boundaries of $\cal T_{p/q}$ is at least $q$ in [@Arnoldcontact] when Broer, Símo and Tatjer have proved that the order of contact of the boundaries of $\cal T_{p/q}$ is exactly $q$ in [@broeretal]. It is not known that this phenomenon is related to the fact that the parabolic multiplicity of $0$ as a fixed point of the map $z\mapsto e^{i2\pi p/q}ze^{\pi z}$ is equal to $1$.

The positive order of contact in the standard family $(S_{t,a})_{(t,a)\in\cal P_S}$ is due to some of its properties: the map $\Pi_a:x\mapsto ae^{i2\pi x}$ semiconjugates $S_{t,a}:\R\to \R$ to $s_{t,a}:C_{|a|}\to C_{|a|}$ where $s_{t,a}:\C^*\to \C^*$ is defined by $$s_{t,a}(z)=e^{i2\pi t}ze^{\pi (z-a^2/z)},$$ and as $a\to 0$, the maps $s_{t,a}$ converge uniformly to $s_t:z\mapsto e^{i2\pi t}ze^{\pi z}$ on compact subsets of $\C^*$. To study the order of contacts in similar families like the standard family, we introduce the notion of and family. Our aim would be to show that the order of contact of the boundaries of $\cal T_{p/q}$ is a multiple of $q$ in an admissible guided family.

Let $I$ and $J$ be open intervals of $\mathbb R$ such that $p/q\in I$ and $0 \in J$. A family $(F_{t,a}:\mathbb R \rightarrow \mathbb R)_{(t,a) \in I \times J}$ is 