for every $q$, thus we have non trivial angle between the boundaries of each rational tongue $\cal T_{p/q}$. We call this the . $\blacklozenge$

![Arnol’d Tongues of the Angle family](anglefamily.pdf){width="5in" height="2.5in"}

Order of contact of the boundaries of the rational tongue {#sec_ooc}
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In the previous section we discussed about the possible angle between the boundaries of the rational tongue in a two parameter family. In many known example the function $\cal A_q(\phi)$ is constant and consequently the angle between these boundaries is zero. This is the case for the standard family $S_{t,a}$ for any rational tongue $\cal T_{p/q}$ with $q\geq 2$. In this situation it is interesting to study the .

Assume that for a certain two parameter family $F_{t,a}:\R\to \R$, the boundaries of the rational tongue $\cal T_{p/q}$ are functions of $a$ and the function $\cal A_q(\p) $ is constant for $q\ge 2$. Then we say that $k$ is the order of contact of the boundaries of $\cal T_{p/q}$ for $q\ge 2$ if $$|\gamma_{p/q}^r(a)-\gamma_{p/q}^l(a)|\underset{a\to 0}=\cal O(a^k)\quad\text{but}\quad |\gamma_{p/q}^r(a)-\gamma_{p/q}^l(a)|\underset{a\to 0}\ne\cal O(a^{k+1}).$$

The order of contact of the boundaries of the rational tongues in the standard family is a known fact. Arnol’d has shown that the order of contact of the 