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Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds
Mathematical Sciences Department, George Mason University, 4400 University Dr., Fairfax, VA 22030, USA |
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.
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. 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. |
[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. 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.
|
[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. |
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. |
[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. 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. |
[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. 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.
|
[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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