of is $\ud{u}^\ab$ and the smallest one is $\ud{a}^\prime\ud{v}^\ab$.

\[pro3.4ter\] Assume that the $\varphi$-expansion is valid.\
1) Let $\ud{u}$ be a solution of , such that $\ud{u}\preceq\sigma^n\ud{u}$ for all $n\geq 1$, and let $\ud{v}$ be a solution of , such that $\sigma^n\ud{v}\preceq\ud{v}$ for all $n\geq 1$. Then $$\ud{u}\preceq\ud{u}^\ab\quad\text{and}\quad
\ud{v}^\ab\preceq\ud{v}\,.$$ 2) Let $\ud{u}$ be a solution of , and let $\ud{u}^\ab=(\ud{a})^\infty$ be periodic with minimal period $p>1$, and suppose that there exists $\ud{w}$ such that $$\forall n\colon\; \ud{u}\preceq\sigma^n\ud{u}\preceq
\ud{w}\quad\text{with}\quad \ophi^\ab(\ud{w})=1\;,\;
\ophi^\ab(\sigma\ud{w})\leq\gamma\,.$$ Then $$\label{3.3.3}
\ud{u}^{\ab}_*\preceq\ud{u}\preceq\ud{u}^\ab\quad\text{where}\quad
\ud{u}^{\ab}_*:=\ud{a}^\prime\ud{v}^\ab\;\text{and}\;
\ud{a}^\p:=(\tp\ud{a})(a_{p-1}-1)\,.$$ Moreover, $\ud{u}=\ud{u}^\ab$ $\iff$ $\ud{a}$ is a prefix of $\ud{u}$ $\iff$ $\ophi^\ab(\sigma^p\ud{u})<1$.\
3) Let $\ud{v}$ be a solution of , and let $\ud{v}^\ab=(\ud{b})^\infty$ be periodic with minimal period $q>1$, and suppose that there exists $\ud{w}$ such that $$\forall n\colon\; \ud{w}\preceq\sigma^n\ud{v}\preceq
\ud{v}\quad\text{with}\quad \ophi^\ab(\ud{w})=0\;,\;
\ophi^\ab(\sigma\ud{w})\geq\alpha\,.$$ Then $$\label{3.3.4}
\ud{v}^\ab\preceq\ud{v}\preceq
\ud{v}^{\ab}_*\quad\text{where}\quad
\ud{v}^{\ab}_*:=\ud{b}^\prime\ud{u}^\ab\;\text{and}\;
\ud{b}^\p:=(\tp\ud{b})(b_{q-1}+1)\,.$$ Moreover, $\ud{v}=\ud{v}^\ab$ $\iff$ $\ud{b}$ is a prefix of $\ud{v}$ $\iff$ $\ophi^\ab(\sigma^q\ud{v})>0$.

Shift space ${\mathbf\Sigma}(\ud{u},\ud{v})$ {#section3}
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Let $\ud{u}\in\tA^{\Z_+}$ and $\ud{v}\in\tA^{\Z_+}$, such that $u_0=0$, $v_0=k-1$ ($k\geq 2$) and holds. These assumptions are valid for the whole section, except subsection \[subsectionalgo\]. We study the shift-space $$\label{3.2.2}
\BSigma(\ud{u},\ud{v}):=\{\ud{x}\in\tA^{\Z_+}\colon \ud{u}\preceq
\sigma^n\ud{x}\preceq \ud{v}\quad \forall\,n\geq 0\}\,.$$ It is useful to extend the relation $\prec$ to words or to words and strings. We do 