Slope gap distribution of saddle connections on the $ 2n $-gon

Figure 1.  The regular decagon, $ O_{10} $. Black vertices are identified together and red vertices are identified together

Figure 2.  A saddle connection $ \gamma $ on $ X_{10} $

Figure 3.  An example of $ \mathcal{S} $ for $ n = 7 $

Figure 4.  Cylinder decomposition of $ O_{2n} $ in the $ \theta = \pi/2n $ direction

Figure 5.  A staircase representative of $ O_{12} $

Figure 6.  The saddle connections of interest for $ \Omega_1 $ (in magenta) on $ \mathcal{S} $ for $ n = 7 $

Figure 7.  An illustration of $ \Omega_2 $ with coordinates in $ \mathbb{R}^2 $, for any $ n $

Figure 8.  Color-coded relevant saddle connections for $ n = 7 $

Figure 9.  The corresponding division of $ \Omega_1 $ for $ n = 7 $ using coordinates in $ \mathbb{R}^2 $. Each saddle connection is color coded to match the region it wins in. The purple region is $ P_1 $, the blue region is $ P_2 $, the green region is $ P_3 $, the yellow region is $ P_4 $, the gray region is $ P_5 $, the orange region is $ P_6 $, and the red region is $ P_7 $

Figure 10.  Numerical computation of the volume of $ {\rm SL}_2(\mathbb{R})/\Gamma_{O_{2n}} $ using our Poincaré section and return time function compared to the exact volume. Within the limits of our computational software, there was essentially no error in the computed volume

Figure 11.  Gap distributions for the $ 2n $-gon for several values of $ n $. Note the changed scaling on the horizontal axis in the last two graphs

Figure 12.  An illustration of $ P_1 $ with coordinates in $ \mathbb{R}^2 $, for any $ n $, along with the hyperbola $ R(x,y) = t $ (in black; here $ t = 3 $)

Figure 13.  In this example, $ n = 7 $. The black line is the hyperbola $ R(x,y) = 3.6 $

Figure 14.  In this example, $ n = 7 $ and $ i = 3 $. The black curve is the hyperbola $ R(x,y) = 3.6 $

Figure 15.  This is the example of $ n = 7 $. The black curve is the equation $ R(x,y) = 2 $

Figure 16.  The region $ P_n $ with $ n = 7 $ with the graph of the equation $ R(x,y) = 3 $

Figure 17.  Time stamps for boundary crossings

Figure 18.  Plot of the upper bound from Theorem 5.1 (olive dashed line) and the actual number of non-differentiable points (pink squares) against $ n $

Figure 19.  Table of the upper bound from Theorem 5.1 and number of non-differentiable points for small $n$

Figure 20.  Labeling of $ \mathcal{S}' $ using $ \sigma $'s and $ \nu $'s for $ n = 7 $

Figure 21.  Labeling of $ \mathcal{S}' $ with $ \lambda $'s for $ n = 7 $

Figure 22.  An example of vertex labeling on $ \mathcal{S'} $ for $ n = 7 $

Figure 23.  If fix $ i = 2 $ in this case, we see that any vector linking $ L_j $ and $ L_{k} $ (black vectors) with $ k \leq i+1 $ while staying in the staircase is either steeper than $ \nu_1 $ (green vector) or $ \nu_2 $ (yellow vector)

Figure 24.  If fix $ i = 2 $ in this case, we see that any vector linking $ L_j $ and $ R_{k} $ with $ k \leq i+1 $ while staying in the staircase (black vectors) is either steeper than $ \nu_1 $ (green vector) or $ \nu_2 $ (yellow vector)

Figure 25.  For the example of $ k = 2 $, we see that any vector connecting some $ R_j $ or $ L_j $ (in this case $ L_0 $) and $ R_2 $ will pass above $ R_1 $ and hence be longer and shallower than $ \sigma_2 $

Figure 26.  For the example of $ k = 1 $, we see that any vector passing through $ v_1 $ is longer and shallower than $ \sigma_2 $

Figure 27.  If fix $ k = 1 $ and $ i = 2 $ in this case, we see that any vector passing through the side of length $ h_1 $ is steeper than $ \nu_1 $ (green vector), which in turn is steeper than $ \nu_2 $

Figure 28.  Vectors ending at $ L_k $ for any $ k $ (shown in black here) are necessarily steeper than the $ \nu $ vector that ends at $ L_k $ (shown in green or yellow), which is in turn steeper than every $ \sigma $ vector

Figure 29.  Let $ n = 7 $ and $ i = 3 $ (so the vector we are trying to show victory for is $ \sigma_3 $). Vectors ending at $ R_k $ for any $ k < i $ (shown in black here) are necessarily shallower and longer than the $ \sigma $ vector that ends at $ R_k $ (in orange), which in turn violates Condition 3

Figure 30.  Let $ n = 7 $ and $ i = 2 $ (so the vector we are trying to show victory for is $ \sigma_2 $). Vectors starting at $ R_j $ for any $ j \leq i $ (shown in black here) are necessarily steeper than the $ \sigma $ vector that starts at $ R_j $ (in red), which in turn violates Condition 2

Figure 31.  Let $ n = 7 $ and $ i = 3 $ (so the vector we are trying to show victory for is $ \sigma_3 $). Vectors starting at some $ L_j $ and ending at $ R_k $ for any $ k < i $ (shown in black here) are necessarily shallower and longer than the $ \sigma $ vector that ends at $ R_k $ (in orange), which in turn violates Condition 3

Figure 32.  Let $ n = 7 $. Vectors starting at $ L_j $ and ending at $ R_{j+1} $ are precisely the diagonals of the large rectangles, which have aspect ratio equal to $ 2 + 2\cos(\pi/n) $

Figure 33.  For instance, any vector passing through the side of length $ h_1 $ is steeper than $ \nu_1 $ (green vector), which in turn is steeper than every $ \sigma $ vector

Figure 34.  Fix $ i = 3 $. For instance, we see that any vector passing through $ v_1 $ is longer and shallower than $ \sigma_2 $, which fails Condition 3