Genus one Birkhoff sections for geodesic flows on orbifolds

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.

💡 Research Summary

The paper studies the geodesic flow on the unit tangent bundle of an orientable hyperbolic 2‑orbifold (\mathcal O). An orbifold of type ((g;p_{1},\dots ,p_{n})) has underlying surface of genus (g) and a finite set of conic points of orders (p_{i}\ge2). The authors focus on “small‑type” orbifolds, namely those with either genus 0 or, if (g\ge1), with at most (2g+6) conic points. For such orbifolds they prove that the geodesic flow (\Phi_{\text{geod}}) admits a Birkhoff section of genus 1 (a torus). This result (Theorem A) generalises Fried’s classical theorem for hyperbolic surfaces and recent constructions for special orbifolds.

A Birkhoff section is an immersed surface whose interior is transverse to the flow, whose boundary consists of finitely many periodic orbits, and such that every orbit meets the surface within a uniform bounded time. If the section is embedded, the first‑return map on the surface is a diffeomorphism; for an Anosov flow this map is itself Anosov. The authors construct explicit embedded genus‑1 sections by a two‑step geometric procedure.

The first tool, introduced in Section 3, is the vertical rectangle (R(\alpha,s)). Given a simple geodesic arc (\alpha) (no interior conic points) and a choice of side (s) (i.e. a co‑orientation), one considers all unit tangent vectors based on points of (\alpha) that point toward the chosen side. This set is a topological disc whose interior is transverse to the flow; its boundary consists of two “horizontal” arcs (the lifts of (\alpha) in both directions) and two vertical arcs lying in the fibers over the endpoints of (\alpha). By gluing four such rectangles around a common endpoint (p) (a conic point or a regular point) and performing a small isotopy near the fiber over (p), one obtains a surface (S_{p}) whose boundary consists of eight horizontal arcs and four vertical arcs that cancel pairwise. When the order of the conic point is 2, this construction yields an embedded torus with all boundary components tangent to the flow (the “negative” case).

The second tool, the butterfly surface, handles conic points of order at least 3. Around a conic point one draws four incident simple arcs (\alpha_{1},\dots ,\alpha_{4}) meeting at the point, colors the four incident sectors alternately, and chooses the side of each arc that corresponds to the white sector. The four associated vertical rectangles glue together into a surface whose boundary again consists of horizontal lifts of the arcs and vertical fiber pieces. After smoothing near the central fiber, the resulting surface is a torus whose boundary consists solely of periodic orbits. By varying the pattern of arcs and the choice of white sectors, the authors treat spheres with many order‑2 points, spheres with order‑≥ 3 points, and higher‑genus orbifolds with a mixture of orders.

In Sections 4 and 5 the authors combine the two constructions. For a genus‑(g) orbifold with (2g+2\le n\le 2g+6) conic points of order ≥ 3 they build a chain of butterfly surfaces linked by vertical rectangles, ensuring that all boundary components glue up to a single torus. When some conic points have order 2, they insert vertical‑rectangle pieces (the “type‑2” pieces) to adjust the Euler characteristic. The final surface is always embedded, has genus 1, and satisfies the Birkhoff conditions: its interior is transverse, its boundary consists of periodic orbits, and every orbit meets it within a uniform time bound.

Having a genus‑1 Birkhoff section implies that the first‑return map on the torus is an Anosov diffeomorphism. The authors show that this map has positive trace and therefore is conjugate to a linear hyperbolic automorphism of the torus. In fact they identify it with the matrix (\begin{pmatrix}2&1\1&1\end{pmatrix}). By a theorem of Minakawa (cited as