Correspondences on Riemann surfaces and non-uniform hyperbolicity

TL;DR

This paper studies certain correspondences on Riemann surfaces, demonstrating a form of hyperbolicity where loops shrink under lifting, and applies this to classify rational maps with four post-critical points via invariant curve collections.

Contribution

It introduces a weak hyperbolicity concept for correspondences on Riemann surfaces and applies it to classify rational maps with four post-critical points using invariant curve collections.

Findings

Long loops get shorter under lifting, indicating weak hyperbolicity.

Finite invariant collections of curves attract all curves under iterated lifting.

Provides a topological normal form for rational maps with four post-critical points.

Abstract

We consider certain correspondences on a Riemann surface, and show that they admit a weak form of hyperbolicity: sufficiently long loops get shorter under lifting at a fixed point and closing. In terms of their algebraic encoding by bisets, this translates to contraction of fundamental group elements along sequences arising from iterated lifting. As an application, we show that apart from the usual Latt\`es counterexamples, for any rational map on $\mathbb P^1$ with $4$ post-critical points, there is a finite invariant collection of isotopy classes of curves into which every curve is attracted under iterated lifting. More generally, among graphs of given complexity, there exists a finite invariant collect ion of isotopy classes of graphs into which every graph is attracted. Applied to sufficiently rich graphs, the graph attr actor provides a finite set of topological normal forms for the rational map. We also present a strategy towards proving the same statements for maps with more than $4$ post-critical points.