_q$.
    Since $\eta^{p-1}(m_{p-1})\cdots \eta^{0}(m_{0})a_0$ is a prefix of
    $\eta^p(a_p)$, we deduce that 
    $u_n = a_0 = \eta^p(a_p)[r] = \eta^p(u_q)[r]$.

    To finish the proof for both cases simultaneously, if $n\geq 1$ ($n\leq -2$), 
    applying Theorem~\ref{thm:dumont-thomas-right-p} 
    (Theorem~\ref{thm:dumont-thomas-for-left-side-p})
    on the $u$-quotient $q$ gives for $\mathtt{d} = \mathtt{0}$ ($\mathtt{d} = \mathtt{1}$)
    $$\rep_{u}(q) = \mathtt{d}, |m_{k}|,|m_{k-1}|,\dots,|m_p|.$$
    As $n\geq 1$ if and only if $q\geq 0$, we have
    \begin{equation*}
        \begin{aligned}
            \rep_{u}(n)
            &=\mathtt{d},\ |m_{k}|,|m_{k-1}|,\dots,|m_p|, |m_{p-1}|, \dots, |m_0| \\
            &= \rep_{u}(q)\cdot |m_{p-1}|, \dots, |m_0| 
            = \rep_{u}(q)\cdot \tail_{\eta,p, u_q}(r).
            \qquad \qed
        \end{aligned}
    \end{equation*}
\end{proof}


\section{Periodic points as Automatic Sequences}

Let $\eta: A^*\to A^*$ be a substitution and let $u\in\Per(\eta)$.
Denote $s = (u_{-1},u_0)$
and $\mathcal{D}=\left\{0,...,\max_{c\in A}|\eta(c)|-1\right\}$.
We associate an automaton $\mathcal{A}_{\eta,s}$ with $(\eta,s)$
by adding a new state $\texttt{start}$ and two additional
edges to the automaton $\mathcal{A}_{\eta, a}$ defined in~\cite{MR2742574}.
The automaton $\mathcal{A}_{\eta,s} = (A\cup\left\{\texttt{start}\right\},\mathcal{D},
\delta, \texttt{start}, A)$ has the transition function 
$\delta:A\cup\{\texttt{start}\}\to A$ such that 
\begin{itemize}
    \item $\delta(\texttt{start},\mathtt{0}) = s_0 =a$, $\quad\delta(\texttt{start},\mathtt{1}) =s_{-1}= b$,
    \item for every $c,d\in A$, 
        every $w=w_0 w_1\dots w_{\ell-1}\in A^\ell$ and every $i\in \mathcal{D}$,
        it holds that
        $\delta(c,i)=d$ if and only if $\eta(c)=w$ and $w_i=d$. 
\end{itemize}

Examples of automata associated to the Fibonacci substitution are shown in
Figure~\ref{fig:automataZ}.
\begin{figure}[h]
	\begin{center}
		\begin{tikzpicture}[auto]
		\begin{scope}[xshift=-.1cm]
	
		\node[draw,circle] (A) at (0,0) {$a$};
		\node[draw,circle] (B) at (1.5,0) {$b$};
	
		\draw[bend left,->] (A) to node {\scriptsize 1} (B);
		\draw[bend left,->] (B) to node {\scriptsize 0} (A);
	
		\draw[loop left,->] (A) to node {\scriptsize 0} (A);
	
		\draw[<-] (A) -- ++ (0,1) node[above] {};
	
		\draw[->] (A) -- ++ (0,-0.5);
		\draw[->] (B) -- ++ (0,-0.5);
		\end{scope}
		\begin{scope}[xshift=3.5cm]
	
		\node[draw,circle] (A) at (0,0) {$a$};
		\node[draw,circle] (B) at (1.5,0) {$b$};
	
		\draw[bend left,->] (A) to node {\scriptsize 1} (B);
		\draw[bend left,->] (B) to node {\scriptsize 0} (A);
	
		\draw[loop left,->] (A) to node {\scriptsize 0} (A);
	
		\draw[<-] (B) -- ++ (0,1) node[above] {};
	
		\draw[->] (A) -- ++ (0,-0.5);
		\draw[->] (B) -- ++ (0,-0.5);
		\end{scope}
		\begin{scope}[xshift=7cm]
	
		\node[draw] (S) at (.75,1.2) {$\texttt{start}$};
		\node[draw,circle] (A) at (0,0) {$a$};
		\node[draw,circle] (B) at (1.5,0) {$b$};
	
		\draw[bend left=20,->] (A) to node {\scriptsize 1} (B);
		\draw[bend left=20,->] (B) to node {\scriptsize 0} (A);
		\draw[bend right,->] (S) to node[left] {\scriptsize 0} (A);
		\draw[bend left,->] (S) to node[right] {\scriptsize 1} (B);
	
		\draw[<-] (S) -- ++ (0,.5) node[above] {};
	
		\draw[loop left,->] (A) to node {\scriptsize 0} (A);
	
		\draw[->] (A) -- ++ (0,-0.5);
		\draw[->] (B) -- ++ (0,-0.5);
		\end{scope}
		\end{tikzpicture}
	\end{center}
	\caption{Automata $\mathcal{A}_{\varphi,a}$, $\mathcal{A}_{\varphi,b}$ and 
		$\mathcal{A}_{\varphi,s}$ for $\varphi:a\mapsto ab,b\mapsto a$ and $s=b.a$.}
	\label{fig:automataZ}
\end{figure}

The automaton $\mathcal{A}_{\eta,s}$
is related to the usual automata
$\mathcal{A}_{\eta,a}$ and $\mathcal{A}_{\eta,b}$ according to the following equalities
for every $w\in A^*$:
\begin{equation}
\label{eq:automata-relations}
    \mathcal{A}_{\eta,s}(\0w) = \mathcal{A}_{\eta,a}(w)
    \qquad
    \text{and}
    \qquad
    \mathcal{A}_{\eta,s}(\1w) = \mathcal{A}_{\eta,b}(w).
\end{equation}
Also if $\mathcal{A}_{\eta,s}(w) = a$ for some $w\in A^+$, then
for every $u\in A^*$
\begin{equation}
\label{eq:automata-relations-2}
    \mathcal{A}_{\eta,a}(u) = \mathcal{A}_{\eta,s}(wu).
\end{equation}

A theorem of Cobham \cite{MR457011} says that 
a sequence $u=(u_n)_{n\geq0}$ is $k$-automatic
with $k\geq2$
if and only if it is the image,
under a coding, of a fixed point of a $k$-uniform morphism 
\cite[\S 6]{MR1997038}.
It was extended to non-uniform morphisms \cite{zbMATH01916667},
see also 
\cite[\S 3]{MR2742574}.
The following extends the above results to the case of two-sided periodic
points of non-uniform substitutions.

\begin{theorem}\label{th:periodic_point_automatic}
    Let $\eta:A^*\to A^*$ be a substitution. 
    Let $u\in \Per(\eta)$ and denote $s = (u_{-1},u_0)$.
	Then for every $n\in\mathbb{Z}$
	\[
		u_n = \mathcal{A}_{\eta,s}(\rep_{u}(n)).
	\]
\end{theorem}
\begin{proof}
    If $n \in\{0,-1\}$ then by definition 
    we have
    $u_n = s_n = \mathcal{A}_{\eta,s}(\rep_{u}(n))$.

    Let $n \in\mathbb{Z}\setminus \{0,-1\}$. 
    Induction hypothesis: for every $m\in\mathbb{Z}$ such that $|m| < |n|$
	it holds that $x_m = \mathcal{A}_{\eta,s}(\rep_{u}(m))$.
	Let $q$ be the $u$-quotient and $r$ the $u$-remainder
	of $n$. As $|q| < |n|$, $q$ fulfils the induction hypothesis, i.e.,
    $u_q=\mathcal{A}_{\eta,s}(\rep_{u}(q))$.
    From Lemma~\ref{lem:rep_recurrence-p} we have
	$u_n  = \eta^p(u_q)[r]$ and 
    $\rep_{u}(n) = \rep_{u}(q)\cdot\tail_{\eta,p,u_q}(r)$. 
    Using Lemma~\ref{lem:automaton_q_r} and 
    Equation~\eqref{eq:automata-relations-2}, we have
	\begin{equation*}
		\begin{aligned}
		u_n &= \eta^p(u_q)[r]
             = \mathcal{A}_{\eta,u_q}(\tail_{\eta,p,u_q}(r)) \\
            &= \mathcal{A}_{\eta,s}(\rep_{u}(q)\cdot \tail_{\eta,p,u_q}(r))
			 = \mathcal{A}_{\eta,s}(\rep_{u}(n)).\qquad \qed
		\end{aligned}
	\end{equation*}
\end{proof}

\section{Numeration systems for $\mathbb{Z}^d$ based on periodic points}\label{sec:NS-for-Zd}

A numeration system for $\mathbb{Z}^d$ can be deduced from the numeration system for
$\mathbb{Z}$ based on a periodic point. Since not all integers are represented by words
of the same length, we propose here a way to pad them to a common length.

Let $\eta:A^*\to A^*$ be a substitution and $u\in \Per(\eta)$ with period $p\geq 1$.
Let ${\mathtt{W}_{\min}}$ and ${\mathtt{W}_{\max}}$ be the following minimum and the maximum element
under the $\tail$ map with particular parameters:
\begin{equation*}
\begin{aligned}
    {\mathtt{W}_{\min}} &= \tail_{\eta,p,u_{0}}(0) = \mathtt{0}^p,\\
    {\mathtt{W}_{\max}} &= \tail_{\eta,p,u_{-1}}(|\eta^p(u_{-1})|-1).
\end{aligned}
\end{equation*}
The words ${\mathtt{W}_{\min}}$ and ${\mathtt{W}_{\max}}$ play the role of neutral words in the
numeration system as illustrated in the next lemma.

\begin{lemma}
    Let $\eta:A^*\to A^*$ be a substitution.
    Let $u\in \Per(\eta)$ with period $p\geq 1$ and denote $s = (u_{-1},u_0)$.
    Let $w\in\mathcal{L}(\mathcal{A}_{\eta,s})$. Then
    \[
		\mathcal{A}_{\eta,s}(w) =
        \begin{cases}
            \mathcal{A}_{\eta,s}(\mathtt{0} ({\mathtt{W}_{\min}})^* v)
            & \text{ if } w=\mathtt{0} v,\\
            \mathcal{A}_{\eta,s}(\mathtt{1} ({\mathtt{W}_{\max}})^* v)
            & \text{ if } w=\mathtt{1} v.
        \end{cases}
    \]
\end{lemma}

\begin{proof}
    It holds that
    $\mathcal{A}_{\eta,s}(u_0, \mathtt{0}^p) = u_0$ 
    and $\mathcal{A}_{\eta,s}(u_{-1}, {\mathtt{W}_{\max}}) = u_{-1}$. \qed 
\end{proof}

It is useful to pad words to a certain length using neutral words as follows
using a pad function.
Let $s = (u_{-1}, u_0)$.
Let $w\in\mathcal{L}_{\ell p+1}(\mathcal{A}_{\eta,s})$ for some $\ell\in\mathbb{N}$.
Let $t\in\mathbb{N}$ such that $t\geq |w|$ and $t \bmod p = 1$.
    We define
    \[
        \pad_t(w)
        =
        \begin{cases}
            \mathtt{0} ({\mathtt{W}_{\min}})^m v  & \text{ if } w=\mathtt{0} v,\\
            \mathtt{1} ({\mathtt{W}_{\max}})^m v  & \text{ if } w=\mathtt{1} v
        \end{cases}
    \]
where $m=(t-|w|)/p$.
The padding map can be used to pad words so that they all
have the same length. For instance, it allows to represent coordinates in
$\mathbb{Z}^d$ in dimension $d\geq1$.

\begin{definition}[Numeration system for $\mathbb{Z}^d$]
	Let ${\boldsymbol{n}}=(n_1,n_2,\dots,n_d)\in\mathbb{Z}^d$.
	We define
	\[
	\rep_u({\boldsymbol{n}})=\left(
	\begin{array}{c}
        \pad_t(\rep_u(n_1))\\
        \pad_t(\rep_u(n_2))\\
        \dots\\
        \pad_t(\rep_u(n_d))
	\end{array}
	\right)
    \in\{\mathtt{0},\mathtt{1}\}^d(\mathcal{D}^d)^*
	\]
    where $t=\max\{|\rep_u(n_i)|\colon 1\leq i\leq d\}$.
\end{definition}


\section{A Total Order}

The radix order is a total order such that $u <_{rad} v$ if and only if
    $|u| < |v|$ or $|u| = |v|$ and $u <_{lex} v$.
We define reversed-radix order as a total order such that $u <_{rev} v$ if and only if
    $|u| > |v|$ or $|u| = |v|$ and $u <_{lex} v$.
We define a total order on $\{\mathtt{0},\mathtt{1}\}\mathcal{D}^*$
and we show that $\rep_{u}$ is an increasing bijection with respect to it.
\begin{definition}[total order $\prec$]\label{def:order}
    For every $u,v \in \{\mathtt{0},\mathtt{1}\}\mathcal{D}^*$,
	we define $u \prec v$ if and only if
	\begin{itemize}
		\item $u\in \mathtt{1} \mathcal{D}^*$ and $v\in \mathtt{0} \mathcal{D}^*$, or
        \item $u,v \in \mathtt{0} \mathcal{D}^*$ and $u <_{rad} v$, or
        \item $u,v \in \mathtt{1} \mathcal{D}^*$ and $u <_{rev} v$.
	\end{itemize}
\end{definition}


\begin{proposition}\label{prop:rep-u-is-increasing}
    Let $\eta:A^*\to A^*$ be a substitution and
    let $u\in\Per(\eta)$ with period $p\geq 1$ and the seed~$s$.
    The map
    $\rep_{u}:\mathbb{Z}  \to  \bigcup_{\ell\in\mathbb{N}}
    \mathcal{L}_{\ell p+1}(\mathcal{A}_{\eta,s}) \setminus \{\mathtt{0}{\mathtt{W}_{\min}}, \mathtt{1}{\mathtt{W}_{\max}}\} \mathcal{D}^*$ is
    an increasing bijection 
    with respect to the order $\prec$.
\end{proposition}

The proof of Proposition~\ref{prop:rep-u-is-increasing} 
is ommited and will be included in an extended version of this article.
It follows the proof of similar results, see
\cite[\S 5]{MR1879663} and \cite[\S 4]{MR2290784}.
In some other work on numeration systems, such increasing bijection 
is not a consequence, but rather a hypothesis.
For example, a bijection $\mathbb{N}\to\mathcal{L}$ serves as the definition of abstract
numeration systems in \cite{MR1799066}.

\vspace{-5pt}
\section{Examples}
\label{sec:fibo-num-system}


Let 
$\psi_{2}: a\mapsto ab, b\mapsto cb, c\mapsto ac$ denote some 2-uniform substitution,
$\psi_{TM}: a\mapsto ab, b\mapsto ba$ denote the Thue-Morse,
$\varphi_{F}: a\mapsto ab, b\mapsto a$ the Fibonacci,
and
$\varphi_{T}: a\mapsto ab, b\mapsto ac, c\mapsto a$ the Tribonacci substitution.
Let also $\rho: a\mapsto ac, b\mapsto cb, c\mapsto c$ be a non-primitive substitution.
We denote $\alpha\in\Per(\psi_{2})$ with the seed $b.a$ and period $1$,
$\beta\in\Per(\psi_{TM})$ with the seed $a.a$ and period $2$,
$\gamma,\delta\in\Per(\varphi_F)$ with the seed $b.a$, $a.a$ and period $2$,
$\tau\in\Per(\varphi_T)$ with the seed $c.a$ and period $3$,
and
$\xi\in\Per(\rho)$ with the seed $b.a$ and period $1$.
The~numeration systems derived from these periodic points
are in Table~\ref{tab:num_systems}.


\begin{table}[h]
\begin{center}
    \scriptsize
    \begin{tabular}{c||r|r|r|r|r|r}
        substitution & 2-uniform & Thue-Morse & Fibonacci & Fibonacci & Tribonacci & non-primitive\\
        images & $(ab,cb,ac)$ & $(ab,ba)$ & $(ab, a)$ & $(ab, a)$ & $(ab,ac,a)$ & $(ac,cb,c)$ \\
        seed & $b.a$ & $a.a$ & $b.a$ & $a.a$ & $c.a$ & $b.a$ \\
        \hline
        $n$ & $\rep_{\alpha}(n)$ & $\rep_{\beta}(n)$ & $\rep_{\gamma}(n)$ & $\rep_{\delta}(n)$ & $\rep_{\tau}(n)$ & $\rep_{\xi}(n)$\\ 
        \hline
        &&&&\\[-2mm]
       
       
       
       
       
       
       
       
       
        10 & \texttt{01010}  &  \texttt{01010}    &  \texttt{0010010}   & \texttt{0010010} & \texttt{0001011} & \texttt{01000000000} \\
        9  & \texttt{01001}  &  \texttt{01001}    &  \texttt{0010001}   & \texttt{0010001} & \texttt{0001010} & \texttt{0100000000} \\
        8  & \texttt{01000}  &  \texttt{01000}    &  \texttt{0010000}   & \texttt{0010000} & \texttt{0001001} & \texttt{010000000} \\
        7  & \texttt{0111}   &  \texttt{00111}    &  \texttt{01010}     & \texttt{01010}   & \texttt{0001000} & \texttt{01000000} \\
        6  & \texttt{0110}   &  \texttt{00110}    &  \texttt{01001}     & \texttt{01001}   & \texttt{0110}    & \texttt{0100000}    \\
        5  & \texttt{0101}   &  \texttt{00101}    &  \texttt{01000}     & \texttt{01000}   & \texttt{0101}    & \texttt{010000}    \\
        4  & \texttt{0100}   &  \texttt{00100}    &  \texttt{00101}     & \texttt{00101}   & \texttt{0100}    & \texttt{01000}    \\
        3  & \texttt{011}    &  \texttt{011}      &  \texttt{00100}     & \texttt{00100}   & \texttt{0011}    & \texttt{0100}    \\
        2  & \texttt{010}    &  \texttt{010}      &  \texttt{010}       & \texttt{010}     & \texttt{0010}    & \texttt{010}    \\
        1  & \texttt{01}     &  \texttt{001}      &  \texttt{001}       & \texttt{001}     & \texttt{0001}    & \texttt{01}    \\
        0  & \texttt{0}      &  \texttt{0}        &  \texttt{0}         & \texttt{0}       & \texttt{0}       & \texttt{0}       \\
       -1  & \texttt{1}      &  \texttt{1}        &  \texttt{1}         & \texttt{1}       & \texttt{1}       & \texttt{1}       \\
       -2  & \texttt{10}     &  \texttt{110}      &  \texttt{100}       & \texttt{101}     & \texttt{1010}    & \texttt{10}    \\
       -3  & \texttt{101}    &  \texttt{101}      &  \texttt{10010}     & \texttt{100}     & \texttt{1001}    & \texttt{100}    \\
       -4  & \texttt{100}    &  \texttt{100}      &  \texttt{10001}     & \texttt{10101}   & \texttt{1000}    & \texttt{1000}    \\
       -5  & \texttt{1011}   &  \texttt{11011}    &  \texttt{10000}     & \texttt{10100}   & \texttt{1010110} & \texttt{10000} \\
       -6  & \texttt{1010}   &  \texttt{11010}    &  \texttt{1001010}   & \texttt{10010}   & \texttt{1010101} & \texttt{100000} \\
       -7  & \texttt{1001}   &  \texttt{11001}    &  \texttt{1001001}   & \texttt{10001}   & \texttt{1010100} & \texttt{1000000} \\
       -8  & \texttt{1000}   &  \texttt{11000}    &  \texttt{1001000}   & \texttt{10000}   & \texttt{1010011} & \texttt{10000000} \\
       -9  & \texttt{10111}  &  \texttt{10111}    &  \texttt{1000101}   & \texttt{1010101} & \texttt{1010010} & \texttt{100000000} \\
       -10 & \texttt{10110}  &  \texttt{10110}    &  \texttt{1000100}   & \texttt{1010100} & \texttt{1010001} & \texttt{1000000000} \\
      
      
      
      
      
      
      
      
      
      
      
    \end{tabular}
\end{center}
    \caption{Numeration systems for periodic points
    $\alpha$, $\beta$, $\gamma$, $\delta$, $\tau$, $\xi$ with given seed.}
    \label{tab:num_systems}
\end{table}


\vspace{-5pt}
\subsection{Two's complement numeration system}

Let $\Sigma = \{\mathtt{0},\mathtt{1}\}$.
In the two's complement representation of integers
the value of a binary word $w=w_{k-1} w_{k-2}\cdots w_0\in\Sigma^{k}$
is $\val_{2c}(w) = \sum_{i=0}^{k-1}w_i 2^i - w_{k-1}2^{k}$, see \cite[\S 4.1]{MR3077153}.
For every $n\in\mathbb{Z}$ there exists a unique word 
$w\in\Sigma^+\setminus \left( \mathtt{0}\0\Sigma^* \cup \mathtt{1}\1\Sigma^* \right)$
such that $n=\val_{2c}(w)$. 
The word $w$ is called the \emph{two's complement representation} of the integer $n$,
and we denote it by $\rep_{2c}(n)$.
It can be shown that the map 
$\rep_{2c}:\mathbb{Z}\to \Sigma^+\setminus \left( \mathtt{0}\0\Sigma^* \cup \mathtt{1}\1\Sigma^* \right)$
is an increasing bijection with respect to the  order~$\prec$.
This implies the following.

\begin{proposition}\label{prop:rep_equality_2c}
    If $\alpha=\psi(\alpha)$ is a two-sided fixed point for some $2$-uniform
    substitution $\psi$, then $\rep_{\alpha}=\rep_{2c}$.
\end{proposition}

\vspace{-5pt}
\subsection{Fibonacci's complement numeration system}

In what follows, the Fibonacci sequence
$(F_n)_{n\geq 0}$,
$F_{n} = F_{n-1} + F_{n-2}$, for all $n \geq 2$,
is defined with the initial conditions
$ F_0 = 1$, $F_1 = 2$.
We denote $\Sigma = \{\mathtt{0},\mathtt{1}\}$.

In \cite{2205.02574}, the Fibonacci's complement numeration system for both nonnegative
and negative integers was defined
from the value map $\val_{\Fcal c}:\Sigma^*\to\mathbb{Z}$ 
defined for every binary word
$w=w_{k-1}\cdots w_0\in\Sigma^{k}$ as
    $\val_{\Fcal c}(w) 
    = \sum_{i=0}^{k-1}w_i F_{i} - w_{k-1}F_{k}$.
It is an analog of the two's complement value map $\val_{2c}$, 
using Fibonacci numbers instead of powers of 2.
It was proved in \cite{2205.02574} that
for every $n\in\mathbb{Z}$ there exists a
unique odd-length word 
$w\in L=\Sigma(\Sigma\Sigma)^*\setminus
    \left( \Sigma^*11\Sigma^* \cup 000\Sigma^* \cup 101\Sigma^*\right)$ 
such that $n=\val_{\Fcal c}(w)$.
It defines the map
$\rep_{\Fcal c}:\mathbb{Z}\to L$ by the rule $n\mapsto w$.


We can show that the numeration system obtained from the two-sided Fibonacci
word is Fibonacci's complement numeration system introduced in~\cite{2205.02574}.

\begin{proposition}\label{prop:rep_equality}
	Let $\varphi_F:a\mapsto ab,b\mapsto a$ be the Fibonacci substitution
    and let $\gamma\in\Per(\varphi_F)$ with the seed $s = b.a$.
    Then $\rep_{\gamma}=\rep_{\Fcal c}$.
\end{proposition}


\begin{proof}
    We observe
    \begin{equation}\label{eq:equality_languages}
        \begin{aligned}
        L 
           
           
            &= (\mathtt{0}(\mathcal{L}(\mathcal{A}_{\varphi_F,a})\setminus\mathtt{0}\0\Sigma^*)\cup
        \mathtt{1}(\mathcal{L}(\mathcal{A}_{\varphi_F,b})\setminus\mathtt{0}\mathtt{1}\Sigma^*))
        \cap \Sigma(\Sigma\Sigma)^*\\
            &= \mathcal{L}(\mathcal{A}_{\eta,s}) \setminus \{\mathtt{0}{\mathtt{W}_{\min}}, \mathtt{1}{\mathtt{W}_{\max}}\} \mathcal{D}^*
        \cap \Sigma(\Sigma\Sigma)^*\\
        \end{aligned}
    \end{equation}
    From \cite{2205.02574}, the map $\rep_{\Fcal c}$ is an increasing bijection $\mathbb{Z} \to
    L$ with respect to the order $\prec$.
   
    By Proposition~\ref{prop:rep-u-is-increasing} and
    Equation~\eqref{eq:equality_languages},
    $\rep_{\gamma}$ is an
    increasing bijection $\mathbb{Z}\to L$ with respect to the order $\prec$.
    Moreover,
    $\rep_{\gamma}(0) = \mathtt{0} = \rep_{\mathcal{F} c}(0)$.
    Since
    there is a unique inscreasing bijection
    $\mathbb{Z}\to L$ such that $0\mapsto\mathtt{0}$,
    $\rep_{\Fcal c} = \rep_{\gamma}$. \qed
\end{proof}



\subsubsection{Acknowledgements.}
This work was partially funded in France by
ANR CODYS (ANR-18-CE40-0007) and ANR IZES (ANR-22-CE40-0011).
The second author acknowledges financial support by the Barrande fellowship programme
and the  Grant Agency of Czech Technical University in Prague
(SGS20/183/OHK4/3T/14).


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