$\gamma_{\alpha} : (a_{\min},a_{\max}) \to \R$; which one has to study to understand the boundaries of the tongues. The existence of the interval $\left[\gamma_{p/q}^l(a),\gamma_{p/q}^r(a)\right]$ and the singleton set $\{\gamma_{\alpha}(a)\}$ are guaranteed by Lemma \[lemma\_transincrease\] and Proposition \[prop\_transincrease\].

Note that when $a=0$, the map $F_{t,a}=F_{t,0}:x\mapsto x+t$ is a translation and any rational tongue $\cal T_{p/q}$ grows (see Figure \[fig\_stdfamily\]) from the level $a=0$ in the parameter plane. A rational tongue $\cal T_{p/q}$ usually looks like union of two horn like regions which are meeting each other at a point at the level $a=0$, whereas an irrational tongue in the parameter space is a curve. Hence to understand how these rational tongues are growing from the level $a=0$, one has to understand well the functions $\gamma_{p/q}^{l}, \gamma_{p/q}^r$; which are the boundaries of $\cal T_{p/q}$. It is interesting to study whether $\gamma_{p/q}^l$ intersects $\gamma_{p/q}^r$ at a non zero angle or they have a common tangency at the level $a=0$. In case of common tangency, for $q\ge2$, we can study the of the boundaries of $\cal T_{p/q}$, which is given by the order of terms upto which the asymptotic expansions of the boundary curves at $a=0$ are identical. This is a difficult question in general, 