$t_0$ and $x_0$ such that $(t_0,a_0)\in \gamma_{p/q}^l$ and $(a_0,t_0,x_0)$ satisfies the following equations $$\begin{aligned}
P(a,t,x) &=S_{t,a}^{\circ q}(x)-x-p=0, \\
Q(a,t,x) &= \frac{\partial}{\partial x}S_{t,a}^{\circ q}(x)-1=0.\label{rel3}\end{aligned}$$ We would try to use the implicit function theorem to obtain that $t$ and $x$ could be expressed as analytic functions of $a$ around $a_0$.

Let’s show that $\ds\frac{\partial}{\partial t}S_{t,a}^{\circ q}(x)\ge 1$ by induction for any triplet $(a,t,x)$. For a fixed value of $a$ define $S(x)=x+a\sin(2\pi x)$ and $S_t(x)=S(x)+t=S_{t,a}(x)$ so that $S_t^{\circ q}(x)=S_{t,a}^{\circ q}(x)$. The statement is true for $q=1$ clearly. Suppose it is true for $q-1$. We note that $$S_t^{\circ q}(x)=S(S_t^{\circ q-1}(x))+t.$$ Thus $$\frac{\partial}{\partial t}S_{t}^{\circ q}(x)=\frac{\partial}{\partial x}S|_{S_t^{\circ q-1}(x)}\cdot \frac{\partial}{\partial t}S_t^{\circ q-1}(x)+1.$$ Since $\ds \frac{\partial}{\partial x}S(x)=\ds\frac{\partial}{\partial x}S_{t,a}(x)>0$ for $S_{t,a}$ being an increasing diffeomorphism and $\ds \frac{\partial}{\partial t}S_t^{\circ q-1}(x) \ge 1$ by induction, it follow that $\ds \frac{\partial}{\partial t}S_t^{\circ q-1}(x)\ge 1$ for any fixed $a$ within its domain. This implies that $\ds \frac{\partial}{\partial t}P(a,t,x)\ge 1$.

We can see that $\ds \frac{\partial}{\partial x}P(a_0,t_0,x_0)=0$ by Equation (\[rel3\]). By choice $x_0$ is a multiple fixed point of $S_{t_0,a_0}^{\circ q}-p$ and according to Proposition \[prop\_boundarystandardmultiple\], the parabolic multiplicity is $\nu=1$: $$S_{t_0,a_0}^{\circ q}(z)-p=z+C(z-z_0)^{2}+\cal O((z-z_0)^{3})\quad \text{with}\quad C\neq 0.$$ Thus $\ds \frac{\partial^2}{\partial x^2}S_{t_0,a_0}^q=2C$ is non zero at $x_0$. Consequently $\ds \frac{\partial Q}{\partial x}$ is non 