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

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  • We study the geodesic flow of a class of 3-manifolds introduced by Benoist which have some hyperbolicity but are non-Riemannian, not CAT(0), and with non-$ C^1 $ geodesic flow. The geometries are nonstrictly convex Hilbert geometries in dimension three which admit compact quotient manifolds by discrete groups of projective transformations. We prove the Patterson–Sullivan density is canonical, with applications to counting, and construct explicitly the Bowen–Margulis measure of maximal entropy. The main result of this work is ergodicity of the Bowen–Margulis measure.

    Mathematics Subject Classification: 37D40.

    Citation:

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  • 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 $

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