this section our aim is to study the boundaries of the rational Arnol’d tongues in a two parameter real analytic family $(F_{t,a}={\rm Id} + t + a\phi)_{(t,a)\in \cal P}$ where $\phi$ is a real analytic function.

To begin, let us fix one parameter $a=a_0\in (a_{\min},a_{\max})$ in the two parameter family and for simplicity, let us write $F_t$ for $F_{t,a_0}$. Then, the family$(F_t)_{t\in \R}$ is a one parameter increasing family in the sense that $F_{t}>F_{t'}$ if $t>t'$.

Let us fix a rational number $p/q$. As mentioned in the previous section, the function $\cal H:t\mapsto \trans(F_t)$ is non decreasing and the set $\cal H^{-1}(p/q)$ is a closed interval $[t^l,t^r]$ by Proposition \[prop\_transincrease\]. We have the following characterization of $t^l$ and $t^r$.

\[lemma\_parabolicNbdry\] We have the following equivalences:

-   $t=t^r$ if and only if $F_t^{\circ q}(x)\geq x+p$ for all $x\in \R$ and $F_t^{circ q}(x_0)= x_0+p$ for some point $x_0\in \R$.

-   $t=t^l$ if and only if $F_t^{\circ q}(x)\leq x+p$ for all $x\in \R$ and $F_t^{circ q}(x_0)= x_0+p$ for some point $x_0\in \R$.

-   $t\in (t^l,t^r)$ if and only if $F_t^{\circ q}-{\rm Id}-p$ takes both positive and negative values.

Set $G_t\eqdef F_t^{\circ q}-{\rm Id}-p$. According to Proposition \[prop\_transratpq\], $t\in [t^l,t^r]$ if and only if 