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

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

Harrison Bray

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

Received  June 15, 2017 Revised  April 13, 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.

Keywords: Hilbert geometry, geodesic flow, nonuniform hyperbolicity, conformal densities, measure of maximal entropy.
Mathematics Subject Classification: 37D40.
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.  Google Scholar

[2]

J. P. Benzécri, Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France, 88 (1960), 229-332.   Google Scholar

[3]

Y. Benoist, Convexes divisibles. I, in Algebraic Groups and Arithmetic, Tata Inst. Fund. Res., Mumbai, 2004,339–374.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[7]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[8]

H. Bray, Geodesic flow of nonstrictly convex Hilbert geometries, to appear, Annales de l'Institute Fourier, 2020. Google Scholar

[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.  Google Scholar

[10]

M. Crampon, Entropies of strictly convex projective manifolds, J. Mod. Dyn., 3 (2009), 511-547.  doi: 10.3934/jmd.2009.3.511.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Crampon, Lyapunov exponents in Hilbert geometry, Ergodic Theory Dynam. Systems, 34 (2014), 501-533.  doi: 10.1017/etds.2012.145.  Google Scholar

[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.  Google Scholar

[15]

E. Franco, Flows with unique equilibrium states, Amer. J. Math., 99 (1977), 486-514.  doi: 10.2307/2373927.  Google Scholar

[16]

E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91 (1939), 261-304.   Google Scholar

[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.  Google Scholar

[18]

G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.  doi: 10.1007/s000390050025.  Google Scholar

[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.  Google Scholar

[20]

A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  doi: 10.2307/1971239.  Google Scholar

[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.  Google Scholar

[22]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.  doi: 10.1007/BF02392046.  Google Scholar

[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.  Google Scholar

[24]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003), vi+96pp. doi: 10.24033/msmf.408.  Google Scholar

[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. Google Scholar

[26]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.   Google Scholar

[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.  Google Scholar

[28]

C. Vernicos, Asymptotic volume in Hilbert geometries, Indiana Univ. Math. J., 62 (2013), 1431-1441.  doi: 10.1512/iumj.2013.62.5138.  Google Scholar

show all references

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.  Google Scholar

[2]

J. P. Benzécri, Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France, 88 (1960), 229-332.   Google Scholar

[3]

Y. Benoist, Convexes divisibles. I, in Algebraic Groups and Arithmetic, Tata Inst. Fund. Res., Mumbai, 2004,339–374.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[7]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[8]

H. Bray, Geodesic flow of nonstrictly convex Hilbert geometries, to appear, Annales de l'Institute Fourier, 2020. Google Scholar

[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.  Google Scholar

[10]

M. Crampon, Entropies of strictly convex projective manifolds, J. Mod. Dyn., 3 (2009), 511-547.  doi: 10.3934/jmd.2009.3.511.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Crampon, Lyapunov exponents in Hilbert geometry, Ergodic Theory Dynam. Systems, 34 (2014), 501-533.  doi: 10.1017/etds.2012.145.  Google Scholar

[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.  Google Scholar

[15]

E. Franco, Flows with unique equilibrium states, Amer. J. Math., 99 (1977), 486-514.  doi: 10.2307/2373927.  Google Scholar

[16]

E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91 (1939), 261-304.   Google Scholar

[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.  Google Scholar

[18]

G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782.  doi: 10.1007/s000390050025.  Google Scholar

[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.  Google Scholar

[20]

A. Manning, Topological entropy for geodesic flows, Ann. of Math. (2), 110 (1979), 567-573.  doi: 10.2307/1971239.  Google Scholar

[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.  Google Scholar

[22]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273.  doi: 10.1007/BF02392046.  Google Scholar

[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.  Google Scholar

[24]

T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), 95 (2003), vi+96pp. doi: 10.24033/msmf.408.  Google Scholar

[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. Google Scholar

[26]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.   Google Scholar

[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.  Google Scholar

[28]

C. Vernicos, Asymptotic volume in Hilbert geometries, Indiana Univ. Math. J., 62 (2013), 1431-1441.  doi: 10.1512/iumj.2013.62.5138.  Google Scholar

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
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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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