The limit set of non-orientable mapping class groups

Figure 1.  A quadratic differential $ q $ on $ \mathscr{S}_2 $ given by the slit torus construction

Figure 2.  A quadratic differential on $ \mathscr{N}_3 $

Figure 3.  Two possibilities for first return to $ \eta_i $: on the left, the arc returns without the local orientation flipping, and on the right, the arc returns with the local orientation flipped

Figure 4.  The curve $ c_i $ is colored blue. Since the leaf from $ p_0 $ returns with the local orientation flipped to both $ p_{i-1} $ and $ p_{i} $, the curve $ c_i $ is two-sided

Figure 5.  Construction of the blue curve $ c_i $ when the leaf always returns with orientation flipped from the "up" or "down" direction

Figure 6.  The arcs $ \widetilde{q_1} $ and $ \widetilde{q_2} $

Figure 7.  Homotopy taking $ q_2 $ to $ q_1 $

Figure 8.  The curves restricted to a pair of pants

Figure 9.  The right angled hexagon obtained by cutting the pants along the seams

Figure 10.  A DQD on $ \mathscr{N}_4 $

Figure 11.  A DQD on $ \mathscr{N}_9 $. To display the gluing maps on the small slits, we have a zoomed in picture in the ellipses