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Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ inducesa 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 thisoperation for Thurston maps with four postcritical points. In this case aThurston obstruction for the map $f$ can be seen as a fixed point of thepull-back operation.  We show that if a Thurston map $f$ with a hyperbolic orbifold and fourpostcritical points has a Thurston obstruction, then one can "blow up" suitablearcs in the underlying $2$-sphere and construct a new Thurston map $\widehat f$for which this obstruction is eliminated. We prove that no other obstructionarises and so $\widehat f$ is realized by a rational map. In particular, thisallows for the combinatorial construction of a large class of rational Thurstonmaps with four postcritical points.  We also study the dynamics of the pull-back operation under iteration. Weexhibit a subclass of our rational Thurston maps with four postcritical pointsfor which we can give positive answer to the global curve attractor problem.

Eliminating Thurston obstructions and controlling dynamics on curves