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 [
Citation: |
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.![]() ![]() ![]() |
|
J. Aaronson
and D. Sullivan
, Rational ergodicity of geodesic flows, Ergodic Theory Dynam. Systems, 4 (1984)
, 165-178.
doi: 10.1017/S0143385700002364.![]() ![]() ![]() |
|
W. Ballmann
, Axial isometries of manifolds of nonpositive curvature, Math. Ann., 259 (1982)
, 131-144.
doi: 10.1007/BF01456836.![]() ![]() ![]() |
|
W. Ballmann
, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985)
, 597-609.
doi: 10.2307/1971331.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
W. Ballmann
and M. Brin
, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math., 82 (1995)
, 169-209.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. Bourdon
, Structure conforme au bord et flot géodésique d'un $\rm CAT(-1)$-espace, Enseign. Math. (2), 41 (1995)
, 63-102.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
K. Burns
and R. Spatzier
, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Etudes Sci. Publ. Math., 65 (1987)
, 35-59.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
E. Hopf, Ergodentheorie, Springer, 1937.
doi: 10.1007/978-3-642-86630-2.![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
G. Knieper
, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal., 7 (1997)
, 755-782.
doi: 10.1007/s000390050025.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
U. Krengel
, Darstellungssätze für Strömungen und Halbströmungen. Ⅰ, Mathematische Annalen, 176 (1968)
, 181-190.
doi: 10.1007/BF02052824.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
G. Link
, M. Peigné
and J.-C. Picaud
, Sur les surfaces non-compactes de rang un, Enseign. Math. (2), 52 (2006)
, 3-36.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
S. J. Patterson
, The limit set of a Fuchsian group, Acta Math., 136 (1976)
, 241-273.
doi: 10.1007/BF02392046.![]() ![]() ![]() |
|
M. Peigné, Autour de l'exposant de Poincaré d'un groupe kleinien, Géométrie ergodique, Monogr. Enseign. Math., Enseignement Math., Geneva, 43 (2013), 25–59.
![]() ![]() |
|
R. Ricks, Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one spaces, PhD Thesis, University of Michigan, 2015.
![]() |
|
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.![]() ![]() ![]() |
|
T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N. S. ), 95 (2003), ⅵ+96pp.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
V. Schroeder
, Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature, Manuscripta Math., 64 (1989)
, 77-105.
doi: 10.1007/BF01182086.![]() ![]() ![]() |
|
V. Schroeder
, Codimension one tori in manifolds of nonpositive curvature, Geom. Dedicata, 33 (1990)
, 251-263.
doi: 10.1007/BF00181332.![]() ![]() ![]() |
|
D. Sullivan
, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979)
, 171-202.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975, Reprinting of the 1959 original.
![]() ![]() |
|
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.![]() ![]() ![]() |