We consider certain analytic correspondences on a Riemann surface, and show that they admit a weak form of expansion. In terms of their algebraic encoding by bisets, this translates to contraction of group elements along sequences arising from iterated lifting. As an application, we show that for every non-exceptional rational map on P1 with 4 post-critical points, there is a finite collection of isotopy classes of curves into which every curve eventually lands under iterated lifting.

Submitted 22 July, 2024; originally announced July 2024.

arXiv:2407.15548