## Abstract

Every Thurston map f : S^{2} → S^{2} on a 2-sphere S^{2} induces a pull-back operation on Jordan curves α ⊂ S^{2 \} P_{f}, where P_{f} is the postcritical set of f. Here the isotopy class [f^{−1}(α)] (relative to P_{f}) only depends on the isotopy class [α]. 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 f̂for which this obstruction is eliminated. We prove that no other obstruction arises and so f̂is 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.

Original language | English |
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Pages (from-to) | 2454-2532 |

Number of pages | 79 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 44 |

Issue number | 9 |

DOIs | |

Publication status | Published - 1 Sept 2024 |

## Keywords

- curve attractor
- intersection numbers
- Lattès maps
- obstructions
- Thurston maps