Thurston's pullback map, invariant covers, and the global dynamics on curves

TL;DR

This paper studies the dynamics of Thurston's pullback map on Teichmüller space for rational maps with invariant sets, revealing conditions under which the dynamics are well-understood and identifying finite global curve attractors.

Contribution

It provides a framework to analyze Thurston's pullback map dynamics using invariant covers with good properties and links these to hyperbolic tessellations, advancing understanding of global dynamics on curves.

Findings

Thurston's pullback map dynamics are tractable with suitable invariant covers.

Existence of finite global curve attractors for certain rational maps.

Invariant tessellations by ideal hyperbolic triangles can be constructed under specific conditions.

Abstract

We consider rational maps $f$ on the Riemann sphere $\widehat {\mathbb{C}}$ with an $f$-invariant set $P\subset \widehat {\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the corresponding Thurston pullback map $\sigma_f$ on the completion $\overline{\mathcal{T}_P}$ of the associated Teichm\"uller space $\mathcal{T}_P$ with respect to the Weil-Petersson metric is easy to understand when $\overline{\mathcal{T}_P}$ admits a cover by sets with good combinatorial and dynamical properties. In particular, the map $f$ has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if $P$ contains all critical points of $f$ and each point in $P$ is periodic, then such a cover of $\overline{\mathcal{T}_P}$ can be obtained from a $\sigma_f$-invariant tessellation by ideal hyperbolic triangles.