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

TL;DR

The paper develops a dynamical-systems framework for infinite-area hyperbolic surfaces by relating surface resonances to the zeros of a graph-derived dynamical zeta function via the spine construction. Using flow-adapted holomorphic IFSes and nuclear operator theory, it proves that in the large-funnel-width regime the surface zeta function $d_X(s,z)$ closely matches its graph counterpart $\widetilde{d}_{\Gamma}(s,z)$ on compact sets, with an exponentially small error in $\eta$. This yields resonance convergence to approximate resonance chains on scaled graphs and demonstrates explicit chain structures when edge lengths are integral or rational. The symmetric three-funneled sphere example illustrates the mechanism and provides quantitative asymptotics for the critical exponent, linking geometric limits to spectral data. The work extends Weich’s three-funneled results and aligns with recent LMPT work, offering a geometric, spine-based route to precise resonance control via cycle expansions.

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.

Figures (4)

• Figure 1: The metric ribbon graph used in Example \ref{['iharaex']}, with labeled directed edges.
• Figure 2: Examples of constructions used in this paper. Top left: a surface core $X_C$ with $g=1$ and $n=1$ (i.e., a one-holed torus). Top right: the same surface core with its spine graph marked. Notice that this metric ribbon graph has a single face. Bottom left: the same surface core, with its ribs added in red and the three corridors shaded. Bottom right: the same surface core, with its intercostals added in magenta and its two sectors shaded.
• Figure 3: An illustration of the map $\sigma_j$ used in the construction of the IFS. On the left, the green shading represent the long sector $H_j$, while the blue shading represents the core surface $X_C$. On the right, the colors represent the images of these regions under $\sigma_j$. Note that while the entirety of the image of $H_j$ is depicted, the entirety of the image of $X_C$ is not shown.
• Figure 4: An illustration of the map $f_{ij}$ and associated objects and quantities, including the quantities $\delta_i$, $\delta_j$, and $\kappa_{ij}$. Here, the green shading on the left represents $D_i$, while the green shading on the right represents its image $D_{ij}=f_{ij}(D_i)$. Note that $f_{ij}$ is not necessarily the unique linear map sending $D_i$ to $D_{ij}$; it may be a more complex isometry.

Theorems & Definitions (36)

• Theorem 1.1: Mazzeo-Melrose MM87, Guillopé-Zworski, Theorem 1 GZ95
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Corollary 1.5
• Corollary 1.6
• Definition 2.1
• Theorem 2.2: Patterson-Perry, Theorem 1.1 EPP01
• Example 2.1
• Theorem 2.3: Bowditch-Epstein BE88