$x_0\in \R$ then $(t,a)\in \cal T_{p/q}$. We will show that $(t,a)$ is in the boundary of $\cal T_{p/q}$ by contradiction. If it were not in the boundary of the tongue, then $S_{t,a}^{\circ q}-p$ would take both positive and negative values. In particular, there would be a point $x_2$ (which a priori might be equal to $x_0$) such that $S_{t,a}^{\circ q}-p$ takes positive values for $x<x_2$ close to $x_2$ and takes negative values for $x>x_2$ close to $x_2$. Then, $x_2$ would be either an attracting fixed point of $S_{t,a}^{\circ q}-p$ or a multiple fixed point of $S_{t,a}^{\circ q}-p$ with parabolic multiplicity $\nu=2$ and $2$ real attracting directions. The latter is not possible. So, $S_{t,a}$ would have an attracting cycle. Again, this is not possible since this attracting cycle would have to attract the orbit of a critical value of $S_{t,a}$ whereas the orbit of the critical orbits of $S_{t,a}$ are attracted by the cycle of $S_{t,a}$ containing $x_0$.

The regularity of boundary curves
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In this section we try to prove that the boundary of $\cal T_{p/q}$ is the union of two Lipschitz curves. Hall has proved this in [@Hall] under the assumption that $\phi$ is $C^1$, using Implicit function theorem. 