April  2019, 39(4): 1731-1744. doi: 10.3934/dcds.2019075

Entropy rigidity and Hilbert volume

1. 

Department of Math & Comp. Sci., Wesleyan University, 265 Church Street, Middletown, CT 06459, USA

2. 

Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA

Received  August 2017 Revised  September 2018 Published  January 2019

Fund Project: The second author was supported in part by NSF RTG grant 1045119

For a closed, strictly convex projective manifold that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson–Courtois–Gallot's entropy rigidity result to Hilbert geometries.

Citation: Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075
References:
[1]

I. Adeboye and D. Cooper, The area of convex projective surfaces and Fock–Goncharov coordinates, Journal of Topology and Analysis, (2018). (To appear). doi: 10.1142/S1793525319500560.  Google Scholar

[2]

W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, volume 61 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1985. doi: 10.1007/978-1-4684-9159-3.  Google Scholar

[3]

D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, volume 200 of Graduate Texts in Mathematics, Springer, New York, 2000. doi: 10.1007/978-1-4612-1268-3.  Google Scholar

[4]

Y. Benoist, Convexes divisibles. I, in Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, (2004), 339–374.  Google Scholar

[5]

Y. Benoist, Convexes hyperboliques et quasiisométries, Geom. Dedicata, 122 (2006), 109-134.  doi: 10.1007/s10711-006-9066-z.  Google Scholar

[6]

Y. Benoist and D. Hulin, Cubic differentials and finite volume convex projective surfaces, Geometry and Topology, 17 (2013), 595-620.  doi: 10.2140/gt.2013.17.595.  Google Scholar

[7]

Y. Benoist and D. Hulin, Cubic differentials and hyperbolic convex sets, Journal of Differential Geometry, 98 (2014), 1-19.  doi: 10.4310/jdg/1406137694.  Google Scholar

[8]

J.-P. Benzecri, Sur les variétés localement affines et localement projectives, Bulletin de la S. M. F., 88 (1960), 229-332.   Google Scholar

[9]

G. BessonG. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., 5 (1995), 731-799.  doi: 10.1007/BF01897050.  Google Scholar

[10]

G. BessonG. Courtois and S. Gallot, Minimal entropy and Mostow's rigidity theorems, Ergodic Theory and Dynamical Systems, 16 (1996), 623-649.  doi: 10.1017/S0143385700009019.  Google Scholar

[11]

J. Boland and F. Newberger, Minimal entropy rigidity for Finsler manifolds of negative flag curvature, Ergodic Theory and Dynamical Systems, 21 (2001), 13-23.  doi: 10.1017/S0143385701001055.  Google Scholar

[12]

H. Bray, Ergodicity of Bowen-Margulis measure for the Benoist 3-manifolds, preprint, arXiv: 1705.08519. Google Scholar

[13]

H. Busemann and P. J. Kelly, Projective Geometry and Projective Metrics, Academic Press Inc., New York, 1953.  Google Scholar

[14]

B. ColboisC. Vernicos and P. Verovic, L'aire des triangles idéaux en géométrie de Hilbert, Enseign. Math., 50 (2004), 203-237.   Google Scholar

[15]

C. Connell and B. Farb, Minimal entropy rigidity for lattices in products of rank one symmetric spaces, Comm. Anal. Geom., 11 (2003), 1001-1026.  doi: 10.4310/CAG.2003.v11.n5.a7.  Google Scholar

[16]

C. Connell and B. Farb, Some recent applications of the barycenter method in geometry, in Topology and Geometry of Manifolds, volume 71 of Proc. Sympos. Pure Math., American Math Society, (2003), 19–50. doi: 10.1090/pspum/071/2024628.  Google Scholar

[17]

C. Connell and B. Farb, The degree theorem in higher rank, Journal of Differential Geometry, 65 (2003), 19-59.  doi: 10.4310/jdg/1090503052.  Google Scholar

[18]

D. CooperD. D. Long and M. B. Thistlethwaite, Flexing closed hyperbolic manifolds, Topology, 11 (2007), 2413-2440.  doi: 10.2140/gt.2007.11.2413.  Google Scholar

[19]

D. CooperD. D. Long and S. Tillmann, On convex projective manifolds and cusps, Adv. Math., 277 (2015), 181-251.  doi: 10.1016/j.aim.2015.02.009.  Google Scholar

[20]

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

[21]

M. Crampon, Dynamics and Entropies of Hilbert Metrics, Thèse, Université de Strasbourg, Strasbourg, 2011.  Google Scholar

[22]

M. Crampon, The boundary of a divisible convex set, Publ. Mat. Urug., 14 (2013), 105-119.   Google Scholar

[23]

P. de la Harpe, On Hilbert's metric for simplices, in Geometric Group Theory, Vol. 1 (Sussex, 1991), volume 181 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, (1993), 97–119. doi: 10.1017/CBO9780511661860.009.  Google Scholar

[24]

R. Feres, The minimal entropy theorem and Mostow rigidity: after G. Besson, G. Courtois and S. Gallot, (available at http://www.math.wustl.edu/ feres/mostow.pdf), 1996. Google Scholar

[25]

W. M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom., 31 (1990), 791-845.  doi: 10.4310/jdg/1214444635.  Google Scholar

[26]

D. Johnson and J. Millson, Deformation spaces associated to compact hyperbolic manifolds, in Discrete Groups in Geometry and Analysis (New Haven, Conn., 1984), volume 67 of Progr. Math., Birkhäuser, Boston, (1987), 48–106. doi: 10.1007/978-1-4899-6664-3_3.  Google Scholar

[27]

V. A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 361-393.   Google Scholar

[28]

M. Kapovich, Convex projective structures on Gromov-Thurston manifolds, Geometry and Topology, 11 (2007), 1777-1830.  doi: 10.2140/gt.2007.11.1777.  Google Scholar

[29]

A. Katok, Entropy and closed geodesics, Ergodic Theory and Dynamical Systems, 2 (1982), 339-365.  doi: 10.1017/S0143385700001656.  Google Scholar

[30]

E. Leuzinger, Entropy of the geodesic flow for metic spaces and Bruhat-Tits buildings, Adv. Geom., 6 (2006), 475-491.  doi: 10.1515/ADVGEOM.2006.029.  Google Scholar

[31]

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

[32]

G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Publications Mathématiques de l'I.H.É.S., 34 (1968), 53-104.   Google Scholar

[33]

X. Nie, On the Hilbert geometry of simplicial Tits sets, Ann. Inst. Fourier (Grenoble), 65 (2015), 1005-1030.  doi: 10.5802/aif.2950.  Google Scholar

[34]

A. Papadopoulos and M. Troyanov, From Funk to Hilbert geometry, in Handbook of Hilbert Geometry, volume 22 of IRMA Lectures in Mathematics and Theoretical Physics, European Mathematical Society, Zurich, (2014), 33–68. Google Scholar

[35]

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

[36]

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

[37]

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

[38]

N. Tholozan, Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into $\text{PSL}(3,\mathbb{R})$, Duke Math. J., 166 (2017), 1377-1403.  doi: 10.1215/00127094-00000010X.  Google Scholar

[39]

T. Zhang, The degeneration of convex $\mathbb{RP}^2$ structures on surfaces, Proc. Lond. Math. Soc., 111 (2015), 967-1012.  doi: 10.1112/plms/pdv051.  Google Scholar

show all references

References:
[1]

I. Adeboye and D. Cooper, The area of convex projective surfaces and Fock–Goncharov coordinates, Journal of Topology and Analysis, (2018). (To appear). doi: 10.1142/S1793525319500560.  Google Scholar

[2]

W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, volume 61 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1985. doi: 10.1007/978-1-4684-9159-3.  Google Scholar

[3]

D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, volume 200 of Graduate Texts in Mathematics, Springer, New York, 2000. doi: 10.1007/978-1-4612-1268-3.  Google Scholar

[4]

Y. Benoist, Convexes divisibles. I, in Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, (2004), 339–374.  Google Scholar

[5]

Y. Benoist, Convexes hyperboliques et quasiisométries, Geom. Dedicata, 122 (2006), 109-134.  doi: 10.1007/s10711-006-9066-z.  Google Scholar

[6]

Y. Benoist and D. Hulin, Cubic differentials and finite volume convex projective surfaces, Geometry and Topology, 17 (2013), 595-620.  doi: 10.2140/gt.2013.17.595.  Google Scholar

[7]

Y. Benoist and D. Hulin, Cubic differentials and hyperbolic convex sets, Journal of Differential Geometry, 98 (2014), 1-19.  doi: 10.4310/jdg/1406137694.  Google Scholar

[8]

J.-P. Benzecri, Sur les variétés localement affines et localement projectives, Bulletin de la S. M. F., 88 (1960), 229-332.   Google Scholar

[9]

G. BessonG. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., 5 (1995), 731-799.  doi: 10.1007/BF01897050.  Google Scholar

[10]

G. BessonG. Courtois and S. Gallot, Minimal entropy and Mostow's rigidity theorems, Ergodic Theory and Dynamical Systems, 16 (1996), 623-649.  doi: 10.1017/S0143385700009019.  Google Scholar

[11]

J. Boland and F. Newberger, Minimal entropy rigidity for Finsler manifolds of negative flag curvature, Ergodic Theory and Dynamical Systems, 21 (2001), 13-23.  doi: 10.1017/S0143385701001055.  Google Scholar

[12]

H. Bray, Ergodicity of Bowen-Margulis measure for the Benoist 3-manifolds, preprint, arXiv: 1705.08519. Google Scholar

[13]

H. Busemann and P. J. Kelly, Projective Geometry and Projective Metrics, Academic Press Inc., New York, 1953.  Google Scholar

[14]

B. ColboisC. Vernicos and P. Verovic, L'aire des triangles idéaux en géométrie de Hilbert, Enseign. Math., 50 (2004), 203-237.   Google Scholar

[15]

C. Connell and B. Farb, Minimal entropy rigidity for lattices in products of rank one symmetric spaces, Comm. Anal. Geom., 11 (2003), 1001-1026.  doi: 10.4310/CAG.2003.v11.n5.a7.  Google Scholar

[16]

C. Connell and B. Farb, Some recent applications of the barycenter method in geometry, in Topology and Geometry of Manifolds, volume 71 of Proc. Sympos. Pure Math., American Math Society, (2003), 19–50. doi: 10.1090/pspum/071/2024628.  Google Scholar

[17]

C. Connell and B. Farb, The degree theorem in higher rank, Journal of Differential Geometry, 65 (2003), 19-59.  doi: 10.4310/jdg/1090503052.  Google Scholar

[18]

D. CooperD. D. Long and M. B. Thistlethwaite, Flexing closed hyperbolic manifolds, Topology, 11 (2007), 2413-2440.  doi: 10.2140/gt.2007.11.2413.  Google Scholar

[19]

D. CooperD. D. Long and S. Tillmann, On convex projective manifolds and cusps, Adv. Math., 277 (2015), 181-251.  doi: 10.1016/j.aim.2015.02.009.  Google Scholar

[20]

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

[21]

M. Crampon, Dynamics and Entropies of Hilbert Metrics, Thèse, Université de Strasbourg, Strasbourg, 2011.  Google Scholar

[22]

M. Crampon, The boundary of a divisible convex set, Publ. Mat. Urug., 14 (2013), 105-119.   Google Scholar

[23]

P. de la Harpe, On Hilbert's metric for simplices, in Geometric Group Theory, Vol. 1 (Sussex, 1991), volume 181 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, (1993), 97–119. doi: 10.1017/CBO9780511661860.009.  Google Scholar

[24]

R. Feres, The minimal entropy theorem and Mostow rigidity: after G. Besson, G. Courtois and S. Gallot, (available at http://www.math.wustl.edu/ feres/mostow.pdf), 1996. Google Scholar

[25]

W. M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom., 31 (1990), 791-845.  doi: 10.4310/jdg/1214444635.  Google Scholar

[26]

D. Johnson and J. Millson, Deformation spaces associated to compact hyperbolic manifolds, in Discrete Groups in Geometry and Analysis (New Haven, Conn., 1984), volume 67 of Progr. Math., Birkhäuser, Boston, (1987), 48–106. doi: 10.1007/978-1-4899-6664-3_3.  Google Scholar

[27]

V. A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 361-393.   Google Scholar

[28]

M. Kapovich, Convex projective structures on Gromov-Thurston manifolds, Geometry and Topology, 11 (2007), 1777-1830.  doi: 10.2140/gt.2007.11.1777.  Google Scholar

[29]

A. Katok, Entropy and closed geodesics, Ergodic Theory and Dynamical Systems, 2 (1982), 339-365.  doi: 10.1017/S0143385700001656.  Google Scholar

[30]

E. Leuzinger, Entropy of the geodesic flow for metic spaces and Bruhat-Tits buildings, Adv. Geom., 6 (2006), 475-491.  doi: 10.1515/ADVGEOM.2006.029.  Google Scholar

[31]

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

[32]

G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Publications Mathématiques de l'I.H.É.S., 34 (1968), 53-104.   Google Scholar

[33]

X. Nie, On the Hilbert geometry of simplicial Tits sets, Ann. Inst. Fourier (Grenoble), 65 (2015), 1005-1030.  doi: 10.5802/aif.2950.  Google Scholar

[34]

A. Papadopoulos and M. Troyanov, From Funk to Hilbert geometry, in Handbook of Hilbert Geometry, volume 22 of IRMA Lectures in Mathematics and Theoretical Physics, European Mathematical Society, Zurich, (2014), 33–68. Google Scholar

[35]

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

[36]

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

[37]

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

[38]

N. Tholozan, Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into $\text{PSL}(3,\mathbb{R})$, Duke Math. J., 166 (2017), 1377-1403.  doi: 10.1215/00127094-00000010X.  Google Scholar

[39]

T. Zhang, The degeneration of convex $\mathbb{RP}^2$ structures on surfaces, Proc. Lond. Math. Soc., 111 (2015), 967-1012.  doi: 10.1112/plms/pdv051.  Google Scholar

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