Eigenvalues of the Thurston operator

TL;DR

This paper investigates the eigenvalues of the Thurston operator associated with postcritically finite rational maps, revealing bounds and algebraic properties for unicritical polynomials with periodic critical points.

Contribution

It characterizes the eigenvalues of the Thurston operator for specific rational maps, especially unicritical polynomials, including bounds on their magnitude and algebraic structure.

Findings

Eigenvalues lie within a specific annulus for unicritical polynomials.

Eigenvalues are contained in the scaled unit group   U.

Provides bounds <||<1 for eigenvalues.

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 $f_*:\mathcal Q(\hat{\mathbb C})\to \mathcal Q(\hat{\mathbb C})$. 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.