Eliminating Thurston obstructions and controlling dynamics on curves

TL;DR

This paper investigates Thurston maps with four postcritical points, showing how to eliminate obstructions via arc modifications, leading to rational maps, and explores the dynamics of the pull-back operation on curves.

Contribution

It introduces a method to remove Thurston obstructions for maps with four postcritical points, enabling the construction of rational maps and analyzing their curve dynamics.

Findings

Obstructions correspond to fixed points of the pull-back operation.

Arc blow-up technique eliminates obstructions, producing realizable rational maps.

Identifies a subclass with positive solutions to the global curve attractor problem.

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.