a multiple fixed point with parabolic multiplicity $\nu =1$.

($\Rightarrow$) It follows from Lemma \[lemma\_parabolicNbdry\] that when $(t,a)$ is on the boundary of the Arnol’ d tongue $\cal T_{p/q}$, then $S_{t,a}^{\circ q}-p$ has a multiple fixed point $x_0\in \R$.

The map $S_{t,a}$ is an entire mapping with finite order of growth. According to a theorem of Ahlfors [@ahlfors], it has at most finitely many asymptotic values. Since $S_{t,a}$ commutes with translation by $1$, $a$ is an asymptotic value of $S_{t,a}$ if and only if $a+1$ is an asymptotic value of $S_{t,a}$. This shows that $S_{t,a}$ has no asymptotic value.

Modulo translation by $1$, the map $S_{t,a}$ has only two critical values. Their orbits under iteration of $S_{t,a}$ are symmetric with respect to the real axis. This shows that modulo translation by $1$, the map $S_{t,a}$ has at most $1$ parabolic cycle in $\R$, and either the parabolic multiplicity is $1$ and the attracting direction is contained in $\R$, or $\nu =2$ and the attracting directions are complex conjugate.

Since the sign of $S_{t,a}^{\circ q}-p$ does not change, there is an attracting direction in $\R$ and so, the parabolic multiplicity is $1$.

($\Leftarrow$) Assume $S_{t,a}^{\circ q}-p$ has a multiple fixed point 