Ergodic geometry for non-elementary rank one manifolds

Previous Article
Exponential stabilization of a structure with interfacial slip
DCDS Home
This Issue
Next Article
Attractors of Hamilton nonlinear PDEs

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

Abstract
Related Papers

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 and 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.
[2]	W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144. doi: 10.1007/BF01456836.
[3]	W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609. doi: 10.2307/1971331.
[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.
[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.
[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.
[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.
[8]	M. Bourdon, Structure conforme au bord et flot géodésique d'un $CAT(-1)$-espace, Enseign. Math. (2), 41 (1995), 63-102.
[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.
[10]	K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math., 35-59.
[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.
[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.
[13]	P. Eberlein, Surfaces of nonpositive curvature, Mem. Amer. Math. Soc., 20 (1979), x+90. 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), 863-877. 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), 57-103. doi: 10.1515/crll.1994.455.57.
[17]	G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782. 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), 291-314. doi: 10.2307/120995.
[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.
[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.
[21]	G. Link, M. Peigné and J.-C. Picaud, Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006), 3-36.
[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.
[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.
[24]	S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273. doi: 10.1007/BF02392046.
[25]	T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), vi+96.
[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.
[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.
[28]	V. Schroeder, Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989), 77-105. doi: 10.1007/BF01182086.
[29]	V. Schroeder, Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990), 251-263. doi: 10.1007/BF00181332.
[30]	D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 171-202.
[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.
[32]	M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original.
[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.

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.
[2]	W. Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982), 131-144. doi: 10.1007/BF01456836.
[3]	W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609. doi: 10.2307/1971331.
[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.
[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.
[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.
[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.
[8]	M. Bourdon, Structure conforme au bord et flot géodésique d'un $CAT(-1)$-espace, Enseign. Math. (2), 41 (1995), 63-102.
[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.
[10]	K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math., 35-59.
[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.
[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.
[13]	P. Eberlein, Surfaces of nonpositive curvature, Mem. Amer. Math. Soc., 20 (1979), x+90. 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), 863-877. 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), 57-103. doi: 10.1515/crll.1994.455.57.
[17]	G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997), 755-782. 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), 291-314. doi: 10.2307/120995.
[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.
[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.
[21]	G. Link, M. Peigné and J.-C. Picaud, Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006), 3-36.
[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.
[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.
[24]	S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273. doi: 10.1007/BF02392046.
[25]	T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), vi+96.
[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.
[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.
[28]	V. Schroeder, Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989), 77-105. doi: 10.1007/BF01182086.
[29]	V. Schroeder, Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990), 251-263. doi: 10.1007/BF00181332.
[30]	D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 171-202.
[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.
[32]	M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original.
[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.

[1]	Gabriele Link. Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5577-5613. doi: 10.3934/dcds.2018245
[2]	Fei Liu, Fang Wang, Weisheng Wu. On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1517-1554. doi: 10.3934/dcds.2020085
[3]	Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711
[4]	Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83
[5]	Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure and Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457
[6]	Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773
[7]	Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012
[8]	Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451
[9]	Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127
[10]	Alexandre A. Rodrigues. Rank-one strange attractors versus heteroclinic tangles. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022097
[11]	Salvador Addas-Zanata, Fábio A. Tal. Support of maximizing measures for typical $\mathcal{C}^0$ dynamics on compact manifolds. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 795-804. doi: 10.3934/dcds.2010.26.795
[12]	Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249
[13]	Fei Liu, Xiaokai Liu, Fang Wang. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4791-4804. doi: 10.3934/dcds.2021057
[14]	I. Baldomá, Àlex Haro. One dimensional invariant manifolds of Gevrey type in real-analytic maps. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 295-322. doi: 10.3934/dcdsb.2008.10.295
[15]	Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191
[16]	William Ott, Qiudong Wang. Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1035-1054. doi: 10.3934/dcds.2010.26.1035
[17]	Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266
[18]	Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004
[19]	Hong Seng Sim, Chuei Yee Chen, Wah June Leong, Jiao Li. Nonmonotone spectral gradient method based on memoryless symmetric rank-one update for large-scale unconstrained optimization. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021143
[20]	Christian Pötzsche. Dichotomy spectra of triangular equations. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 423-450. doi: 10.3934/dcds.2016.36.423

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Gabriele Link Jean-Claude Picaud
on Google Scholar
Gabriele Link Jean-Claude Picaud

[Back to Top]