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

Abstract

We consider rational maps $f$ on the Riemann sphere $\hat{\mathbb{C}}$ with an $f$-invariant set $P\subset \hat{\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the corresponding Thurston pullback map $\ddot{I}{ƒ}_f$ on the completion $\overline{{\mathcal{T}}_P}$ of the associated Teichmüller 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 $\ddot{I}{ƒ}_f$-invariant tessellation by ideal hyperbolic triangles.

12 pages