Ergodic Geodesic Flows and First Kind Flute Surfaces

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.

💡 Research Summary

The paper studies a special class of infinite‑type hyperbolic Riemann surfaces called flute surfaces. A flute surface is obtained by gluing together an infinite chain of pairs of pants so that the isolated ends (punctures) accumulate to a single non‑isolated end. Such a surface is completely described by its Fenchel–Nielsen coordinates ((\ell_n,t_n)_{n\ge1}), where (\ell_n>0) are the cuff lengths and (t_n\in