Eliminating Thurston obstructions and controlling dynamics on curves

Abstract

Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha\subset S^2\setminus P_f$, where $P_f$ is the postcritical set of $f$. Here the isotopy class $[f^{-1}(\alpha)]$ (relative to $P_f$) only depends on the isotopy class $[\alpha]$. We study this operation for Thurston maps with four postcritical points. In this case a Thurston obstruction for the map $f$ can be seen as a fixed point of the pull-back operation. We show that if a Thurston map $f$ with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can "blow up" suitable arcs in the underlying $2$-sphere and construct a new Thurston map $\widehat f$ for which this obstruction is eliminated. We prove that no other obstruction arises and so $\widehat f$ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.