Eigenvalues of the Thurston operator

Abstract

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 $f_{\ast }:\mathcal{Q}(\hat{\mathbb{C}})\to \mathcal{Q}(\hat{\mathbb{C}})$. 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 $f_{\ast }:\mathcal{Q}(\hat{\mathbb{C}})\to \mathcal{Q}(\hat{\mathbb{C}})$ are contained in the annulus $\{\frac{1}{4D}<|\lambda |<1\}$ and belong to $\frac{1}{D}\mathbb{U}$ where $\mathbb{U}$ is the group of algebraic units.