Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$ -sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha \subset S^2\smallsetminus {P_f}$ , 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.

中文翻译：

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消除瑟斯顿障碍并控制曲线动力学

每个瑟斯顿地图 $f\冒号 S^2\右箭头 S^2$ 在一个 $2$ -领域 $S^2$ 引起乔丹曲线上的回拉操作 $\alpha \子集 S^2\smallsetminus {P_f}$ ， 在哪里 ${P_f}$ 是后临界集F。这里是同位素类 $[f^{-1}(\alpha)]$ （关系到 ${P_f}$ ) 仅取决于同位素类别 $[\阿尔法]$ 。我们研究了具有四个后临界点的 Thurston 映射的此操作。在这种情况下，地图的瑟斯顿障碍物F可以看作是固定点的回拉操作。我们证明如果瑟斯顿地图F具有双曲线轨道折叠和四个后临界点具有瑟斯顿障碍，那么可以在底层“炸毁”合适的弧 $2$ -球体并构建新的瑟斯顿地图 $\宽帽 f$ 为此，该障碍被消除。我们证明没有其他障碍出现，所以 $\宽帽 f$ 是通过有理图来实现的。特别是，这允许组合构造一大类具有四个后临界点的有理瑟斯顿图。我们还研究了迭代下回拉操作的动力学。我们展示了理性瑟斯顿图的一个子类，其中有四个后临界点，我们可以对全局曲线吸引子问题给出肯定的答案。