$f$ is said to be if the multiplier $f'(z_0)$ is a root of unity, i.e. $f'(z_0)=e^{i2\pi p/q}$ for some integers $p$ and $q$, and if $f^{\circ q}$ is not the equal to the identity near $z_0$. It is a fixed point if the multiplier is $1$.

It is well known (see [@milnor] Section 10 for example) that when $z_0$ is a parabolic fixed point of $f$ with multiplier $e^{i2\pi p/q}$, then there exists an integer $\nu \geq 1$ such that $$f^{\circ q}(z)=z+ C(z-z_0)^{\nu q+1}+\cal O((z-z_0)^{\nu q+2})\quad \text{with}\quad C\neq 0.$$ The integer $\nu$ is called the of $z_0$ as a fixed point of $f$. The map $f^{\circ q}$ has $\nu q$ attracting petals which are forward invariant and on which the sequence of iterates $f^{\circ nq}$ converges locally uniformly to $z_0$. Those form $\nu$ cycles of attracting petals. When $f:\C\to \C$ is an entire map, each cycle of attracting petals must attract the orbit of a critical value or an asymptotic value of $f$. In particular, the map $f$ has at least $\nu$ critical or asymptotic values.

\[prop\_boundarystandardmultiple\] In the standard family $(S_{t,a})_{(t,a)\in \cal P_S}$, a point $(t_0,a_0)\in \cal P_S$ is on the boundary of $\cal T_{p/q}$ $\iff$ $S_{t_0,a_0}^{\circ q}-p$ has 