Advertisement

Dynamical Systems with Transitive Symmetry Group. Geometric and Statistical Properties

  • A. V. Safonov
  • A. N. Starkov
  • A. M. Stepin
Chapter
  • 656 Downloads
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 66)

Abstract

The main topic of discussion in this survey is the geometric and metric properties of geodesic and horocycle flows on surfaces of constant curvature and their multidimensional analogues, the so-called G-induced actions (that is, restrictions to a suitable subgroup F of the natural action of G on a homogeneous space of it). One usually takes as F one-parameter or cyclic subgroups of G and the horospherical subgroups corresponding to them (the details are given in Sect. 1). However, in the course of the exposition of our main topic we sometimes mention results relating to the case when F is an n-parameter subgroup of G or a lattice. This class of dynamical systems consists precisely of group actions admitting an extension to a transitive action of a Lie group. We shall not be considering in our survey (or, at any rate, not systematically) the following themes which are close to our main subject matter:
  • Topological dynamics and metric properties of general group actions (in particular, the work of Zimmer on semisimple group actions (Zimmer 1984);

  • Group automorphisms;

  • Affine transformations of homogeneous spaces;

  • G-induced flows on spaces with an infinite invariant measure. The papers (Hopf 1939), (Sullivan 1981) and the recent book (Nicholls 1989) are devoted to the connections between G-induced flows (and also geodesic flows) and the theory of discrete groups of motions of a hyperbolic space;

  • The actions of subgroups on the double cosets related to geodesic (for n > 2) and horocycle (for n > 3) flows on an n-dimensional manifold of constant negative curvature (see (Gel’fand and Fomin 1952) and, in particular, (Mautner 1957)). Flows of this type in somewhat more general form differing from geodesic and horocycle flows have not been investigated.

Keywords

Homogeneous Space Geodesic Flow Homogeneous Flow Constant Negative Curvature Unique Ergodicity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alekseev, V.M. (1976): Symbolic Dynamics (Eleventh Mathematical School). Inst. Mat. Akad. Nauk Ukr. SSRGoogle Scholar
  2. Anosov, D.V. (1967): Geodesic flows on closed Riemannian manifolds of negative curvature. Tr. Mat. Inst. Steklova 90 (209 pp.). [English transi.: Proc. Steklov Inst. Math. 90 (1969)] Zbl. 163,436Google Scholar
  3. Anosov, D.V., Sinai, Ya.G. (1967): Some smooth ergodic systems. Usp. Mat. Nauk 22, No. 5, 107–172. [English transi.: Russ. Math. Surv. 22,No. 5, 103–167 (1967)] Zbl. 177,420Google Scholar
  4. Auslander, L. (1973): An exposition of the structure of solvmanifolds. Bull. Am. Math. Soc. 79, No. 2, 227–285. Zbl. 265.22016–017MathSciNetzbMATHCrossRefGoogle Scholar
  5. Auslander, L., Brezin, J. (1968): Almost algebraic Lie algebras. J. Algebra 8, No. 1, 295–313. Zbl. 197,30MathSciNetzbMATHCrossRefGoogle Scholar
  6. Auslander, L., Green, L., Hahn, F. (1963): Flows on homogeneous spaces. Ann. Math. Stud. 53. Princeton Univ. Press, Princeton N.J. Zbl. 106,368Google Scholar
  7. Ballmann, W., Brin, M. (1982): On the ergodicity of geodesic flows. Ergodic Theory Dyn. Syst. 2, No. 3–4, 311–315. Zbl. 519.58036MathSciNetzbMATHCrossRefGoogle Scholar
  8. Benardete, D. (1988): Topological equivalence of flows on homogeneous spaces and divergence of one-parameter subgroups of Lie groups. Trans. Am. Math. Soc. 306, 499–527. Zbl. 652.58036MathSciNetzbMATHCrossRefGoogle Scholar
  9. Borel, A., Prasad, G. (1992): Values of isotropic quadratic forms at S-integral points. Compos. Math. 83, No. 3, 347–372. Zbl. 777.11008MathSciNetzbMATHGoogle Scholar
  10. Bourbaki, N. (1971): Groupes et algèbres de Lie. Ch. 1,2. Eléments Math. No. 26, Hermann, Paris. Zbl. 213,41Google Scholar
  11. Bowen, R. (1971): Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153,401–414MathSciNetzbMATHCrossRefGoogle Scholar
  12. Bowen, R. (1971): Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 181 (1973), 509–510. Zbl. 212,292MathSciNetGoogle Scholar
  13. Bowen, R. (1976): Weak mixing and unique ergodicity on homogeneous spaces. Isr. J. Math. 23, No. 34, 267–273. Zbl. 338.43014MathSciNetzbMATHGoogle Scholar
  14. Bowen R. (1979): Methods of Symbolic Dynamics Mir, Moscow (Russian). (Russian translation of several papers by Bowen not listed in this bibliography)Google Scholar
  15. Bowen, R., Marcus, B. (1977): Unique ergodicity for horocycle foliations. Isr. J. Math. 26, No. 1, 43–67. Zbl. 346.58009MathSciNetzbMATHGoogle Scholar
  16. Brezin, J., Moore, C.C. (1981): Flows on homogeneous spaces: a new look. Am. J. Math. 103,571–613. Zbl. 506.22008MathSciNetzbMATHCrossRefGoogle Scholar
  17. Bunimovich, L.A., Pesin, Ya.B., Sinai, Ya.G., Yakobson, M.V. (1985): Dynamical Systems II: Ergodic Theory of Smooth Dynamical Systems. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 2, 113–232. [English transl. in: Encycl. Math. Sci. 2, 99–206. Springer-Verlag, Berlin Heidelberg New York 1989] Zbl. 781.58018Google Scholar
  18. Chumak, M.L., Stepin, A.M. (1989): Partial hyperbolicity of G-induced flows. Dynamical Systems and Ergodic Theory. Banach Cent. Publ. 23, 475–479. Zbl. 705.58040MathSciNetGoogle Scholar
  19. Dani, S.G. (1976a): Bernoullian translations and minimal horospheres on homogeneous spaces. J. Indian Math. Soc. 40, No. 1, 245–284. Zbl. 435.28015MathSciNetzbMATHGoogle Scholar
  20. Dani, S.G. (1976b): Kolmogorov automorphisms on homogeneous spaces 5. Am. J. Math. 98, 119–163. Zbl. 325.22011MathSciNetzbMATHCrossRefGoogle Scholar
  21. Dani, S.G. (1977a): Spectrum of an affine transformation. Duke Math. J. 44, No. 1, 129–155. Zbl. 351.22005MathSciNetzbMATHGoogle Scholar
  22. Dani, S.G. (1977b): Sequential entropy of generalized horocycles. J. Indian Math. Soc. 41, No. 12, 17–25. Zbl. 445.28017MathSciNetzbMATHGoogle Scholar
  23. Dani, S.G. (1977c): Some mixing multi-parameter dynamical systems. J. Indian Math. Soc. 41, No. 12, 1–15. Zbl. 447.28018MathSciNetzbMATHGoogle Scholar
  24. Dani, S.G. (1980): Strictly non-ergodic actions on homogeneous spaces. Duke Math. J. 47, No. 3, 633–639. Zbl. 447.28019MathSciNetzbMATHGoogle Scholar
  25. Dani, S.G. (1981): Invariant measures and minimal sets of horospherical flows. Invent. Math. 64, 357–385. Zbl. 498.58013MathSciNetzbMATHCrossRefGoogle Scholar
  26. Dani, S.G. (1982): On uniformly distributed orbits of certain horocycle flows. Ergodic Theory Dyn. Syst. 2, No. 2, 139–158. Zbl. 504.22006MathSciNetzbMATHCrossRefGoogle Scholar
  27. Dani, S.G. (1985): Divergent trajectories of flows on homogeneous spaces and diophantine approximation. J. Reine Angew. Math. 359, 55–89. Zbl. 578.22012MathSciNetzbMATHGoogle Scholar
  28. Dani, S.G. (1986a): Orbits of horospherical flows. Duke Math. J. 53, No. 1, 177–188. Zbl. 609.58038MathSciNetzbMATHGoogle Scholar
  29. Dani, S.G. (1986b): On orbits of unipotent flows on homogeneous spaces II. Ergodic Theory Dyn. Syst. 6, No. 2, 167–182. Zbl. 601.22003MathSciNetzbMATHGoogle Scholar
  30. Dani, S.G. (1986c): Bounded orbits of flows on homogeneous spaces. Comment. Math. Helv. 61, 636–660. Zbl. 627.22013MathSciNetzbMATHCrossRefGoogle Scholar
  31. Dani, S.G., Keane, M. (1979): Ergodic invariant measures for actions of SL(2,Z). Ann. Inst. Henri Poincaré, New Ser., Sect. B 15, No. 1, 79–84. Zbl. 392.22018MathSciNetzbMATHGoogle Scholar
  32. Dani, S.G., Margulis, G.A. (1989): Values of quadratic forms at primitive integral points. Invent. Math. 98,405–424. Zbl. 682.22008MathSciNetzbMATHGoogle Scholar
  33. Dani, S.G., Margulis, G.A. (1992): On the limit distributions of orbits of unipotent flows and integral solutions of quadratic inequalities. C. R. Acad. Sci. Paris, Ser. I 314, 699–704. Zbl. 780.22002MathSciNetGoogle Scholar
  34. Ellis, R., Perrizo, W. (1978): Unique ergodicity of flows on homogeneous spaces. Isr. J. Math. 29, No. 23, 276–284. Zbl. 383.22004MathSciNetzbMATHGoogle Scholar
  35. Feldman, J., Ornstein, D.E. (1987): Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature. Ergodic Theory Dyn. Syst. 7, No. 1, 49–72. Zbl. 633.58024MathSciNetzbMATHCrossRefGoogle Scholar
  36. Flaminio, L. (1987): An extension of Ratner’s rigidity theorem to n-dimensional hyperbolic space. Ergodic Theory Dyn. Syst. 7, 73–92. Zbl. 623.57030MathSciNetzbMATHCrossRefGoogle Scholar
  37. Flaminio, L., Spatzier, R.J. (1990): Geometrically finite groups, Patterson-Sullivan measures and Ratner’s rigidity theorem. Invent. Math. 99, No. 3, 601–626. Zbl. 667.57014MathSciNetzbMATHGoogle Scholar
  38. Furstenberg, H. (1963): The structure of distal flows. Am. J. Math. 85, 477–515. Zbl. 199,272MathSciNetzbMATHCrossRefGoogle Scholar
  39. Furstenberg, H. (1973): The unique ergodicity of the horocycle flow. In A. Beck (ed.): Recent Advances in Topological Dynamics, Proc. Conf. 1972. Lect. Notes Math. 318, 95–115. Zbl. 256.58009Google Scholar
  40. Gel’fand, I.M., Fomin, S.V. (1952): Geodesic flows on manifolds of constant negative curvature. Usp. Mat. Nauk 7, No. 1, 118–137. Zbl. 48,92MathSciNetzbMATHGoogle Scholar
  41. Green, L. (1961): Spectra of nifflows. Bull. Am. Math. Soc. 67, 414–415. Zbl. 99,391zbMATHCrossRefGoogle Scholar
  42. Gurevich, B.M. (1961): Entropy of a horocycle flow. Dokl. Akad. Nauk SSSR 136, 768–770. [English transl.: Sov. Math., Dokl. 2, 124–126 (1961)] Zbl. 134,338Google Scholar
  43. Hedlund, G.A. (1936): Fuchsian groups and transitive horocycles. Duke Math. J. 2, No. 1, 530–542. Zbl. 15,102MathSciNetGoogle Scholar
  44. Hedlund, G.A. (1939): The dynamics of geodesic flows. Bull. Am. Math. Soc. 45, No. 4, 241–260. Zbl. 20,403MathSciNetzbMATHCrossRefGoogle Scholar
  45. Hopf, E. (1936): Fuchsian groups and ergodic theory. Trans. Am. Math. Soc. 39, 299–314. Zbl. 14,83MathSciNetCrossRefGoogle Scholar
  46. Hopf, E. (1939): Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sachs. Akad. Wiss., Leipzig, Math.-Nat. Kl. 91, 261–304. Zbl. 24,80MathSciNetGoogle Scholar
  47. Jacobson, N. (1962): Lie algebras. Interscience, New York London. Zbl. 121,275zbMATHGoogle Scholar
  48. Kornfel’d, I.P., Sinai, Ya.G., Fomin, S.V. (1980): Ergodic Theory. Nauka, Moscow. [English transl.: Springer-Verlag, Berlin Heidelberg New York 1982] Zbl. 493.28007Google Scholar
  49. Kushnirenko, A.G. (1965): An estimate from above of the entropy of a classical dynamical system. Dokl. Akad. Nauk SSSR 161, 37–38. [English transl.: Soy. Math., Dokl. 6, 360–362 (1965)] Zbl. 136,429Google Scholar
  50. Marié, R. (1981): A proof of Pesin’s formula. Ergodic Theory Dyn. Syst. 1, No. 1, 95–102; (1983): 3, No. 1, 159–160. Zbl. 489.58018; Zbl. 522.58031Google Scholar
  51. Marié, R. (1987): Ergodic Theory and Differentiable Dynamics. Springer-Verlag, Berlin Heidelberg New York. Zbl. 616.28007Google Scholar
  52. Marcus, B. (1975): Unique ergodicity of the horocycle flow: the variable negative curvature case. Isr. J. Math. 21, No. 23, 133–144. Zbl. 314.58013zbMATHGoogle Scholar
  53. Marcus, B. (1978): The horocycle flow is mixing of all degrees. Invent. Math. 46, No. 3, 201–209. Zbl. 395.28012MathSciNetzbMATHGoogle Scholar
  54. Marcus, B. (1983): Topological conjugacy of horocycle flows. Am. J. Math. 105, 623–632. Zbl. 533.58029zbMATHCrossRefGoogle Scholar
  55. Margulis, G.A. (1971): On the action of unipotent groups in a space of lattices. Mat. Sb., Nov. Ser. 86, 552–556. [English transl.: Math. USSR, Sb. 15, 549–554 (1972)] Zbl. 257.54037Google Scholar
  56. Margulis, G.A. (1987): Formes quadratiques indéfinies et flots unipotents sur les espaces homogènes. C. R. Acad. Sci. Paris, Ser. I 304, No. 10, 249–253. Zbl. 624.10011MathSciNetzbMATHGoogle Scholar
  57. Margulis, G.A. (1989): Compactness of minimal closed invariant sets of actions of unipotent groups. [Preprint Math. No. 78, Inst. Hautes Etud. Sci., Bures sur Yvette;] see also Geom. Dedicata 37, No, 1, 1–7 (1991). 733.22005Google Scholar
  58. Margulis, G.A. (1991): Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory. Proc. ICM, Kyoto Japan 1990, Vol. I, 193–215. Zbl. 747.58017Google Scholar
  59. Margulis, G.A., Tomanov, G.M. (1992): Measure rigidity for algebraic groups over local fields. C. R. Acad. Sci., Paris, Ser. I 315,No. 12, 1221–1226MathSciNetzbMATHGoogle Scholar
  60. Mautner, F.J. (1957): Geodesic flows on symmetric Riemann spaces. Ann. Math., II. Ser. 65, 416–431. Zbl. 84,375MathSciNetzbMATHCrossRefGoogle Scholar
  61. Millionshchikov, V.M. (1976): A formula for the entropy of a smooth dynamical system. Differ. Uravn. 12, 2188–2192. [English transl.: Differ. Equations 12, 1527–1530 (1977)]Google Scholar
  62. Zbl. 357.34043Google Scholar
  63. Moore, C.C. (1966): Ergodicity of flows on homogeneous spaces. Am. J. Math. 88, 154–178. Zbl. 148,379zbMATHCrossRefGoogle Scholar
  64. Moore, C.C. (1980): The Mautner pheonomenon for general unitary representations. Pac. J. Math. 86, No. 1, 155–169. Zbl. 446.22014MathSciNetzbMATHGoogle Scholar
  65. Morse, H.M. (1921): Recurrent geodesics on a surface of negative curvature. Trans. Am. Math. Soc. 22, 84–100. FdM 48,786MathSciNetzbMATHCrossRefGoogle Scholar
  66. Mozes, S. (1992): Mixing of all orders of Lie group actions. Invent. Math. 107, No. 2, 235–241. Zbl. 783.28011MathSciNetzbMATHGoogle Scholar
  67. Nicholls, P.J. (1989): The Ergodic Theory of Discrete Groups. Lond. Math. Soc. Lect. Note Ser. 143, Cambridge. Zbl. 674.58001Google Scholar
  68. Ornstein, D.S. (1974): Ergodic Theory, Randomness, and Dynamical Systems. Yale Univ. Press, New Haven London. Zbl. 296.28016zbMATHGoogle Scholar
  69. Ornstein, D.S., Weiss, B. (1973): Geodesic flows are Bernoullian. Isr. J. Math. 14, No. 1, 184–198. Zbl. 256.58006MathSciNetzbMATHGoogle Scholar
  70. Parasyuk, O.S. (1953): Horocycle flows on surfaces of constant negative curvature. Usp. Mat. Nauk 8, No. 3, 125–126. Zbl. 52,342MathSciNetzbMATHGoogle Scholar
  71. Parry, W. (1969): Ergodic properties of affine transformations and flows on nilmanifolds. Am. J. Math. 91, 751–777. Zbl. 183,515MathSciNetCrossRefGoogle Scholar
  72. Parry, W. (1971): Metric classification of ergodic nilflows and unipotent affines. Am. J. Math. 93, 819–828. Zbl. 222.22010MathSciNetzbMATHCrossRefGoogle Scholar
  73. Pesin, Ya.B. (1977): Characteristic Lyapunov exponents and smooth ergodic theory. Usp. Mat. Nauk 32, No. 4, 55–111. [English transi.: Russ. Math. Surv. 32, No. 4, 55–114 (1977)] Zbl. 359.58010Google Scholar
  74. Pesin, Ya.B. (1981): Geodesic flows with hyperbolic behaviour of the trajectories and objects associated with them. Usp. Mat. Nauk 36, No. 4, 3–51. [English transi.: Russ. Math. Surv. 36, No. 4, 1–59 (1981)] Zbl. 482.58002Google Scholar
  75. Raghunathan, M.S. (1972): Discrete Subgroups of Lie Groups. Springer-Verlag, Berlin Heidelberg New York. Zbl. 254.22005zbMATHCrossRefGoogle Scholar
  76. Ratner, M. (1978): Horocycle flows are loosely Bernoullian. Isr. J. Math. 31, No. 2, 122–132. Zbl. 417.58013MathSciNetzbMATHGoogle Scholar
  77. Ratner, M. (1979): The Cartesian square of the horocycle flow is not loosely Bernoullian. Isr. J. Math. 34, No. 12, 72–96. Zbl. 437.28010MathSciNetzbMATHGoogle Scholar
  78. Ratner, M. (1982a): Rigidity of horocycle flows. Ann. Math.. II. Ser. 115, 597–614. Zbl. 506.58030MathSciNetzbMATHCrossRefGoogle Scholar
  79. Ratner, M. (1982b): Factors of horocycle flows. Ergodic Theory Dyn. Syst. 2, No. 34, 465–485. Zbl. 536.58029MathSciNetzbMATHGoogle Scholar
  80. Ratner, M. (1983): Horocycle flows, joinings and rigidity of products. Ann. Math., II. Ser. 118, 277–313. Zbl. 556.28020MathSciNetzbMATHCrossRefGoogle Scholar
  81. Ratner, M. (1987): The rate of mixing for geodesic and horocycle flows. Ergodic Theory Dyn. Syst. 7, No. 2, 267–288. Zbl. 623.22008MathSciNetzbMATHGoogle Scholar
  82. Ratner, M. (1986): Rigidity of time changes for horocycle flows. Acta Math. 156, No. 12, 1–32. Zbl. 694.58036MathSciNetzbMATHCrossRefGoogle Scholar
  83. Ratner, M. (1990): Invariant measures for unipotent translations on homogeneous spaces. Proc. Natl. Acad. Sci. USA, 4309–4311Google Scholar
  84. Ratner, M.(1991a): On Raghunathan’s measure conjecture. Ann. Math., II. Ser. 134, 545–607. Zbl. 763.28012MathSciNetzbMATHCrossRefGoogle Scholar
  85. Ratner, M.(1991b): Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63, No. 1, 235–280. Zbl. 733.22007MathSciNetzbMATHGoogle Scholar
  86. Ratner, M. (1993): to appear in Bull. Am. Math. SocGoogle Scholar
  87. Ryzhikov, V. (1991): Mixing property of a flow and its connection with the isomorphism between the transformation in the flow. Mat. Zametki 49, No. 6, 98–106. [English transi.: Math. Notes 49, 621–627 (1991)1 Zbl. 777.28008Google Scholar
  88. Safonov, A.V. (1982): On the spectral type of ergodic G-induced flows. Izv. Vyssh. Uchebn. Zaved., Mat. 1982, No. 6, (241), 42–47. [English transi.: Sov. Math. 26, No. 6, 47–54 (1982)] Zbl. 498.22008Google Scholar
  89. Schmidt, K. (1980): Asymptotically invariant sequences and an action of SL(2,Z) on the 2-sphere. Isr. J. Math. 37, No. 3, 193–208. Zbl. 485.28018MathSciNetzbMATHGoogle Scholar
  90. Schmidt, W.M. (1966): On badly approximable numbers and certain games. Trans. Am. Math. Soc. 123,178–199. Zbl. 232.10029zbMATHCrossRefGoogle Scholar
  91. Shah, N. (1991): Uniformly distributed orbits of certain flows on homogeneous spaces. Math. Ann. 289, 315–334. Zbl. 702.22014MathSciNetzbMATHCrossRefGoogle Scholar
  92. Sinai, Ya.G. (1960): Geodesic flows on manifolds of constant negative curvature. Dokl. Akad. Nauk SSSR 131, 752–755. [English transi.: Soy. Math., Dokl. 1, 335–339 (1960)] Zbl. 129,311Google Scholar
  93. Sinai, Ya.G. (1961): Geodesic flows on compact surfaces of negative curvature. Dokl. Akad. Nauk SSSR 136, 549–552.MathSciNetGoogle Scholar
  94. Sinai, Ya.G.: English transi.: Sov. Math., Dokl. 2, 106–109 (1961) Zbl. 133,110MathSciNetzbMATHGoogle Scholar
  95. Sinai, Ya.G. (1966): Classical dynamical systems with Lebesgue spectrum of countable multiplicity II. Izv. Akad. Nauk SSSR, Ser. Mat. 30, No. 1, 15–68.MathSciNetGoogle Scholar
  96. Sinai, Ya.G.: English transl.: Am. Math. Soc., Transl., II. Ser. 68, 34–88 (1968) Zbl. 213,340Google Scholar
  97. Starkov, A.N. (1983): Ergodic behaviour of flows on homogeneous spaces. Dokl. Akad. Nauk SSSR 273, 538–540.MathSciNetGoogle Scholar
  98. Sinai, Ya.G.: English transl.: Sov. Math., Dokl. 28, 675–676 (1983) Zbl. 563.43010Google Scholar
  99. Starkov, A.N. (1984): Flows on compact solvmanifolds. Mat. Sb., Nov. Ser. 123, 549–556.MathSciNetGoogle Scholar
  100. Sinai, Ya.G.: English transl.: Math USSR, Sb. 51, 549–556 (1985) Zbl. 545.28013CrossRefGoogle Scholar
  101. Starkov, A.N. (1986): On spaces of finite volume. Vestn. Mosk. Univ., Ser. I 1986, No. 5, 64–66.MathSciNetGoogle Scholar
  102. Sinai, Ya.G.: English transi.: Mosc. Univ. Math. Bull. 41, No. 5, 56–58 (1986) Zbl. 638.22007MathSciNetGoogle Scholar
  103. Starkov, A.N. (1987a): On an ergodicity criterion for G-induced flows. Usp. Mat. Nauk 42, No. 3, 197–198.MathSciNetzbMATHGoogle Scholar
  104. Starkov, A.N.: English transi.: Russ. Math. Surv. 42, No. 3, 233–234 (1987) Zbl. 629.22009MathSciNetzbMATHCrossRefGoogle Scholar
  105. Starkov, A.N. (1987b): Solvable homogeneous flows. Mat. Sb., Nov. Ser. 134, 242–259.Google Scholar
  106. Starkov, A.N.: English transi.: Math. USSR, Sb. 62, No. 1, 243–260 (1989) Zbl. 653.58034MathSciNetzbMATHCrossRefGoogle Scholar
  107. Starkov, A.N. (1989): Ergodic decomposition of flows on homogeneous spaces of finite volume. Mat. Sb. 180,1614–1633.Google Scholar
  108. Starkov, A.N.: English transl.: Math. USSR, Sb. 68, No. 2, 438–502 (1989)MathSciNetGoogle Scholar
  109. Starkov, A.N. (1990): Structure of orbits of homogeneous flows and Raghunathan’s conjecture. Usp. Mat. Nauk 45, No. 2, 219–220.MathSciNetGoogle Scholar
  110. Starkov, A.N.: English transl.: Russ. Math. Surv. 45, No. 2, 227–228 (1990) Zbl. 722.58034MathSciNetzbMATHCrossRefGoogle Scholar
  111. Starkov, A.N. (1991): Horospherical flows on homogeneous spaces of finite volume. Mat. Sb. 182,No. 5, 774–784.zbMATHGoogle Scholar
  112. Starkov, A.N.: English transi.: Math. USSR, Sb. 73, No. 1, 161–170 (1992) Zbl. 742.58042MathSciNetCrossRefGoogle Scholar
  113. Starkov, A.N. (1992): Rigidity criterion for lattices in solvable Lie groups. Workshop in Ergodic Theory and Lie groups, MSRI, April 1992, 59–60Google Scholar
  114. Starkov, A.N. (1993): On mixing of higher orders of homogeneous flows. To appear in Dokl. Ross. Akad. NaukGoogle Scholar
  115. Stepin, A.M. (1969): Flows on solvmanifolds. Usp. Mat. Nauk 14, No. 5, 241–242. Zbl. 216,345MathSciNetGoogle Scholar
  116. Stepin, A.M. (1973): Dynamical systems on homogeneous spaces of semisimple Lie groups. Izv. Akad. Nauk SSSR, Ser. Mat. 37, No. 5, 1091–1107.MathSciNetGoogle Scholar
  117. Starkov, A.N.: English transl.: Math. USSR, Izv. 7, 1089–1104 (1975) Zbl. 314.28014Google Scholar
  118. Sullivan, D. (1981): On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. Riemann surfaces and related topics. Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 465–496. Zbl. 567.58015Google Scholar
  119. Sullivan, D. (1982): Discrete conformal groups and measurable dynamics. Bull. Am. Math. Soc., New Ser. 6, No. 1, 57–73. Zbl. 489.58027CrossRefGoogle Scholar
  120. Tomter, P. (1975): On the classification of Anosov flows. Topology 14, No. 2, 179–189. Zbl. 365.58013MathSciNetzbMATHCrossRefGoogle Scholar
  121. Veech, W.A. (1977): Unique ergodicity of horospherical flows. Am. J. Math. 99, 827–859. Zbl. 365.28012MathSciNetzbMATHCrossRefGoogle Scholar
  122. Vinberg, Eh.B., Gorbatsevich, V.V., Shvartsman, O.V. (1988): Lie Groups and Lie Algebras II: Discrete Subgroups of Lie Groups. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 21, 5–120. [English transi. in: Encycl. Math. Sci. 21. Springer-Verlag, Berlin Heidelberg New York, in preparation] Zbl. 656.22004Google Scholar
  123. Witte, D. (1985): Rigidity of some translations on homogeneous spaces. Bull. Am. Math. Soc., New Ser. 12, No. 1, 117–119 [see also Invent. Math. 81, 1–27 (1985)] Zbl. 558.58025Google Scholar
  124. Witte, D. (1987): Zero-entropy affine maps on homogeneous spaces. Am. J. Math. 109,927–961. Zbl. 653.22005MathSciNetzbMATHCrossRefGoogle Scholar
  125. Witte, D. (1989): Rigidity of horospherical foliations. Ergodic Theory Dyn. Syst. 9, 191–205. Zbl. 667.58053MathSciNetzbMATHCrossRefGoogle Scholar
  126. Witte, D. (1990): Topological equivalence of foliations of homogeneous spaces. Trans. Am. Math. Soc. 317,143–166. Zbl. 689.22006MathSciNetzbMATHCrossRefGoogle Scholar
  127. Witte, D. (1992): Measurable quotients of unipotent translations on homogeneous spaces. To appear in Trans. Am. Math. SocGoogle Scholar
  128. Zimmer, R.J. (1984): Ergodic Theory and Semisimple Groups. Birkhäuser, Boston. Zbl. 571.58015zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • A. V. Safonov
  • A. N. Starkov
  • A. M. Stepin

There are no affiliations available

Personalised recommendations