could use instead of the intervals $I_j$ the intervals $I^\prime_j:=[a_j,a_{j+1})$, $j\in\tA$. In that case $S_0=\{a_k\}$ and $S_j=\emptyset$ for all $j\geq 1$. This would correspond to the setting of Parry’s paper [@P2].

We define a map $\varphi$ on the disjoint union $${\rm dom}\varphi:=\bigcup_{j=0}^{k-1}j+J_j\subset \R\,,$$ by setting $$\label{2.3}
\varphi(x):= f^{-1}_j(t) \quad \text{if  $x=j+t$ and $t\in
J_j$}\,.$$ The map $\varphi$ is continuous, injective with range $X_1$. On $X_1$ the inverse map is $$\varphi^{-1}(x)=j+Tx\quad\text{if $x\in I_j$}\,.$$ For each $j$, such that $f_j$ is increasing, we define $\overline{\varphi}^j$ on $j+[0,1]$ (using the extension of $f_j$ to $[a_j,a_{j+1}]$) by $$\label{2.4}
\overline{\varphi}^j(x):=\begin{cases} a_j & \text{if $x=j+t$ and
$t\leq
f_j(a_j)$}\\
f^{-1}_j(t) & \text{if  $x=j+t$ and $t\in J_j$}\\
a_{j+1} & \text{if  $x=j+t$ and $f_j(a_{j+1})\leq t$.}
\end{cases}$$ For each $j$, such that $f_j$ is decreasing, we define $\overline{\varphi}^j$ on $j+[0,1]$ by $$\label{2.5}
\overline{\varphi}^j(x):=\begin{cases} a_{j+1} & \text{if $x=j+t$
and $t\leq
f_j(a_{j+1})$}\\
f^{-1}_j(t) & \text{if  $x=j+t$ and $t\in J_j$}\\
a_{j} & \text{if  $x=j+t$ and $f_j(a_{j})\leq t$.}
\end{cases}$$

It is convenient below to consider the family of maps $\overline{\varphi}^j$ as a single map defined on $[0,k]$, which is denoted by $\overline{\varphi}$. In order to avoid ambiguities at integers, where the map may be multi-valued, we always write a point of $[j,j+1]$ as $x=j+t$, 