Genus one Birkhoff sections for geodesic flows on orbifolds

TL;DR

This paper demonstrates that geodesic flows on certain hyperbolic 2-orbifolds have genus one Birkhoff sections, linking their dynamics to a well-understood suspension flow on the torus.

Contribution

It proves the existence of genus one Birkhoff sections for geodesic flows on specific hyperbolic orbifolds, extending the understanding of their dynamical structure.

Findings

Existence of genus one Birkhoff sections for the flows.

Flow is almost equivalent to a suspension flow on the torus.

Results apply to orbifolds with up to 2g+6 conic points.

Abstract

For $\mathcal{O}$ a hyperbolic orientable 2-orbifold of genus $g$ with at most $2g+6$ conic points, we prove that the geodesic flow on the unitary tangent bundle$\mathrm{T}^1\mathcal{O}$ admits a Birkhoff section whose genus is one. Together with a result of Minakawa, this implies that this flow is almost equivalent to the suspension flow of the $(\begin{smallmatrix}2\&1\\1\&1\end{smallmatrix})$-map on the torus.