$t\in[0,1]$, so that $$\overline{\varphi}(j+t)\equiv\overline{\varphi}(x):=\overline{\varphi}^j(t)\,.$$ We define the [coding map]{} $\i:X\backslash S\ra \tA^{\Z_+}$ by $$\i(x):=(\i_0(x),\i_1(x),\ldots)\quad\text{with $\i_n(x):=j$ if
$T^nx\in I_j$}\,.$$ The [$\varphi$-code]{} of $x\in X\backslash S$ is the string $\i(x)$, and we set $$\Sigma=\{\ud{x}\in\tA^{\Z_+}\colon \text{$\ud{x}=\i(x)$ for some
$x\in X\backslash S$}\}\,.$$ For $x\in X\backslash S$ and any $n\geq 0$, $$\label{2.7}
\varphi^{-1}(T^nx)=\i_n(x)+T^{n+1}x\quad\text{and}\quad
\i(T^nx)=\sigma^n\i(x) \,.$$ Let $z_j\in\tA$, $1\leq j \leq n$, and $t\in [0,1]$; we set $$\overline{\varphi}_1(z_1+t):=\overline{\varphi}(z_1+t)$$ and $$\label{formule}
\overline{\varphi}_n(z_1,\ldots,z_n+t):=
\overline{\varphi}_{n-1}(z_1,\ldots,z_{n-1}+\overline{\varphi}(z_n+t))\,.$$

For $n\geq 1$ and $m\geq 1$ we have $$\label{formulegenerale}
\overline{\varphi}_{n+m}(z_1,\ldots,z_{n+m}+t) =
\overline{\varphi}_n(z_1,\ldots,z_{n}+
\overline{\varphi}_m(z_{n+1},\ldots,z_{n+m}+t))\,.$$

The map $t\mapsto \overline{\varphi}_n(x_0,\ldots,x_{n-1}+t)$ is increasing if $\delta(x_0\cdots x_{n-1})=1$ and decreasing if $\delta(x_0\cdots x_{n-1})=-1$. We also write $\overline{\varphi}_n(\ud{x})$ for $\overline{\varphi}_n(x_0,\ldots,x_{n-1})$.

\[defn2.2\] The real number $s$ has a [$\varphi$-expansion]{} $\ud{x}\in\tA^{\Z_+}$ if the following limit exists, $$s=\lim_{n\ra\infty}\overline{\varphi}_n(\ud{x})\equiv
\overline{\varphi}\big(x_0+\overline{\varphi}(x_1+\ldots)\big)\equiv
\overline{\varphi}_\infty(\ud{x})\,.$$ The [$\varphi$-expansion is well-defined]{} if for all $\ud{x}\in\tA^{\Z_+}$, $\lim_{n\ra\infty}\overline{\varphi}_n(\ud{x})=
\overline{\varphi}_\infty(\ud{x})$ exists.\
The [$\varphi$-expansion is valid]{} if for all $x\in
X\backslash S$ the $\varphi$-code $\i(x)$ of $x$ is a [$\varphi$-expansion]{} of $x$.

If the $\varphi$-expansion is valid, then for $x\in X\backslash
S$, using , and the continuity of the maps $\overline{\varphi}^j$, $$\begin{aligned}
\label{2.8}
x&=\lim_{n\ra\infty}
\overline{\varphi}_n\big(\i_0(x),\ldots,\i_{n-1}(x)\big)\nonumber\\
&= \lim_{m\ra\infty}
\overline{\varphi}_{n}\big(\i_0(x),\ldots,\i_{n-1}(x)+
\overline{\varphi}_{m}\big(\i_n(x),\ldots,\i_{n+m-1}(x)\big)\big)
\\
&= \overline{\varphi}_{n}\big(\i_0(x),\ldots,\i_{n-1}(x)+
\overline{\varphi}_\infty(\i(T^nx)\big)\,.\nonumber\end{aligned}$$ The basic and elementary fact of the $\varphi$-expansion is $$\label{2.9}
\text{$a,b\in[0,1]$ and $x_0<x_0^\prime$} \implies
\overline{\varphi}(x_0+a)\leq \overline{\varphi}(x_0^\prime+b)\,.$$

We begin with two lemmas on 