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Published online by Cambridge University Press:  05 July 2011

Anatole Katok
Affiliation:
Pennsylvania State University
Viorel Niţică
Affiliation:
West Chester University, Pennsylvania
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References

[1] R. L., Adler, B., Weiss. Entropy, a complete metric invariant for automorphisms of the torus. Proc. Nat. Acad. Sci. 57 (1967) 1575–1576.Google Scholar
[2] D. V., Anosov. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Stek. Inst. 90 (1967) 1–235.Google Scholar
[3] N., Aoki. A simple proof of Bernoullicity of automorphisms of compact abelian groups. Isr. J. Math. 38 (1981) 189–198.Google Scholar
[4] L., Auslander, J., Scheuneman. On certain automorphisms of nilpotent Lie groups. Proc. Symp. Pure Math. 14 (1970) 9–15.Google Scholar
[5] H., Bercovici, V., Niţică. A Banach algebra version of the Livšic theorem. Discrete Contin. Dynam. Systems 4 (1998) 523–534.Google Scholar
[6] L., Barreira, Ya., Pesin. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge: Cambridge University Press, 2007.
[7] D., Berend. Multi-invariant sets on tori. Trans. Amer. Math. Soc. 280 (1983) 509–532.Google Scholar
[8] Z. I., Borevich, I. R., Shafarevich. Number Theory. New York: Academic Press, 1966.
[9] R., Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. 470. New York: Springer, 1975.
[10] G. E., Bredon. Introduction to Compact Transformations Groups. New York: Academic Press, 1972.
[11] M. I., Brin. Topological transitivity of a class of dynamical systems and frame flow on manifolds of negative curvature. Func. Anal. and Appl. 9 (1975) 9–19.Google Scholar
[12] M. I., Brin, Y. A., Pesin. Partially hyperbolic dynamical systems. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 170–212.Google Scholar
[13] K. S., Brown. Cohomology of Groups. Graduate Texts in Math. 87. Berlin: Springer-Verlag, 1982.
[14] S., Campanato. Proprieta di una famiglia di spazi functionali. Ann. Scuola Norm. Sup. Pisa 18 (1964) 137–160.Google Scholar
[15] C., Chevalley. Deux théorèmes d'Arithmétique. J. Math. Soc. of Japan 3 (1951) 36–44.Google Scholar
[16] H., Cohen. A Course in Computational Algebraic Number Theory. Berlin-Heidelberg-New York: Springer, 1996.
[17] P., Collet, H., Epstein, G., Gallavotti. Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties. Com. Math. Phys. 95 (1984) 61–112.Google Scholar
[18] M., Cowling. Sur les Coeficients des Representations Unitaires des Groupes de Lie Simple. Lecture Notes in Math. 739. Berlin: Springer-Verlag, 1979, pp. 132–178.Google Scholar
[19] D., Damjanovic. Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions. J. Mod. Dyn. 1 (2007) 665–688.Google Scholar
[20] D., Damjanovic, A., Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Res. Announce. Amer. Math. Soc. 10 (2004) 142–154.Google Scholar
[21] D., Damjanovic, A., Katok. Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic actions. Discrete Contin. Dynam. Systems 13 (2005) 985–1005.Google Scholar
[22] D., Damjanovic, A., Katok. Local rigidity of partially hyperbolic actions I. KAM method and ℤk actions on the torus. Annals Math. (2010), to appear.Google Scholar
[23] D., Damjanovic, A., Katok. Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl chamber flows on SL(n, ℝ)/ Г, available online at http://www.math.psu.edu/katok a.
[24] D., Damjanovic, A., Katok. Local rigidity of homogeneous parabolic actions: I. A model case, preprint, 2010.
[25] S. G., Dani, M. G., Mainkar. Anosov automorphisms on compact nilmanifolds associated with graphs. Trans Amer. Math. Soc. 357 (2005) 2235–2251.Google Scholar
[26] P., Didier. Stability of accessibility. Ergodic Theory Dynam. Systems 23 (2003) 1717–1731.Google Scholar
[27] D., Dolgopyat, A., Wilkinson. Stable accessibility is C1 dense. Geometric methods in dynamics. II. Astérisque 287 (2003) 33–60.Google Scholar
[28] M., Einsiedler, A., Katok. Invariant measures on G/Г for split simple Lie Groups G. Comm. Pure. Appl. Math. (Moser memorial issue) 56 (2003) 1184–1221.Google Scholar
[29] M., Einsiedler, A., Katok. Rigidity of measures – the high entropy case and non-commuting foliations. Israel Math. J. 148 (2005) 169–238.Google Scholar
[30] M., Einsiedler, A., Katok, E., Lindenstrauss. Invariant measures and the set of exceptions to Littlewood's conjecture. Annals of Math. 164 (2006) 513–560.Google Scholar
[31] M., Einsiedler, E., Lindenstrauss. Rigidity properties of ℤd -actions on tori and solenoids. Electronic Res. Announce. Math. Soc. 9 (2004) 99–110.CrossRefGoogle Scholar
[32] R., Feres, A., Katok. Ergodic theory and dynamics of G-spaces, in Handbook of Dynamical Systems, vol. 1A. Amsterdam: Elsevier, 2002, pp. 665–763.
[33] S., Ferleger, A., Katok. Non-commutative first cohomology rigidity of the Weyl chamber flows, preprint, 1997.
[34] D., Fisher. Local rigidity of group actions: past, present, future, in Dynamics, Ergodic Theory and Geometry. Cambridge: Cambridge University Press, 2007.
[35] D., Fisher, G., Margulis. Almost isometric actions, property T, and local rigidity. Inventiones Math. 162 (2005) 19–80.CrossRefGoogle Scholar
[36] D., Fisher, G., Margulis. Local rigidity of affine actions of higher rank Lie groups and their lattices. Annals Math. 170 (2009) 67–122.Google Scholar
[37] L., Flaminio, G., Forni. Invariant distributions and time averages for horocycle flows. Duke Math. J. 119 (2003) 465–526.Google Scholar
[38] G., Forni. Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. Math. 146 (1997) 295–344.Google Scholar
[39] J., Franks. Anosov diffeomorphisms on tori. Trans. Amer. Math. Soc. 145 (1969) 117–124.CrossRefGoogle Scholar
[40] H., Furstenberg. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967) 1–49.Google Scholar
[41] E., Goetze, R. J., Spatzier. Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices. Ann. Math. 150 (1999) 743–773.Google Scholar
[42] E., Goetze, R. J., Spatzier. On Livsic's theorem, superrigidity, and Anosov actions of semisimple Lie groups. Duke Math. J. 88 (1997) 1–27.Google Scholar
[43] V., Guillemin, D., Kazhdan. Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980) 301–313.Google Scholar
[44] R., Hamilton. The inverse limit theorem of Nash and Moser. Bull. Amer. Math. Soc. 7 (1982) 65–222.Google Scholar
[45] P., de la Harpe, A., Valette. La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque 175 (1989).Google Scholar
[46] B., Hasselblatt. Hyperbolic dynamical systems, in Handbook of Dynamical Systems, vol. 1A. Amsterdam: Elsevier, 2002, pp. 239–319.
[47] B., Hasselblatt, A., Katok. Principal structures, in Handbook of Dynamical Systems, vol. 1A. Amsterdam: Elsevier, 2002, pp. 1–203.
[48] S., Helgason. Differential Geometry, Lie Groups and Symmetric Spaces. New York: Academic Press, 1978.
[49] F. R., Hertz, M. A. R., Hertz, R., Ures. Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1d-center bundle. Inventiones Math. 172 (2008) 353–381.Google Scholar
[50] M., Hirsch. Differential Topology. Graduate Texts in Math. 33. New York: Springer-Verlag 1976.
[51] M., Hirsch, C., Pugh, M., Shub. Invariant Manifolds. Lecture Notes in Math. 583. Berlin: Springer Verlag, 1977.
[52] L., Hörmander. Hypoelliptic second order differential equations. Acta Mathematica 119 (1967) 147–171.Google Scholar
[53] R., Howe. A notion of rank for unitary representations of the classical groups, in A., Figa Talamanca (ed.), Harmonic Analysis and Group Representations. Firenze, Italy: CIME, 1980.
[54] S., Hurder. Affine Anosov actions. Michigan Math. J. 40 (1993) 561–575.Google Scholar
[55] S., Hurder. Rigidity of Anosov actions of higher rank lattices. Annals Math. 135 (1992) 361–410.Google Scholar
[56] S., Hurder, A., Katok. Differentiability, rigidity and Godbillon–Vey classes for Anosov flows. Publ. Math. IHES 72 (1990) 5–61.CrossRefGoogle Scholar
[57] H. C., Im Hof. An Anosov action on the bundle of Weyl chambers. Ergodic Theory Dynam. Systems 5 (1985) 587–593.Google Scholar
[58] J.-L., Journé. A regularity lemma for functions of several variables. Revista Matematica Iberoamericana 4 (1988) 187–193.Google Scholar
[59] B., Kalinin. Livšic theorem for matrix cocycles. Annals of Math, to appear.
[60] B., Kalinin, A., Katok. Invariant measures for actions of higher rank abelian groups, in Smooth Ergodic Theory and its Applications. Proc. Symp. Pure Math 69. Providence, RI: Amer. Math. Soc., 2001, pp. 593–637.
[61] B., Kalinin, A., Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori. J. Mod. Dyn. 1 (2007) 123–146.Google Scholar
[62] B., Kalinin, A., Katok, F., Rodriguez Hertz. Nonuniform measure rigidity. Annals of Math., to appear.
[63] B., Kalinin, V., Sadovskaya. On local and global rigidity of quasi-conformal Anosov diffeomorphisms. J. Inst. Math. Jussieu 2–4 (2003) 567–582.Google Scholar
[64] B., Kalinin, R., Spatzier. On the classification of Cartan actions. Geom. Func. Anal. 17 (2007) 468–490.CrossRefGoogle Scholar
[65] M., Kanai. Rigidity of Weil chamber flow, and vanishing theorems of Matsushima and Weil. Ergod. Theory Dynam. Systems 29 (2009) 1273–1288.CrossRefGoogle Scholar
[66] A., Katok. Combinatorial Constructions in Ergodic Theory and Dynamics. University Lecture Series 30. Providence, RI: Agmerican Mathematical Society, 2003.
[67] A., Katok, B., Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press, 1995.
[68] A., Katok, S., Katok. Higher cohomology for abelian groups of toral automorphisms. Ergodic Theory Dynam. Systems 15 (1995) 569–592.Google Scholar
[69] A., Katok, S., Katok. Higher cohomology for abelian groups of toral automorphisms. II. The partially hyperbolic case, and corrigendum. Ergodic Theory Dynam. Systems 25 (2005) 1909–1917.CrossRefGoogle Scholar
[70] A., Katok, S., Katok, K., Schmidt. Rigidity of measurable structure for ℤd -actions by automorphisms of a torus. Comm. Math. Helvetici 77 (2002) 718–745.Google Scholar
[71] A., Katok, A., Kononenko. Cocycles' stability for partially hyperbolic systems. Math. Res. Lett. 3 (1996) 191–210.Google Scholar
[72] A., Katok, V., Niţică. Rigidity of higher rank abelian cocycles with values in diffeomorphism groups. Geometriae Dedicata. 124 (2007) 109–131.Google Scholar
[73] A., Katok, V., Niţică, A., Török. Non-abelian cohomology of abelian Anosov actions. Ergodic Theory Dynam. Systems 2 (2000) 259–288.Google Scholar
[74] A., Katok, F., Rodriguez Hertz. Uniqueness of large invariant measures for ℤk actions with Cartan homotopy data. J. Mod. Dyn. 1 (2007) 287–300.Google Scholar
[75] A., Katok, F., Rodriguez Hertz. Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher rank abelian groups. J. Mod. Dyn. 4 (2010), to appear.Google Scholar
[76] A., Katok, F., Rodriguez Hertz. Rigidity of real-analytic actions of SL(n, ℤ) on Tn: A case of realization of Zimmer program. Discrete Contin. Dynam. Systems 27 (2010) 609–615.CrossRefGoogle Scholar
[77] A., Katok, K., Schmidt. The cohomology of expansive ℤd -actions by automorphisms of compact, abelian groups. Pacific J. Math 170 (1995) 105–142.CrossRefGoogle Scholar
[78] A., Katok, R., Spatzier. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Inst. Hautes Études Sci. Publ. Math. 79 (1994) 131–156.Google Scholar
[79] A., Katok, R., Spatzier. Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math. Res. Letters 1 (1994) 193–202.Google Scholar
[80] A., Katok, R., Spatzier. Invariant measures for higher rank hyperbolic Abelian actions. Ergodic Theory Dynam. Systems 16 (1996) 751–778.Google Scholar
[81] A., Katok, R., Spatzier. Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Tr. Mat. Inst. Steklova 216 (1997) 292–319.Google Scholar
[82] S., Katok. Finite spanning sets for cusp forms and a related geometric result. J. Reine Angew. Math. 395 (1989) 186–195.Google Scholar
[83] Y., Katznelson. Ergodic automorphisms on Tn are Bernoulli shifts. Israel J. Math. 10 (1971) 186–195.Google Scholar
[84] D., Kleinbock, N., Shah, A., Starkov. Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory in: Handbook of Dynamical Systems, vol. 1A. Amsterdam: Elsevier, 2002, pp. 813–930.
[85] A. W., Knapp. Representation Theory of Semisimple Groups: an Overview Based on Examples. Princeton, NJ: Princeton University Press, 2001.
[86] A., Kononenko. Twisted cocycles and rigidity problems. Electron. Res. Announc. Amer. Math. Soc. 1 (1995) 26–34.CrossRefGoogle Scholar
[87] N., Kopell. Commuting diffeomorphisms. Proc. Symp. Pure Math. 14 (1970) 165–184.CrossRefGoogle Scholar
[88] S., Krantz. Lipschitz spaces, smoothness of functions and approximation theory. Expo. Math. 3 (1983) 193–260.Google Scholar
[89] R., Krikorian. Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on T × SL(2, ℝ), preprint.
[90] S., Lang. Algebra. Reading, MA: Addison-Wesley, 1984.
[91] S., Lang. Introduction to Differentiable Manifolds. New York: Interscience, 1962.
[92] J., Lauret. Examples of Anosov diffeomorphisms. J. Algebra 262 (2003) 201–209.Google Scholar
[93] F., Ledrappier. Un champ markovien peut être d'entropie nulle et mélangeant. C. R. Acad. Sci. Paris Sér. A-B 287 (1978) A561–A563.Google Scholar
[94] K. B., Lee, F., Raymond. Geometric realization of group extensions by the Seifert construction. Contemporary Math. AMS 33 (1984) 353–411.Google Scholar
[95] E., Lindenstrauss. Rigidity of multiparameter actions. Israel Math. J. 149 (2005) 199–226.Google Scholar
[96] A., Livšic. Homology properties of U-systems. Math. Zametki 10 (1971) 758–763.Google Scholar
[97] A., Livšic. Cohomology of dynamical systems. Math. USSR Izvestija 6 (1972) 1278–1301.Google Scholar
[98] R., de la Llave. Invariants for smooth conjugacy of hyperbolic dynamical systems. I. Comm. Math. Phys. 109 (1987) 369–378.Google Scholar
[99] R., de la Llave. Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic dynamical systems. Commun. Math. Phys. 150 (1992) 289–320.Google Scholar
[100] R., de la Llave. Analytic regularity of solutions of Livšic's cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems. Ergodic Theory Dynam. Systems 17 (1997) 649–662.Google Scholar
[101] R., de la Llave. Remarks on Sobolev regularity in Anosov systems. Ergodic Theory Dynam. Systems 21 (2001) 1139–1180.Google Scholar
[102] R., de la Llave. Tutorial on KAM theory, in Smooth Ergodic Theory and its Applications Proc. Symp. Pure Math 69. RI: American Mathematical Society, Providence, 2001, pp. –.
[103] R., de la Llave. Bootstrap of regularity for integrable solutions of cohomology equations, in Modern Dynamical Systems and Applications, M., Brin, B., Hasselblatt, Ya. B., Pesin (eds). Cambridge: Cambridge University Press, 2004, pp. 405–418.
[104] R., de la Llave, J., Marco, R., Moriyon. Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann. of Math. 123 (1986) 537–611.Google Scholar
[105] R., de la Llave, R., Moriyon. Invariants for smooth conjugacy of hyperbolic dynamical systems IV. Comm. Math. Phys. 116 (1988) 185–192.Google Scholar
[106] C., Lobry. Controllability of nonlinear systems on compact manifolds. SIAM J. Control 12 (1974) 1–4.Google Scholar
[107] A., Malćev. On a class of homogenous spaces. Transl. Amer. Math. Soc. 1 (1962) 276–307.Google Scholar
[108] W., Malfait. An obstruction to the existence of Anosov diffeomorphisms on infra-nilmanifolds. Contemporary Math. 262 (2000) 233–251.Google Scholar
[109] A., Manning. There are no new Anosov diffeomorphisms on tori. Amer. J. Math. 96 (1974) 422–429.Google Scholar
[110] G. A., Margulis. Discrete Subgroups of Semisimple Lie Groups.Berlin: Springer Verlag, 1991.
[111] G. A., Margulis, N., Qian. Local rigidity of weakly hyperbolic actions of higher rank real Lie groups and their lattices, Ergodic Theory Dynam. Systems 21 (2001), 121–164.Google Scholar
[112] H., Matsumoto. Sur les sous-groupes arithmétiques des groupes semi-simples dÉployÉs. Ann. Sci. Éc. Norm. Sup. 4, serie 2 (1969) 1–62.Google Scholar
[113] D., Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. J. Modern Dynam. 1 (2007) 61–92.Google Scholar
[114] G., Metivier. Function spectrale et valeur propres d'une classe d'operateurs non elliptiques. Communi. PDE 1 (1976), 467–519.Google Scholar
[115] J., Milnor. Introduction to Algebraic K-theory. Princeton, NJ: Princeton University Press, 1971.
[116] D., Montgomery, L., Zippin. A theorem on Lie groups. Bull. Amer. Math. Soc. 48 (1942) 448–452.CrossRefGoogle Scholar
[117] C. C., Moore. Exponential decay of correlation coefficients for geodesic flows, in C. C., Moore (ed.), Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics. Proceedings of a Conference in Honor of George Mackey. MSRI publications, Springer Verlag, New York: 1987.
[118] C. C., Moore. Decomposition of unitary representations defined by discrete subgroups of nilpotent groups. Annals of Math. 82 (1965), 146–182.Google Scholar
[119] N., Mok, Y. T., Siu, S. K., Yeung. Geometric superrigidity. Invent. Math. 113 (1993), 57–83.Google Scholar
[120] D., Montgomery, L., Zippin, Topological Transformation Groups. New York: Interscience Publishers, 1955.
[121] M. H. A., Newman. A theorem on periodic transformations of spaces. Quart. J. Math. Oxford Ser. 2 (1931), 1–9.Google Scholar
[122] M., Nicol, M., Pollicott. Measurable cocycle rigidity for some non-compact groups. Bull. London Math. Soc. 311 (1999) 529–600.Google Scholar
[123] M., Nicol, M., Pollicott. Livšic's theorem for semisimple Lie groups. Ergodic Theory Dynam. Systems 21 (2001) 1501–1509.Google Scholar
[124] M., Nicol, A., Török. Whitney regularity for the solutions of the coboundary equations on Cantor sets. Math. Phys. Electronic J. 13 (2007) paper 6.Google Scholar
[125] V., Niţică, A., Török. Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher rank lattices. Duke Math. J. 79 (1995) 751–810.Google Scholar
[126] V., Niţică, A., Török. Regularity results for the solutions of the Livshits cohomology equation with values in diffeomorphism groups. Ergodic Theory Dynam. Systems 16 (1996) 325–333.Google Scholar
[127] V., Niţică, A., Török. Regularity of the transfer map for cohomologous cocycles. Ergodic Theory Dynam. Systems 18 (1998) 1187–1209.CrossRefGoogle Scholar
[128] V., Niţică, A., Török. On the cohomology of Anosov actions, in Rigidity in Dynamics and Geometry. Berlin: Springer, 2000, pp. 345–361.
[129] V., Niţică, A., Török. An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one. Topology 40 (2001) 259–278.Google Scholar
[130] V., Niţică, A., Török. Cocycles over abelian TNS actions. Geometriae Dedicata 102 (2003) 65–90.Google Scholar
[131] V., Niţică. Journé's theorem for Cn,ω regularity. Discrete Contin. Dynam. Systems 22 (2008) 413–425.CrossRefGoogle Scholar
[132] A. L., Onishchik, E. B., Vinberg. Lie Groups and Lie Algebras.Berlin: Springer-Verlag, 1994.
[133] D., Ornstein. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970) 337–352.Google Scholar
[134] http://pari.math.u-bordeaux.fr/
[135] W., Parry. The Livšic periodic point theorem for non-abelian cocycles. Ergodic Theory Dynam. Systems 19 (1999) 687–701.Google Scholar
[136] W., Parry, M., Pollicott. The Livšic cocycle equation for compact Lie group extensions of hyperbolic systems. J. London Math. Soc. 56 (1997) 405–416.Google Scholar
[137] W., Parry, M., Pollicott. Skew-products and Livsic theory, in Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, V. A., Kaimanovich, A., Lodkin (eds). Adv. Math. Sci. Series 2, 217. Providence, RI: American Mathematical Society, 2006.
[138] M., Pollicott, C. P., Walkden. Livšic theorems for connected Lie groups. Trans. Amer. Math. Soc. 353 (2001) 2879–2895.Google Scholar
[139] M., Pollicott, M., Yuri. Regularity of solutions to the measurable Livshits equation. Trans. Amer. Math. Soc. 351 (1999) 559–568.Google Scholar
[140] H. L., Porteous. Anosov difeomorphisms of flat manifolds. Topology 11 (1972) 307–315.Google Scholar
[141] M., Postnikov. Lie Groups and Lie Algebras. Moskow: Mir Publishers, 1986.
[142] G., Prasad, M. S., Raghunathan. Cartan subgroups and lattices in semi-simple groups. Ann. Math. 96 (1972) 296–317.Google Scholar
[143] C., Pugh, M., Shub. Ergodicity of Anosov actions. Invent. Math. 15 (1972) 1–23.CrossRefGoogle Scholar
[144] C., Pugh, M., Shub. Stable ergodicity and julienne quasi-conformality. J. Eur. Math. Soc. (JEMS) 2 (2000) 1–52.CrossRefGoogle Scholar
[145] N., Qian. Rigidity Phenomena of group actions on a class of nilmanifolds and Anosov ℝn actions. Unpublished Ph.D. thesis, California Insitute of Technology, 1992.
[146] M. S., Raghunathan. Discrete Subgroups of Lie Groups. Berlin: Springer-Verlag, 1972.
[147] M., Ratner. The rate of mixing for geodesic and horocycle flows. Ergodic Theory Dynam. Systems 7 (1987) 267–288.Google Scholar
[148] M., Ratner. On Raghunathan's measure conjecture. Ann. of Math. 134 (1991) 545–607.Google Scholar
[149] B. L., ReinhartDifferential Geometry of Foliations. Berlin: Springer-Verlag, 1983.
[150] C., Rockland. Hypoellipticity on the Heisenberg group: representation theoretic criteria. Trans. Amer. Math. Soc. 240 (1978) 1–52.Google Scholar
[151] F., Rodriguez Hertz. Global rigidity of certain abelian actions by toral automorphisms. J. Modern Dynam., to appear.
[152] L. P., Rothschild. A criterion for hypoellipticity of operators constructed from vector fields. Commun. PDE 4 (1979) 645–699.Google Scholar
[153] L. P., Rothschild, E., Stein. Hypoelliptic differential operators and nilpotent groups. Acta Math. 1976 247–320.
[154] D., Rudolph. ×2 and ×3 invariant measures and entropy. Ergodic Theory Dynam. Systems 10 (1990) 395–406.Google Scholar
[155] K., Schmidt. The cohomology of higher-dimensional shifts of finite type. Pacific J. Math. 170 (1995) 237–269.Google Scholar
[156] K., Schmidt. Cohomological rigidity of algebraic ℤd -actions. Ergodic Theory Dynam. Systems 15 (1995) 759–805.Google Scholar
[157] K., Schmidt. Dynamical Systems of Algebraic Origin. Basel-Berlin-Boston: Birkhäuser Verlag, 1995.
[158] K., Schmidt. Remarks on Livšic' theory for non-abelian cocycles. Ergodic Theory Dynam. Systems 19 (1999) 703–721.Google Scholar
[159] K., Schmidt, T., Ward. Mixing automorphisms of compact groups and a theorem of Schlickewei. Inventiones Math. 111 (1993) 69–76.Google Scholar
[160] M., Shub. Endomorphisms of compact differentiable manifolds. Amer. J. Math. 96 (1974) 422–429.Google Scholar
[161] Y. G., Sinai. Gibbs measures in ergodic theory. Russ. Math. Surv. 27 (1972) 21–70.Google Scholar
[162] S., Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967) 747–817.Google Scholar
[163] R., Spatzier. Harmonic Analysis in Rigidity Theory. Ergodic theory and its connections with harmonic analysis. London Math. Soc. Lecture Notes Ser. 205. Cambridge: Cambridge University Press, 1995.
[164] A., Starkov. First cohomology group, mixing and minimal sets of commutative group of algebraic action on torus. J. Math. Sci (New York) 95 (1999) 2576–2582.Google Scholar
[165] N., Steenrod. The Topology of Fiber Bundles.Princeton, NJ: Princeton University Press, 1951.
[166] R., Steinberg. Générateurs, relations et revêtements de groupes algébraiques. Colloq. Theorie des groupes algebraiques, Bruxelles (1962) 113–127.Google Scholar
[167] E. M., Stein, G., Weiss. Introduction to Fourier Analysis on Fourier Spaces.Princeton, NJ: Princeton University Press, 1971.
[168] A., Unterberger, J., Unterberger. Hölder estimates and hypoellipticity. Ann. Inst. Fourier 26 (1976) 35–54.CrossRefGoogle Scholar
[169] W. A., Veech. Periodic points and invariant pseudomeasures for toral endomorphisms. Ergodic Theory Dynam. Systems 6 (1986) 449–473.Google Scholar
[170] C. P., Walkden. Solutions to the twisted cocycle equation over hyperbolic systems. Discrete Contin. Dynam. Systems 6 2000, 935–946.Google Scholar
[171] Z. J., Wang. Local rigidity of partially hyperbolic actions. J. Mod. Dyn. 4 (2010) 271–327.Google Scholar
[172] Z. J., Wang. New cases of differentiable rigidity for partially hyperbolic actions: symplectic groups and resonance directions. J. Mod. Dyn., to appear.
[173] G., Warner. Harmonic Analysis on Semisimple Lie Groups I. Berlin: Springer Verlag, 1972.
[174] A., Weil. On discrete subgroups of Lie groups I. Annals of Math. 72 (1960) 369–384.Google Scholar
[175] A., Weil. On discrete subgroups of Lie groups II. Annals of Math. 75 (1962) 578–602.Google Scholar
[176] A., Weil. Adels and Algebraic Groups. Progress in Mathematics 23. Boston, MA: Birkhäuser,
[177] E., Weiss. Algebraic Number Theory. New York: Chelsea Publishing Company, 1963.
[178] M. D., Witte. Ratner's Theorems on Unipotent flows. Chicago Lectures in Mathematics, 2005.
[179] R., Zimmer. Ergodic Theory and Semisimple Groups. Boston, MA: Birkhäuser, 1984.

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  • References
  • Anatole Katok, Pennsylvania State University, Viorel Niţică, West Chester University, Pennsylvania
  • Book: Rigidity in Higher Rank Abelian Group Actions
  • Online publication: 05 July 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511803550.008
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  • References
  • Anatole Katok, Pennsylvania State University, Viorel Niţică, West Chester University, Pennsylvania
  • Book: Rigidity in Higher Rank Abelian Group Actions
  • Online publication: 05 July 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511803550.008
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Anatole Katok, Pennsylvania State University, Viorel Niţică, West Chester University, Pennsylvania
  • Book: Rigidity in Higher Rank Abelian Group Actions
  • Online publication: 05 July 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511803550.008
Available formats
×