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

Ergodic geometry for non-elementary rank one manifolds

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 $\Gamma$.
Citation: Gabriele Link, Jean-Claude Picaud. Ergodic geometry for non-elementary rank one manifolds. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1017/S0143385799126592.

[2]

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

[3]

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

[4]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar,, Birkhäuser Verlag, (1995). doi: 10.1007/978-3-0348-9240-7.

[5]

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

[6]

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

[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.

[8]

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

[9]

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

[10]

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

[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. doi: 10.2307/2154747.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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. doi: 10.2307/120995.

[19]

U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics,, Walter de Gruyter & Co., (1985). doi: 10.1515/9783110844641.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. doi: 10.1007/BF02785371.

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

M. Tsuji, Potential Theory in Modern Function Theory,, Chelsea Publishing Co., (1975).

[33]

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

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. doi: 10.1017/S0143385799126592.

[2]

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

[3]

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

[4]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar,, Birkhäuser Verlag, (1995). doi: 10.1007/978-3-0348-9240-7.

[5]

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

[6]

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

[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.

[8]

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

[9]

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

[10]

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

[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. doi: 10.2307/2154747.

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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. doi: 10.2307/120995.

[19]

U. Krengel, Ergodic Theorems, vol. 6 of de Gruyter Studies in Mathematics,, Walter de Gruyter & Co., (1985). doi: 10.1515/9783110844641.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. doi: 10.1007/BF02785371.

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

M. Tsuji, Potential Theory in Modern Function Theory,, Chelsea Publishing Co., (1975).

[33]

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

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