1. Thurston's pullback map, invariant covers, and the global dynamics on curves
- Author
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Bonk, Mario, Hlushchanka, Mikhail, and Lodge, Russell
- Subjects
Mathematics - Dynamical Systems ,Mathematics - Complex Variables ,37F10, 37F20 - 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., Comment: 12 pages
- Published
- 2024