Correspondences on Riemann surfaces and non-uniform hyperbolicity

TL;DR

The paper develops a comprehensive framework of admissible complex correspondences on finite-area hyperbolic Riemann surfaces and proves a weak form of hyperbolicity for the pullback dynamics. By encoding dynamics via X-rays and bisets, it shows that for rational maps with four postcritical points (excluding Lattès examples) there exists a finite attractor for the iterated pullback of curves, trees, and graphs, yielding finite topological normal forms. The core technique combines hyperbolic geometry (thick-thin decompositions, cusp behavior) with an algebraic/biset formalism to control lengths and demonstrate contraction, culminating in a robust approach to Thurston-type obstruction questions. The results illuminate invariant finite structures (graphs, multicurves) capturing the dynamics and provide a roadmap toward extending the theory to higher postcritical cardinalities and dimensions. The work thus bridges complex dynamics, moduli/Teichmüller theory, and combinatorial topology to advance understanding of non-uniform contraction phenomena in Thurston-type settings.

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 Lattès 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.

Figures (3)

• Figure 1: Modeling the behavior of $F^{(-n)}$ near the cusp $x$, with image $y$. Arrows labeled $\cong$ are isomorphisms and those whose tails are curved are inclusions. Following the arrows along the left-hand side from bottom-right to top yields a univalent, single-valued map $h\coloneqq h^x_y$ locally of the form $z \mapsto az(1+zg(z))$ with $|g|<1/2$ if $\delta'$ is chosen sufficiently small.
• Figure 2:
• Figure 3: Top: dependence on index $m$ suppressed. The geodesic $p$ joins the cusps $x_i$ and $x_{i+1}$. The symbols involving $\delta$ are circumferences of cusp neighborhoods. Some traces of a homotopy between a subsegment of $p$ and $\ell_i$ are indicated in thick gray. Bottom: the lifts under $F^{-n}$ are indicated with under-tilde's.

Theorems & Definitions (58)

• Theorem 1: Finite Attractor for ${\sf X}$-rays
• Corollary 2
• proof : Outline of proof of Corollary \ref{['cor:attractor']}
• Theorem 3
• Conjecture 4
• Proposition 3.1
• Proposition 4.1: $(\delta,\zeta)$-thick-thin decomposition for pointed surfaces
• Proposition 4.2: $(\delta,\zeta)$-thick-thin decomposition for pointed loops
• Lemma 4.3: Winding numbers
• proof