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November  2018, 38(11): 5577-5613. doi: 10.3934/dcds.2018245

Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry

Gabriele Link ,

Institut für Algebra und Geometrie, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76 131 Karlsruhe, Germany

* Corresponding author: Gabriele Link

Received  November 2017 Revised  June 2018 Published  August 2018

Let $X$ be a proper Hadamard space and $\Gamma <{\text{Is}}(X)$ a non-elementary discrete group of isometries with a rank one isometry. We discuss and prove Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on the set of parametrized geodesics of the quotient $\Gamma \backslash X$ and with respect to Ricks' measure introduced in [35]. This generalizes previous work of the author and J. C. Picaud on Hopf-Tsuji-Sullivan dichotomy in the analogous manifold setting and with respect to Knieper's measure.

Keywords: Rank one space, Hopf-Tsuji-Sullivan dichotomy, geodesic currents, Patterson-Sullivan measures.
Mathematics Subject Classification: Primary: 37D40, 20F67; Secondary: 37D25.
Citation: Gabriele Link. Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5577-5613. doi: 10.3934/dcds.2018245
References:
[1]

J. Aaronson and M. Denker, The Poincaré series of $\mathbb C\setminus \mathbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20. doi: 10.1017/S0143385799126592. Google Scholar

[2]

J. Aaronson and D. Sullivan, Rational ergodicity of geodesic flows, Ergodic Theory Dynam. Systems, 4 (1984), 165-178. doi: 10.1017/S0143385700002364. Google Scholar

[3]

W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144. doi: 10.1007/BF01456836. Google Scholar

[4]

W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609. doi: 10.2307/1971331. Google Scholar

[5]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin. doi: 10.1007/978-3-0348-9240-7. Google Scholar

[6]

W. Ballmann and M. Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math., 82 (1995), 169-209. Google Scholar

[7]

W. Ballmann, M. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature. Ⅰ, Ann. of Math. (2), 122 (1985), 171-203. doi: 10.2307/1971373. Google Scholar

[8]

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

[9]

V. Bangert and V. Schroeder, Existence of flat tori in analytic manifolds of nonpositive curvature, Ann. Sci. École Norm. Sup. (4), 24 (1991), 605-634. doi: 10.24033/asens.1638. Google Scholar

[10]

M. Bourdon, Structure conforme au bord et flot géodésique d'un $\rm CAT(-1)$-espace, Enseign. Math. (2), 41 (1995), 63-102. Google Scholar

[11]

M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9. Google Scholar

[12]

D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033. Google Scholar

[13]

K. Burns, Hyperbolic behaviour of geodesic flows on manifolds with no focal points, Ergodic Theory Dynam. Systems, 3 (1983), 1-12. doi: 10.1017/S0143385700001796. Google Scholar

[14]

K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Etudes Sci. Publ. Math., 65 (1987), 35-59. Google Scholar

[15]

P-E. Caprace and K. Fujiwara, Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal., 19 (2010), 1296-1319. doi: 10.1007/s00039-009-0042-2. Google Scholar

[16]

M. Coornaert and A. Papadopoulos, Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d'isométries des arbres, Trans. Amer. Math. Soc., 343 (1994), 883-898. doi: 10.2307/2154747. Google Scholar

[17]

F. Dal'bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124. doi: 10.1007/BF02803518. Google Scholar

[18]

U. Hamenstädt, Rank-one isometries of proper CAT(0)-spaces, Discrete Groups and Geometric Structures, Contemporary Mathematics,, American Mathematical Society, Providence, RI, 501 (2009), 43–59. doi: 10.1090/conm/501/09839. Google Scholar

[19]

E. Hopf, Ergodentheorie, Springer, 1937. doi: 10.1007/978-3-642-86630-2. Google Scholar

[20]

E. Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc., 77 (1971), 863-877. doi: 10.1090/S0002-9904-1971-12799-4. Google Scholar

[21]

V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math., 455 (1994), 57-103. doi: 10.1515/crll.1994.455.57. Google Scholar

[22]

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

[23]

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

[24]

U. Krengel, Darstellungssätze für Strömungen und Halbströmungen. Ⅰ, Mathematische Annalen, 176 (1968), 181-190. doi: 10.1007/BF02052824. Google Scholar

[25]

U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1985, With a supplement by Antoine Brunel. doi: 10.1515/9783110844641. Google Scholar

[26]

G. Link, Asymptotic geometry and growth of conjugacy classes of nonpositively curved manifolds, Ann. Global Anal. Geom., 31 (2007), 37-57. doi: 10.1007/s10455-006-9016-x. Google Scholar

[27]

G. Link, Asymptotic geometry in products of Hadamard spaces with rank one isometries, Geometry and Topology, 14 (2010), 1063-1094. doi: 10.2140/gt.2010.14.1063. Google Scholar

[28]

G. Link, M. Peigné and J.-C. Picaud, Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006), 3-36. Google Scholar

[29]

G. Link and J.-C. Picaud, Ergodic geometry for non-elementary rank one manifolds, Discrete and Continuous Dyn. Syst. A, no. 11, 36 (2016), 6257–6284. doi: 10.3934/dcds.2016072. Google Scholar

[30]

P. J. Nicholls, The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511600678. Google Scholar

[31]

J.-P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Math. J., 125 (2004), 15-44. doi: 10.1215/S0012-7094-04-12512-6. Google Scholar

[32]

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

[33]

M. Peigné, Autour de l'exposant de Poincaré d'un groupe kleinien, Géométrie ergodique, Monogr. Enseign. Math., Enseignement Math., Geneva, 43 (2013), 25–59. Google Scholar

[34]

R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one spaces, PhD Thesis, University of Michigan, 2015.Google Scholar

[35]

R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces, Ergodic Theory Dynam. Systems, no. 3, 37 (2017), 939–970. doi: 10.1017/etds.2015.78. Google Scholar

[36]

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

[37]

T. Roblin, Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math., 147 (2005), 333-357. doi: 10.1007/BF02785371. Google Scholar

[38]

V. Schroeder, Existence of immersed tori in manifolds of nonpositive curvature, J. Reine Angew. Math., 390 (1988), 32-46. doi: 10.1515/crll.1988.390.32. Google Scholar

[39]

V. Schroeder, Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989), 77-105. doi: 10.1007/BF01182086. Google Scholar

[40]

V. Schroeder, Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990), 251-263. doi: 10.1007/BF00181332. Google Scholar

[41]

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

[42]

M. E. Taylor, Measure Theory and Integration, vol. 76 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/gsm/076. Google Scholar

[43]

M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original. Google Scholar

[44]

C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005. doi: 10.1090/S0002-9947-96-01614-5. Google Scholar

show all references

References:
[1]

J. Aaronson and M. Denker, The Poincaré series of $\mathbb C\setminus \mathbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20. doi: 10.1017/S0143385799126592. Google Scholar

[2]

J. Aaronson and D. Sullivan, Rational ergodicity of geodesic flows, Ergodic Theory Dynam. Systems, 4 (1984), 165-178. doi: 10.1017/S0143385700002364. Google Scholar

[3]

W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144. doi: 10.1007/BF01456836. Google Scholar

[4]

W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609. doi: 10.2307/1971331. Google Scholar

[5]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin. doi: 10.1007/978-3-0348-9240-7. Google Scholar

[6]

W. Ballmann and M. Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math., 82 (1995), 169-209. Google Scholar

[7]

W. Ballmann, M. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature. Ⅰ, Ann. of Math. (2), 122 (1985), 171-203. doi: 10.2307/1971373. Google Scholar

[8]

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

[9]

V. Bangert and V. Schroeder, Existence of flat tori in analytic manifolds of nonpositive curvature, Ann. Sci. École Norm. Sup. (4), 24 (1991), 605-634. doi: 10.24033/asens.1638. Google Scholar

[10]

M. Bourdon, Structure conforme au bord et flot géodésique d'un $\rm CAT(-1)$-espace, Enseign. Math. (2), 41 (1995), 63-102. Google Scholar

[11]

M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9. Google Scholar

[12]

D. Burago, Y. Burago and S. Ivanov, A course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033. Google Scholar

[13]

K. Burns, Hyperbolic behaviour of geodesic flows on manifolds with no focal points, Ergodic Theory Dynam. Systems, 3 (1983), 1-12. doi: 10.1017/S0143385700001796. Google Scholar

[14]

K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Etudes Sci. Publ. Math., 65 (1987), 35-59. Google Scholar

[15]

P-E. Caprace and K. Fujiwara, Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal., 19 (2010), 1296-1319. doi: 10.1007/s00039-009-0042-2. Google Scholar

[16]

M. Coornaert and A. Papadopoulos, Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d'isométries des arbres, Trans. Amer. Math. Soc., 343 (1994), 883-898. doi: 10.2307/2154747. Google Scholar

[17]

F. Dal'bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel J. Math., 118 (2000), 109-124. doi: 10.1007/BF02803518. Google Scholar

[18]

U. Hamenstädt, Rank-one isometries of proper CAT(0)-spaces, Discrete Groups and Geometric Structures, Contemporary Mathematics,, American Mathematical Society, Providence, RI, 501 (2009), 43–59. doi: 10.1090/conm/501/09839. Google Scholar

[19]

E. Hopf, Ergodentheorie, Springer, 1937. doi: 10.1007/978-3-642-86630-2. Google Scholar

[20]

E. Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc., 77 (1971), 863-877. doi: 10.1090/S0002-9904-1971-12799-4. Google Scholar

[21]

V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math., 455 (1994), 57-103. doi: 10.1515/crll.1994.455.57. Google Scholar

[22]

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

[23]

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

[24]

U. Krengel, Darstellungssätze für Strömungen und Halbströmungen. Ⅰ, Mathematische Annalen, 176 (1968), 181-190. doi: 10.1007/BF02052824. Google Scholar

[25]

U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1985, With a supplement by Antoine Brunel. doi: 10.1515/9783110844641. Google Scholar

[26]

G. Link, Asymptotic geometry and growth of conjugacy classes of nonpositively curved manifolds, Ann. Global Anal. Geom., 31 (2007), 37-57. doi: 10.1007/s10455-006-9016-x. Google Scholar

[27]

G. Link, Asymptotic geometry in products of Hadamard spaces with rank one isometries, Geometry and Topology, 14 (2010), 1063-1094. doi: 10.2140/gt.2010.14.1063. Google Scholar

[28]

G. Link, M. Peigné and J.-C. Picaud, Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006), 3-36. Google Scholar

[29]

G. Link and J.-C. Picaud, Ergodic geometry for non-elementary rank one manifolds, Discrete and Continuous Dyn. Syst. A, no. 11, 36 (2016), 6257–6284. doi: 10.3934/dcds.2016072. Google Scholar

[30]

P. J. Nicholls, The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511600678. Google Scholar

[31]

J.-P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Math. J., 125 (2004), 15-44. doi: 10.1215/S0012-7094-04-12512-6. Google Scholar

[32]

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

[33]

M. Peigné, Autour de l'exposant de Poincaré d'un groupe kleinien, Géométrie ergodique, Monogr. Enseign. Math., Enseignement Math., Geneva, 43 (2013), 25–59. Google Scholar

[34]

R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one spaces, PhD Thesis, University of Michigan, 2015.Google Scholar

[35]

R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces, Ergodic Theory Dynam. Systems, no. 3, 37 (2017), 939–970. doi: 10.1017/etds.2015.78. Google Scholar

[36]

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

[37]

T. Roblin, Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math., 147 (2005), 333-357. doi: 10.1007/BF02785371. Google Scholar

[38]

V. Schroeder, Existence of immersed tori in manifolds of nonpositive curvature, J. Reine Angew. Math., 390 (1988), 32-46. doi: 10.1515/crll.1988.390.32. Google Scholar

[39]

V. Schroeder, Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989), 77-105. doi: 10.1007/BF01182086. Google Scholar

[40]

V. Schroeder, Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990), 251-263. doi: 10.1007/BF00181332. Google Scholar

[41]

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

[42]

M. E. Taylor, Measure Theory and Integration, vol. 76 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/gsm/076. Google Scholar

[43]

M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original. Google Scholar

[44]

C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005. doi: 10.1090/S0002-9947-96-01614-5. Google Scholar

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