Eigenvalues of the Thurston operator

Let $f:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ be a postcritically finite rational map. Let $\mathcal{Q}(\widehat{\mathbb{C}})$ be the space of meromorphic quadratic differentials on $\widehat{\mathbb{C}}$ with simple poles. We study the set of eigenvalues of the pushforward operator $f_:\mathcal Q(\hat{\mathbb C})\to \mathcal Q(\hat{\mathbb C})$\mathrm{.}Inparticular,weshowthatwhen$f:\mathbb C \to \mathbb C$isaunicriticalpolynomialofdegree$D$ with periodic critical point, the eigenvalues of $f_:\mathcal Q(\hat{\mathbb C})\to \mathcal Q(\hat{\mathbb C})$arecontainedintheannulus$\bigl{\frac{1}{4D}<|\lambda|<1\bigr}$andbelongto$\frac{1}{D} \mathbb U$where$\mathbb U$ is the group of algebraic units.