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November  2016, 36(11): 6257-6284. doi: 10.3934/dcds.2016072

Ergodic geometry for non-elementary rank one manifolds

Gabriele Link 1, and  Jean-Claude Picaud 2,

1. 

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

LMPT, UMR 6083, Faculté des Sciences et Techniques, Parc de Grandmont, 37 200 Tours, France

Received  February 2013 Revised  May 2016 Published  August 2016

Let $X$ be a Hadamard manifold, and $\Gamma\subset Is(X)$ a non-elementary discrete subgroup of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/\Gamma$ to the behavior of the Poincaré series of $\Gamma$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $\Gamma$-invariant conformal densities supported on the geometric limit set of $\Gamma$.
Keywords: Patterson-Sullivan measures., Rank one manifolds, Hopf-Tsuji-Sullivan dichotomy.
Mathematics Subject Classification: Primary: 37D40, 28D20; Secondary: 37D25, 20F6.
Citation: Gabriele Link, Jean-Claude Picaud. Ergodic geometry for non-elementary rank one manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6257-6284. doi: 10.3934/dcds.2016072
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]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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.  Google Scholar

[8]

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

[9]

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

[10]

K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings,, Inst. Hautes Études Sci. Publ. Math., (): 35.   Google Scholar

[11]

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

[12]

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

[13]

P. Eberlein, Surfaces of nonpositive curvature, Mem. Amer. Math. Soc., 20 (1979), x+90. doi: 10.1090/memo/0218.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

T. Roblin, Ergodicité et équidistribution en courbure négative,, Mém. Soc. Math. Fr. (N.S.), ().   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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.  Google Scholar

[8]

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

[9]

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

[10]

K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings,, Inst. Hautes Études Sci. Publ. Math., (): 35.   Google Scholar

[11]

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

[12]

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

[13]

P. Eberlein, Surfaces of nonpositive curvature, Mem. Amer. Math. Soc., 20 (1979), x+90. doi: 10.1090/memo/0218.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

T. Roblin, Ergodicité et équidistribution en courbure négative,, Mém. Soc. Math. Fr. (N.S.), ().   Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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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