Ergodic Geodesic Flows and First Kind Flute Surfaces

TL;DR

This work analyzes flute surfaces through Fenchel–Nielsen coordinates $X=(\ell_n,t_n)_{n\in\mathbb{N}^*}$ and establishes precise divergences of cosh-weighted sums as criteria for two global geometric behaviors. It extends prior results by Pandazis–Šarić to twists in $t_n\in\{0,\tfrac{1}{2}\}$ and general twists, using restricted patchworks to connect local geometry with global type (parabolic vs. first kind) and geodesic completeness. A central contribution is a divergence criterion involving an explicit sequence $(u_n)$ such that if $\sum_{n=1}^{\infty}(e^{-\ell_{n+1}/2}+e^{-\ell_n/2})\cosh(u_n\ell_n+...+u_1\ell_1)=\infty$, the ideal vertices of the lift accumulate at a single boundary point, making the surface of the first kind; a parallel criterion characterizes parabolicity in the symmetric twist setting. The paper then generalizes to broader patchworks with $(v'_n)$ and $(w_n)$, yielding a comprehensive condition for first-kindness across all orthorays, and discusses the containment and limitations of parabolicity within this framework, including symmetry-driven equivalences and counterexamples. Overall, the results illuminate how cuff-length sequences $(\ell_n)$ and twist data $(t_n)$ govern global dynamical and analytic properties of flute surfaces, providing concrete, testable criteria for both parabolicity and first-kindness with potential extensions to wider twist regimes.

Abstract

We study flute surfaces and extend results of Pandazis and Šarić giving necessary and sufficient conditions on the Fenchel-Nielsen coordinates of the surface to be of the first kind. As a consequence of the first result, we characterize parabolic flute surfaces (i.e. flute surfaces with ergodic geodesic flow) with twist parameters in {0,1/2}, extending the work of Pandazis and Šarić.

Figures (22)

• Figure 1: Orientation of the $\alpha_n$ in $X$.
• Figure 2: Examples of upward and downward pentagons.
• Figure 3: Examples of forbidden configurations.
• Figure 4: Trirectangle
• Figure 5: Lift $\Tilde{X^*}$ of $X^*$ in the universal covering $\mathbb{H}$.

Theorems & Definitions (32)

• Theorem 1.1
• Conjecture 1
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Definition 2.7
• Lemma 2.8