\ud{u}\preceq\sigma^n\ud{u}\preceq \ud{v}\quad \forall\,n\geq 0\quad\text{and}\quad \ud{u}\preceq \sigma^n\ud{v}\preceq \ud{v}\quad \forall\,n\geq 0\,.$$ If $k=0$ we also assume that $\sigma\ud{u}\preceq\sigma\ud{v}$. Let $\bar{\alpha}$ and $\bar{\beta}$ be 