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.
Categories: math.DS math.CV