the standard family (Theorem \[theo\_bdanalytic\]). We can also derive the order of contact of the boundaries of the rational tongues in the Blaschke fraction family using this technique of admissible and guided family.

\[fig\_stdfamily\] ![Some rational Arnol’d tongues in the standard family.](sinefamily3.png "fig:"){width="5.5in" height="2.5in"}

Preliminaries
=============

Before we proceed further, we recall some basic facts about translation and rotation numbers. By definition the rotation number is real, so it is either rational or irrational. The two cases and the corresponding dynamics are discussed in the following results. We deal with the case of rational rotation number first. Whenever we write $p/q$ for a rational number, we implicitly assume that $p$ and $q$ are coprime.

\[prop\_transratpq\] If $\rho(f)\in \Q/\Z$ then $f:\T\to \T$ has a periodic point. More precisely, if $\trans(F) =p/q\in \Q$ then there is a point $a\in \R$ such that $F^{\circ q}(a)=a+p$.

Note that $G\eqdef F^{\circ q}-{\rm Id}-p$ vanishes on the whole $F$-orbit of $a$, in particular on the set $\{a, F(a), \ldots,
F^{\circ (q-1)}(a)\}$ with $q$ points whose image in $\T$ is a cycle of $f$. We say that such a cycle has rotation number $p/q$. The derivative of $G$ is constant along the orbit of $a$ under iteration 