  Let $f:\hat{\mathbb C}\to \hat{\mathbb C}$ be a postcritically finite
rational map. Let $\mathcal Q(\hat{\mathbb C})$ be the space of meromorphic
quadratic differentials on $ \hat{\mathbb C}$ with simple poles. We study the
set of eigenvalues of the pushforward operator $f_*:\mathcal Q(\hat{\mathbb
C})\to \mathcal Q(\hat{\mathbb C})$. In particular, we show that when
$f:\mathbb C \to \mathbb C$ is a unicritical polynomial of degree $D$ with
periodic critical point, the eigenvalues of $f_*:\mathcal Q(\hat{\mathbb C})\to
\mathcal Q(\hat{\mathbb C})$ are contained in the annulus
$\bigl\{\frac{1}{4D}<|\lambda|<1\bigr\}$ and belong to $\frac{1}{D} \mathbb U$
where $\mathbb U$ is the group of algebraic units.
