# American Institute of Mathematical Sciences

2020, 16: 305-329. doi: 10.3934/jmd.2020011

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

 Mathematical Sciences Department, George Mason University, 4400 University Dr., Fairfax, VA 22030, USA

Received  June 15, 2017 Revised  April 13, 2020 Published  November 2020

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.

Citation: Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011
##### References:
 [1] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, with an appendix by Misha Brin, DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9240-7. [2] J. P. Benzécri, Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France, 88 (1960), 229-332. [3] Y. Benoist, Convexes divisibles. I, in Algebraic Groups and Arithmetic, Tata Inst. Fund. Res., Mumbai, 2004,339–374. [4] Y. Benoist, Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math., 164 (2006), 249-278.  doi: 10.1007/s00222-005-0478-4. [5] T. Barthelmé, L. Marquis and A. Zimmer, Entropy rigidity of Hilbert and Riemannian metrics, Int. Math. Res. Not. IMRN, 2017 (2017), 6841-6866.  doi: 10.1093/imrn/rnw209. [6] R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X. [7] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. [8] H. Bray, Geodesic flow of nonstrictly convex Hilbert geometries, to appear, Annales de l'Institute Fourier, 2020. [9] M. Crampon and L. Marquis, Le flot géodésique des quotients géométriquement finis des géométries de Hilbert, Pacific J. Math., 268 (2014), 313-369.  doi: 10.2140/pjm.2014.268.313. [10] M. Crampon, Entropies of strictly convex projective manifolds, J. Mod. Dyn., 3 (2009), 511-547.  doi: 10.3934/jmd.2009.3.511. [11] M. Crampon, Dynamics and Entropies of Hilbert Metrics, Institut de Recherche Mathématique Avancée, Université de Strasbourg, Strasbourg, 2011. Thèse, Université de Strasbourg, Strasbourg, 2011. [12] M. Crampon, The geodesic flow of Finsler and Hilbert geometries, in Handbook of Hilbert Geometry, IRMA Lect. Math. Theor. Phys., 22, Eur. Math. Soc., Zürich, 2014,161–206. [13] M. Crampon, Lyapunov exponents in Hilbert geometry, Ergodic Theory Dynam. Systems, 34 (2014), 501-533.  doi: 10.1017/etds.2012.145. [14] P. de la Harpe, On Hilbert's metric for simplices, in Geometric Group Theory, Vol. 1 (Sussex, 1991), London Math. Soc. Lecture Note Ser., 181, Cambridge Univ. Press, Cambridge, 1993, 97–119. doi: 10.1017/CBO9780511661860.009. [15] E. Franco, Flows with unique equilibrium states, Amer. J. Math., 99 (1977), 486-514.  doi: 10.2307/2373927. [16] E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91 (1939), 261-304. [17] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. [18] G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.  doi: 10.1007/s000390050025. [19] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. of Math. (2), 148 (1998), 291-314.  doi: 10.2307/120995. [20] A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  doi: 10.2307/1971239. [21] L. Marquis, Around groups in Hilbert geometry, in Handbook of Hilbert Geometry, IRMA Lect. Math. Theor. Phys., 22, Eur. Math. Soc., Zürich, 2014,207–261. [22] S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.  doi: 10.1007/BF02392046. [23] R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces, Ergodic Theory Dynam. Systems, 37 (2017), 939-970.  doi: 10.1017/etds.2015.78. [24] T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003), vi+96pp. doi: 10.24033/msmf.408. [25] É. Socié-Méthou, Comportements Asymptotiques et Rigidité en Géométrie de Hilbert, Institut de Recherche Mathématique Avancée, Université de Strasbourg, Strasbourg, 2000. Thèse, Université de Strasbourg, Strasbourg, 2000. Available from: http://irma.math.unistra.fr/annexes/publications/pdf/00044.pdf. [26] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202. [27] N. Tholozan, Volume entropy of Hilbert metrics and length spectrum of {H}itchin representations into PSL$(3, \mathbb{R})$, Duke Math. J., 166 (2017), 1377-1403.  doi: 10.1215/00127094-00000010X. [28] C. Vernicos, Asymptotic volume in Hilbert geometries, Indiana Univ. Math. J., 62 (2013), 1431-1441.  doi: 10.1512/iumj.2013.62.5138.

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##### References:
 [1] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, with an appendix by Misha Brin, DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9240-7. [2] J. P. Benzécri, Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France, 88 (1960), 229-332. [3] Y. Benoist, Convexes divisibles. I, in Algebraic Groups and Arithmetic, Tata Inst. Fund. Res., Mumbai, 2004,339–374. [4] Y. Benoist, Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math., 164 (2006), 249-278.  doi: 10.1007/s00222-005-0478-4. [5] T. Barthelmé, L. Marquis and A. Zimmer, Entropy rigidity of Hilbert and Riemannian metrics, Int. Math. Res. Not. IMRN, 2017 (2017), 6841-6866.  doi: 10.1093/imrn/rnw209. [6] R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X. [7] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. [8] H. Bray, Geodesic flow of nonstrictly convex Hilbert geometries, to appear, Annales de l'Institute Fourier, 2020. [9] M. Crampon and L. Marquis, Le flot géodésique des quotients géométriquement finis des géométries de Hilbert, Pacific J. Math., 268 (2014), 313-369.  doi: 10.2140/pjm.2014.268.313. [10] M. Crampon, Entropies of strictly convex projective manifolds, J. Mod. Dyn., 3 (2009), 511-547.  doi: 10.3934/jmd.2009.3.511. [11] M. Crampon, Dynamics and Entropies of Hilbert Metrics, Institut de Recherche Mathématique Avancée, Université de Strasbourg, Strasbourg, 2011. Thèse, Université de Strasbourg, Strasbourg, 2011. [12] M. Crampon, The geodesic flow of Finsler and Hilbert geometries, in Handbook of Hilbert Geometry, IRMA Lect. Math. Theor. Phys., 22, Eur. Math. Soc., Zürich, 2014,161–206. [13] M. Crampon, Lyapunov exponents in Hilbert geometry, Ergodic Theory Dynam. Systems, 34 (2014), 501-533.  doi: 10.1017/etds.2012.145. [14] P. de la Harpe, On Hilbert's metric for simplices, in Geometric Group Theory, Vol. 1 (Sussex, 1991), London Math. Soc. Lecture Note Ser., 181, Cambridge Univ. Press, Cambridge, 1993, 97–119. doi: 10.1017/CBO9780511661860.009. [15] E. Franco, Flows with unique equilibrium states, Amer. J. Math., 99 (1977), 486-514.  doi: 10.2307/2373927. [16] E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91 (1939), 261-304. [17] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. [18] G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.  doi: 10.1007/s000390050025. [19] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. of Math. (2), 148 (1998), 291-314.  doi: 10.2307/120995. [20] A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  doi: 10.2307/1971239. [21] L. Marquis, Around groups in Hilbert geometry, in Handbook of Hilbert Geometry, IRMA Lect. Math. Theor. Phys., 22, Eur. Math. Soc., Zürich, 2014,207–261. [22] S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.  doi: 10.1007/BF02392046. [23] R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces, Ergodic Theory Dynam. Systems, 37 (2017), 939-970.  doi: 10.1017/etds.2015.78. [24] T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003), vi+96pp. doi: 10.24033/msmf.408. [25] É. Socié-Méthou, Comportements Asymptotiques et Rigidité en Géométrie de Hilbert, Institut de Recherche Mathématique Avancée, Université de Strasbourg, Strasbourg, 2000. Thèse, Université de Strasbourg, Strasbourg, 2000. Available from: http://irma.math.unistra.fr/annexes/publications/pdf/00044.pdf. [26] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202. [27] N. Tholozan, Volume entropy of Hilbert metrics and length spectrum of {H}itchin representations into PSL$(3, \mathbb{R})$, Duke Math. J., 166 (2017), 1377-1403.  doi: 10.1215/00127094-00000010X. [28] C. Vernicos, Asymptotic volume in Hilbert geometries, Indiana Univ. Math. J., 62 (2013), 1431-1441.  doi: 10.1512/iumj.2013.62.5138.
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
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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