Ergodic Geodesic Flows and First Kind Flute Surfaces

TL;DR

This paper characterizes flute surfaces with ergodic geodesic flows by analyzing Fenchel-Nielsen coordinates, extending previous results to identify conditions for surfaces of the first kind and specific parabolic flute surfaces.

Contribution

It provides necessary and sufficient conditions for flute surfaces to be of the first kind and characterizes parabolic flute surfaces with specific twist parameters.

Findings

Conditions for first kind flute surfaces established

Characterization of parabolic flute surfaces with twist 0 or 1/2

Extension of previous work on ergodic geodesic flows

Abstract

We study flute surfaces and extend results of Pandazis and \v{S}ari\'c 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 \v{S}ari\'c.