Dynamical zeta functions and resonance chains for infinite-area hyperbolic surfaces with large funnel widths

TL;DR

This paper investigates the resonance structures of infinite-area hyperbolic surfaces with large funnels, establishing a connection with metric graphs and demonstrating the existence of approximate resonance chains in specific geometric regimes.

Contribution

It introduces a novel approach linking resonance sets of hyperbolic surfaces to metric graphs using the spine graph construction, and proves the existence of resonance chains in the long-boundary-length limit.

Findings

Resonance sets of hyperbolic surfaces relate to those of metric graphs.

Existence of approximate resonance chains in long-boundary-length regimes.

Generalization of previous results for three-funneled spheres.

Abstract

We quantitatively relate the resonance sets of topologically finite infinite-area hyperbolic surfaces with no cusps to the resonance sets of certain metric graphs via the spine graph construction. In particular, we prove the existence of approximate resonance chains in resonance sets of these surfaces in the long-boundary-length regime. Our results are similar in spirit to those obtained in recent independent work by Li-Matheus-Pan-Tao, although our perspective and hypotheses are somewhat different. Our results also generalize older results obtained for three-funneled spheres by Weich. We primarily make use of transfer operators for holomorphic iterated function schemes, along with certain geometric bounds.