Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds

Figure 3.1.  For the proof of Lemma 3.2. In the left panel, we take the 2-dimensional intersection of $ \Omega $ with the projective plane $ P $ determined by $ x, \xi, $ and $ y $. In the right panel, we take a sequence 2-dimensional intersections of $ \Omega $ with the projective plane $ P_n $ determined by the projective lines $ \overline{x z_n} $ and $ \overline{{y z_n}} $, and see that $ \beta_{z_n}(x, y) = \beta_{z_n}(x, x_n) = \frac12\log[x_n^-:x:x_n:x_n^+] $ where $ x_n $ is as pictured. In Lemma 3.2 we confirm that if $ \xi $ is smooth, then the image on the right converges to the image on the left, and $ \beta_\xi(x, y) = \frac12\log[x^-:x:\bar{x}:\xi] $ as pictured in the left panel

Figure 3.2.  For the proof of Lemma 3.4. For clarity and simplicity, the figure only depicts the case where $ \Omega $ is two-dimensional and $ x, y $ are such that $ \beta_\xi(x, y) = 0 $. By Lemma 3.2, to show that the Busemann functions $ \beta_{\xi_n}(x, y) $ converge to $ \beta_\xi(x, y) $ as $ \xi_n $ converges to $ \xi $, it suffices to show that $ q_n $ converges to $ q $