Chapter 9 Ergodic theory and dynamics of G-spaces (with special emphasis on rigidity phenomena)

Handbook of Dynamical Systems

Volumn 1, Issue PART A, 2002, Pages 665-763

  Feres, Renato ^a   Katok, Anatole ^b  

^a WASHINGTON UNIVERSITY   (United States)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33847612266     PISSN: 1874575X     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-575X(02)80011-X     Document Type: Article

References (143)

1. Chernov N. Invariant measures for hyperbolic dynamical systems. In: Hasselblatt A.B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1 (2002), Elsevier, Amsterdam 321-407
2. Franks J., and Misiurewicz M. Topological methods in dynamics. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical vSystems Vol. 1A (2002), Elsevier, Amsterdam 547-598
3. Furman A. Random walks on groups and random transformations. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 931-1014
4. Hasselblatt B. Hyperbolic dynamical systems. In: Hassel- blatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier 239-319
5. Hasselblatt B., and Katok A. Principal structures. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 1-203
6. Holer H., Wysocki K., and Zehnder E. Pseudoholomorphic curves and dynamics in three dimensions. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 1129-1188
7. Jakobson M., and Świa{ogonek}tek G. One-dimensional maps. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 599-664
8. Kleinbock D., Shah N., and Starkov A. Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 813-930
9. Knieper G. Hyperbolic dynamics and Riemannian geometry. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 453-545
10. Lind D., and Schmidt K. Symbolic and algebraic dynamical systems. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 765-812
11. Masur H., and Tabachnikov S. Rational billiards and flat structures. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 1015-1089
12. Pollicott M. Periodic orbits and zeta functions. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 409-452
13. Rabinowitz P. Variational methods for Hamiltonian systems. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 1091-1127
14. Thouvenot J.-P. Entropy isomorphism and equivalence in ergodic theory. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 205-238
15. Abramov L.M. Metric automorphisms with quasi-discrete spectrum. Izv. Akad. Sci. SSSR Ser. Mat. 26 (1962) 513-530
16. English translation:. Abramov L.M. Metric automorphisms with quasi-discrete spectrum. Amer. Math. Soc. Transl. 39 (1964) 37-56
17. Adams S., and Stuck G. The isometry group of a compact Lorentz manifold I, II. Invent. Math. 129 2 (1997) 239-261
18. Adams S., and Stuck G. The isometry group of a compact Lorentz manifold I, II. Invent. Math. 129 2 (1997) 263-287
19. A'Campo N., and Burger M. Réseaux arithmétiques et commensurateur d'après G.A. Margulis. Invent. Math. 116 (1994) 1-25
20. Genève. Anantharaman-Delaroche C., and Renault J. Amenable Groupoids. Monographie n. 36 de l'Enseignement Mathématique (2000)
21. Benoist Y. Orbites des structures rigides (d'après Gromov), Integrable Systems and Foliations (Montpel- lier, 1995). Progr. Math. Vol. 145 (1997), Birkhäuser, Boston, MA 1-17
22. Benoist Y., and Labourie F. Sur les espaces homogènes modèles de variétés compactes. Inst. Hautes Études Sci. Publ. Math. 76 (1992) 99-109
23. Benoist Y., and Labourie F. Sur les difféomorphismes d'Anosov affines feuilletages stable et instable différentiables. Invent. Math. 111 2 (1993) 285-308
24. Benoist Y., Foulon P., and Labourie F. Flots d'Anosov à distributions stable et instable différentiables. J. Amer. Math. Soc. 5 1 (1992) 33-74
25. Benveniste J. Rigidity of isometric lattice actions on compact Riemannian manifolds. Geom. Funct. Anal. 10 3 (2000) 516-542
26. Borel A. Introduction aux Groupes Arithmétiques (1969), Hermann, Paris
27. Borel A. Compact Clifford-Klein forms of symmetric spaces. Topology 2 (1963) 111-122
28. Borel A., and Harish-Chandra. Arithmetic subgroups of algebraic groups. Ann. Math. 75 (1962) 485-535
29. Burger M., and Monod N. Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. 1 2 (1999) 199-235
30. Burger M., and Monod N. Erratum. J. Eur. Math. Soc. 1 3 (1999) 338
31. Connes A., Feldman J., and Weiss B. An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynamical Systems 1 (1981) 431-450
32. van Danzig D., and van der Waerden B.L. Über metrisch homogene Räume. Abh. Math. Sem. Univ. Hamburg 6 (1928) 374-376
33. Dixmier J. Les C* -Algèbres et leur Représentations (1964), Gauthier-Viliars, Paris
34. Effros E.G. Transformation groups and C*-algebras. Ann. of Math. 81 (1965) 38-55
35. Ellis R., and Nerurkar M. Weakly almost periodic flows. Trans. Amer. Math. Soc. 313 1 (1989) 103-119
36. Farb B., and Shalen P. Real analytic actions of lattices. Invent. Math. 135 (1999) 273-296
37. Feldman J., Hahn P., and Moore C.C. Orbit structures and countable sections for actions of continuous groups. Adv. Math. 28 (1978) 186-230
38. Feres R. Actions of discrete linear groups and Zimmer's conjecture. J. Differential Geometry 42 3 (1995) 554-576
39. Feres R. The invariant connection of a half-pinched Anosov diffeomorphism and rigidity. Pacific J. Math. 171 1 (1995) 139-155
40. Feres R. Dynamical Systems and Semisimple Groups (1998), Cambridge Univ. Press
41. Feres R., and Katok A. Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows. Ergodic Theory Dynamical Systems 9 3 (1989) 427-432
42. Feres R., and Labourie F. Topological superrigidity and Anosov actions of lattices. Ann. Scient. École Norm. Sup. 4e. série 31 (1998) 599-629
43. In preparation. Fisher D., and Margulis G.A. Local rigidity for partially hyperbolic and isometric actions of lattices (2001)
44. Furman A. Orbit equivalence rigidity. Ann. Math 150 (1999) 1083-1108
45. Furman A. Gromov's measure equivalence and rigidity of higher rank lattices. Ann. Math. 150 (1999) 1059-1081
46. D. Fisher and R. Zimmer, Geometric Lattice Actions, Entropy and Fundamental Groups, Preprint.
47. Furstenberg H. A Poisson formula for semi-simple Lie groups. Ann. Math. 77 (1963) 335-386
48. Furstenberg H. Noncommuting random products. Trans. Amer. Math. Soc. 108 (1963) 377-428
49. Furstenberg H. Random walks and discrete subgroups of Lie groups. Advances in Probability Vol. 1 (1970), Dekker, New York 3-63
50. Furstenberg H. Boundary theory and stochastic processes on homogeneous spaces. Proc. Symp. Pure Math. 26 (1974) 193-226
51. 80. Furstenberg H. Rigidity and cocycles for ergodic actions of semisimple groups [after G.A. Margulis and R. Zimmer]. Seminaire Bourbaki Vol. 559 (1979)
52. Ghys E. Actions de réseaux sur le cercle. Invent. Math. 137 (1999) 199-231
53. Ghys E. Rigidité differentiable des groupes fuchsiens. Inst. Hautes Études Sci. Publ. Math. 78 (1993) 163-185
54. Glimm J. Locally compact transformation groups. Trans. Amer. Math. Soc. 101 (1961) 124-138
55. Goetze E., and Spatzier R. On Livsic's theorem superrigidity and Anosov actions of semisimple Lie groups. Duke Math. J. 88 (1997) 1-27
56. Goetze E., and Spatzier R. Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices. Ann. Math. 150 (1999) 743-773
57. Guysinsky M., and Katok A. Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations. Math. Res. Lett. 5 (1998) 149-163
58. Greenleaf F. Invariant Means on Topological Groups (1969), van Nostrand, New York
59. Greschonig G., and Schmidt K. Ergodic decomposition of quasi-invariant probability measures. Colloq. Math. 84 85 (2000) 495-514
60. Gromov M. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53 (1981) 53-73
61. Gromov M. Rigid transformation groups. Géométrie Différentielle, Travaux en cours Vol. 33 (1988), Hermann, Paris
62. de la Harpe P., and Valette A. La Propriété (T) de Kazhdan pour les Groupes Localement Compacts. Astérisque Vol. 175 (1989), Société Mathématique de France
63. Howe R., and Moore C.C. Asymptotic properties of unitary representations. J. Funct. Anal. 32 (1979) 72-96
64. Hurder S. Rigidity for Anosov actions of higher rank lattices. Ann. of Math. 134 (1992) 361-410
65. d Actions (Warwick, 1993-1994). London Math. Soc. Lecture Note Ser. Vol. 228 (1996), Cambridge UnivPress, Cambridge 127-176
66. Kanai M. Geodesic fiows of negatively curved manifolds with smooth stable and unstable foliations. Ergodic Theory Dynamical Systems 8 (1988) 215-240
67. Kanai M. A new approach to the rigidity of discrete group actions. Geom. Funct. Anal. 6 6 (1996) 943-1056
68. Katok A., and Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems (1995), Cambridge University Press
69. d actions by automorphisms of a torus, to appear.
70. Katok A., and Lewis J. Local rigidity of certain groups of toral automorphisms. Israel J. Math. 75 (1991) 203-241
71. Katok A., and Lewis J. Global rigidity results for lattice actions on tori and new examples of volume-preserving actions. Israel J. Math. 93 (1996) 253-280
72. Katok A., and Spatzier R. First cohomology of Anosov actions of higher rank Abelian groups and applications to rigidity. Inst. Hautes Études Sci. Publ. Math. 79 (1994) 131-156
73. Katok A., and Spatzier R. Subelliptic estimates of polynomial differential operators and applications to rigidity of Abelian actions. Math. Res. Lett. 1 (1994) 193-202
74. Katok A., and Spatzier R. Invariant measures for higher rank hyperbolic Abelian actions. Ergodic Theory Dynamical Systems 16 (1996) 751-778
75. Katok A., and Spatzier R. Invariant measures for higher rank hyperbolic Abelian actions. Errata: Ergodic Theory Dynamical Systems 18 (1998) 503-507
76. Katok A., and Spatzier R. Differential rigidity of Anosov actions of higher rank Abelian groups and algebraic lattice actions. Dynamical Systems and Related Topics (Volume dedicated to D.V. Anosov). Proc. Steklov Math. Inst. Vol. 216 (1997) 287-314
77. Kazhdan D. Connection of the dual space of a group with the structure of its closed subgroups. Funct. Anal. Appl. 1 (1967) 63-65
78. Knapp A.W. Representation Theory of Semisimple Groups - An Overview Based on Examples (1986), Princeton University Press, Princeton, NJ
79. Knapp A. Lie Groups Beyond an Introduction (1996), Birkhäuser, Basel
80. Kobayashi S. Transformation Groups in Differential Geometry (1972), Springer, Berlin
81. Kowalsky N. Noncompact simple automorphism groups of Lorentz manifolds and other geometric manifolds. Ann. of Math.(2) 144 3 (1996) 611-640
82. 2(ø) (1975), Addison-Wesley, Reading, MA
83. J. Lewis, The algebraic hull of a measurable cocycle, Preprint.
84. Lubotzky A., and Zimmer R. A canonical arithmetic quotient for actions of simple Lie groups. In: Dani S. (Ed). Proc. Internat. Coll. on Lie Groups and Ergodic Theory. TIFR, Mumbai, 1996 (1998), Narosa Publ. 131-142
85. A. Lubotzky and R. Zimmer, Arithmetic structure of fundamental groups and actions of semisimple Lie groups, Topology, To appear.
86. Mackey G.W. Ergodic transformation groups with a pure point spectrum. Illinois J. Math. 8 (1964) 593-600
87. Mackey G.W. Unitary Group Representations (1989), Addison-Wesley, Reading, MA
88. Mackey G.W. The Theory of Unitary Group Representations (1976), University of Chicago Press, Chicago
89. Margulis G.A. Quotient groups of discrete subgroups and measure theory. Funct. Anal. Appl. 12 (1978) 295-305
90. Margulis G.A. Arithmeticity of the irreducible lattices in the semi-simple groups of rank greater than I. Invent. Math. 76 (1984) 93-120
91. Margulis G.A. Discrete Subgroups of Semisimple Lie Groups (1991), Springer-Verlag, Berlin
92. Margulis G.A., and Qian N. Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices. Ergodic Theory Dynamical Systems 21 (2001) 121-164
93. Montgomery D., and Zippin L. Topological Transformation Groups (1955), Wiley, New York
94. Mostow G.D. Strong Rigidity of Locally Symmetric Spaces. Annals of Mathematical Studies Vol. 78 (1973), Princeton University Press, Princeton, NJ
95. Myers S.B., and Steenrod N. The group of isometries of a Riemannian manifold. Ann. of Math. 40 (1939) 400-416
96. A. Nevo and R. Zimmer, A structure theorem for actions of semisimple Lie groups, Preprint.
97. Nevo A., and Zimmer R. Homogeneous projective factors for actions of semisimple Lie groups. Invent. Math. 138 (1999) 229-252
98. A. Nevo and R. Zimmer, Rigidity of Furstenherg entropy for actions of semisimple Lie groups, Ann. Sci. École Norm. Sup., to appear.
99. Palais R. A Global Formulation of the Lie Theory of Transformation Groups. Mem. Amer. Math. Soc. 22 (1957)
100. Paterson A. Amenability. Mathematical Surveys and Monographs Vol. 29 (1988), Amer. Math. Soc.
101. A.L. Onishchik E.B. Vinberg Lie Groups and Lie Algebras II Encyclopaedia of Mathematical Sciences Vol. 21, Berlin, Springer.
102. Onishchik A.L., and Vinberg E.B. Lie Groups and Algebraic Groups (1990), Springer-Verlag
103. Phelps R. Lectures on Choquet's Theorem (1966), Van Nostrand, Princeton
104. Platonov V., and Rapinchuk A. Algebraic Groups and Number Theory (1994), Academic Press
105. Pollicott M. Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds. London Mathematical Society Lecture Note Series Vol. 180 (1993), Cambridge University Press, Cambridge
106. Raghunathan M.S. Discrete Subgroups of Lie Groups (1970), Springer-Verlag, New York
107. Ratner M. On Raghunathan's measure conjecture. Ann. of Math. 134 (1991) 545-607
108. Reeb G., and Schweitzer P. Un théorème de Thurston établi au moyen de l'analyse non standard. Differential Topology Foliations and Gelfand-Fuks Cohomology. Lectures Notes Math Vol. 652 (1978) 138
109. Renault J. A Groupoid Approach to C*-Algebras. Lecture Notes in Math. Vol. 793 (1980)
110. Rosenlicht M. A remark on quotient spaces. Anais da Academia Brasileira de Ciencias 35 4 (1963) 487-489
111. Ruelle D. Thermodynamic Formalism (1978), Addison-Wesley, Reading, MA
112. W. Schachermayer, Une modification standard de la demonstration non standard de Reeb et Schweitzer, 139-140.
113. Schmidt K. Amenability Kazhdan's property Tstrong ergodicity and invariant means for ergodic group actions. Ergodic Theory Dynamical Systems 1 (1981) 223-236
114. In: Schweitzer P.A. (Ed). Differential Topology, Foliations and Gelfand-Fuks Cohomology, Proc. Sympos., Pontifícia Univ. Católica. Rio de Janeiro, January 1976. Lecture Notes in Math. Vol. 652 (1978), Springer, New York
115. Segal I.E. Ergodic subgroups of the orthogonal group on a real Hilbert space. Ann. of Math. 66 2 (1957)
116. Sherman T. A weight theory for unitary representations. Canadian J. Math. 18 (1966) 159-168
117. Thurston W. A generalization of the Reeb stability theorem. Topology 13 (1974) 347-352
118. Varadarajan V.S. Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963) 191-220
119. Varadarajan V.S. Lie Groups, Lie Algebras, and Their Representation Theory (1984), Springer-Verlag
120. Varadarajan V.S. Geometry of Quantum Theory (1985), Springer-Verlag
121. Russian. Vershik A.M. A measurable realization of continuous groups of automorphisms of a unitary ring. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965) 127-136
122. English transl.:. Vershik A.M. A measurable realization of continuous groups of automorphisms of a unitary ring. Transl., II. Ser. Amer. Math. Soc. 84 (1969) 69-81
123. Weil A. Adeles and Algebraic Groups (1982), Birkhäuser, Basel
124. Witte D. Arithmetic groups of higher -rank. Proc. Amer. Math. Soc. 122 (1994) 333-340
125. Witte D., and Zimmer R. Actions of semisimple Lie groups on circle bundles. Geometriae Dedicata 87 (2001) 91-121
126. S. Yaskolko, Rigidity of actions of nonuniform lattices on boundaries, Preprint.
127. Zeghib A. The identity component of the isometry group of a compact Lorentz manifold. Duke Math. J. 92 2 (1998) 321-333
128. Zimmer R. Orbit spaces of unitary representations ergodic theory and simple Lie groups. Ann. of Math. 106 (1977) 573-588
129. Zimmer R. Induced and amenable actions of Lie groups. Ann. Sci. École Norm. Sup. 11 (1978) 407-428
130. Zimmer R. On the cohomology of ergodic actions of semisimple Lie groups and discrete subgroups. Amer. J. Math. 103 (1981) 937-950
131. Zimmer R. Ergodic Theory and Semisimple Groups. Monographs in Mathematics (1984), Birkhäuser, Basel
132. Zimmer R. Actions of lattices in semisimple groups preserving a G-structure of finite type. Ergodic The-ory Dynamical Systems 5 (1985) 301-306
133. Zimmer R. Actions of semisimple groups and discrete subgroups. Proc. Int. Congr. of Mathematicians. Berkeley, CA. Amer. Math. Soc., Providence RI (1986) 1247-1258
134. Zimmer R. On the automorphism group of a compact Lorentz manifold and other geometric manifolds. Invent. Math. 83 (1986) 411-424
135. Zimmer R. Ergodic theory and the automorphism group of a G-structure. Proceedings of a Conference in Honor of G.W. Mackey. Berkeley, 1984 (1986), MSRI Publications, Springer-Verlag
136. Zimmer R. Lattices in semisimple groups and invariant geometric structures on compact manifolds. Proceedings of a conference in Honor of G.D. Mostow. New Haven, 1984. Discrete Groups in Geometry and Analysis (1986), Birkhäuser, Boston
137. Zimmer R. Ergodic theory and the automorphism group of a G-structure. In: Moore C.C. (Ed). Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics (1987), Springer, New York 247-278
138. Zimmer R. On the algebraic hull of an automorphism group of a principal bundle. Comment. Math. Helv. 65 (1990) 375-387
139. Zimmer R. Superrigidity Ratner's theorem and fundamental groups. Israel J. Math. 74 2-3 (1991) 199-207
140. Class notes. Zimmer R. Topological superrigidity (1992)
141. Part 3. Zimmer R. Automorphism groups and fundamental groups of geometric manifolds. Proc. Sympos. Pure Math Vol. 54 (1993), Amer. Math. Soc., Providence, RI 693-710
142. R. Zimmer, Entropy and arithmetic quotients for simple automorphism groups of geometric manifolds. Preprint.
143. June. Zimmer R. lectures on Ergodic Theory, Groups, and Geometry. Notes taken by D. Witte. NSF-CBMS Regional Conference in the Mathematical Sciences (1998), University of Minnesota