Puncture-Forgetting Maps for Measured Foliations and Applications in Teichmüller Space and Complex Dynamics

We introduce puncture-forgetting maps for measured foliations and investigate their relations with mapping class groups, Teichm\"uller spaces, and extremal length. To this end, we develop the notions of cube complexes of pre-homotopic multicurves and tree coordinate systems on CAT(0) cube complexes. As an application to the dynamics of post-critically finite rational maps on the Riemann sphere, we obtain a partial result toward the finite curve attractor conjecture posed by Kevin Pilgrim. We also apply these methods to uncover a relation between horospheres and geodesic flows in the universal curve over Teichm\"uller space.