notation $\ud{w}=w_0\cdots w_{n-1}$. The [length of $\ud{w}$]{} is $|\ud{w}|=n$. A [$n$-word]{} is a word of length $n$. There is a single word of length $0$, the [empty-word]{} $\epsilon$. The set of all words is $\tA^*$. The shift-map $\sigma\colon \tA^{\Z_+}\ra\tA^{\Z_+}$ is defined by $$\sigma(\ud{x}):=(x_1,x_2,\ldots)\,.$$ We define two operations $\tp$ and $\s$ on $\tA^*\backslash\{\epsilon\}$, $$\begin{aligned}
\tp{\ud{w}}&:=\begin{cases} w_0\cdots w_{n-2}& \text{if
$\ud{w}=w_0\cdots w_{n-1}$ and $n\geq 2$}\label{opp}\\
\epsilon & \text{if $\ud{w}=w_0$}
\end{cases}\\
\s{\ud{w}}&:=\begin{cases} w_1\cdots w_{n-1}& \text{if
$\ud{w}=w_0\cdots w_{n-1}$ and $n\geq 2$}\label{ops}\\
\epsilon & \text{if $\ud{w}=w_0$.}
\end{cases}\end{aligned}$$ On $\tA^{\Z_+}$ we define a total order, denoted by $\prec$. We set $$\delta(j):=\begin{cases} +1 &\text{if $f_j$ is increasing}\\
-1 & \text{if $f_j$ is decreasing,}
\end{cases}$$ and for a word $\ud{w}$, $$\delta(\ud{w}):=\begin{cases} \delta(w_0)\cdots \delta(w_{n-1})
&\text{if
$\ud{w}=w_0\cdots w_{n-1}$}\\
1& \text{if $\ud{w}=\epsilon$.}
\end{cases}$$ Let $\ud{x}^\prime\not =\ud{x}^{\prime\prime}$ belong to $\tA^{\Z_+}$; define $j$ as the smallest integer with $x^\prime_j\not=x^{\prime\prime}_j$. By definition $$\ud{x}^\prime\prec\ud{x}^{\prime\prime}\iff
\begin{cases}
x^\prime_j<x^{\prime\prime}_j &\text{if  $\delta(x^\prime_0\cdots
x^\prime_{j-1})=1$}\\
x^\prime_j>x^{\prime\prime}_j &\text{if  $\delta(x^\prime_0\cdots
x^\prime_{j-1})=-1$}\,.
\end{cases}$$ As usual $\ud{x}^\prime\preceq \ud{x}^{\prime\prime}$ if and only if $\ud{x}^\prime\prec\ud{x}^{\prime\prime}$ or $\ud{x}^\prime=\ud{x}^{\prime\prime}$. When all maps $f_j$ are increasing this order is the lexicographic order.

$\varphi$-expansion {#subsection2.2}
-------------------

We give an alternative description of a piecewise monotone dynamical system as in Parry’s paper [@P2]. In this description, when all maps $f_j$ are increasing, one 