Genus one Birkhoff sections for geodesic flows on orbifolds

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.

Figures (17)

• Figure 1: On the left, a geodesic simple arc $\alpha$ on a 2-orbifold $\mathcal{O}$, a choice of a side $s$, and the corresponding rectangle $R(\alpha, s)$ in $\mathrm{T}^1\mathcal{O}$. In the center, the intersection of two geodesics on $\mathcal{O}$ forms a graph with four arcs adjacent to one vertex. Considering a (local) checkerboard coloring and the sides of the arcs determined by the white face, we obtain a (local) topological surface in $\mathrm{T}^1\mathcal{O}$ made of four rectangles. On the right, a smoothing of that surface that turns it into a smooth surface transverse to the geodesic flow.
• Figure 2: On the left, the $2g{+}2$-gon $A$, and its images $B, D, C$ under the symmetries $s_h$ and $s_v$ and the $\pi$-rotation $r$. On the right, the hyperbolic surface $\Sigma_g$ obtained after identifications of the sides of $A, B, C, D$, with the $2g{+}2$ closed geodesics $\alpha_1, \dots, \alpha_{2g+2}$ on it. These curves yield a graph $G_g$ on $\Sigma_g$.
• Figure 3: The dual graph $G^*_g$. Every depicted edge has in fact many parallel copies with the same origin and destination. Given a geodesic $\gamma$ on $\Sigma_g$, the faces and edges visited by $\gamma$ describe a path in $G^*_g$. Every time this path uses a bold edge corresponds to an intersection point of the orbit $\overset\rightarrow{\alpha}$ with the surface $S_g$ in $\mathrm{T}^1\Sigma_g$.
• Figure 4: The orbifold $\mathcal{O}_{0; 2, \dots, 2}$, with the $n$ conic points $b_1, \dots, b_n$ of order $2$ on the equator.
• Figure 5: In the case $p_{i-1}=p_i=p_{i+1}=2$, the surfaces $R(\beta_{i-1}, s_{i-1})$ and $R(\beta_{i}, s_{i})$ around the fiber $\mathrm{T}^1 b_i$ (right) and in the degree-2 cover (left).

Theorems & Definitions (14)

• Theorem A
• Remark 1.1
• Corollary B
• Definition 2.1: orientable hyperbolic 2-orbifold
• Theorem 2.2: see e.g. SKatok_FuchsianGroups
• Definition 2.3: geodesic flow
• Theorem 2.5
• proof
• Definition 3.1
• Remark 3.2