References. 1. V.I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (W.A. Benjamin, 1968)

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1 References 1. V.I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (W.A. Benjamin, 1968) 2. J.P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617 (1985) 3. V.I. Arnold, Mathematical Methods of Classical Mechanics, (Springer- Verlag, 1980) 4. R. Abraham and J.E. Marsden, Foundations of Mechanics, (Benjamin- Cummings, 1978) 5. Jurgen Moser, Stable and random motions in dynamical systems, with special emphasis on celestial mechanics, (Princeton University Press, 1973) 6. O. Lanford, Introduction to Hyperbolic Sets, in Erice 1983 Proceedings, G. Velo and A.S. Wightman editors 7. A.S. Wightman, Regular and Chaotic Motions in Dynamical Systems, Introduction to the Problems, in Erice 1983 Proceedings, G. Velo and A.S. Wightman editors 8. I.P. Cornfeld, S.V. Fomin, and Ya.G. Sinai, Ergodic Theory, (Springer- Verlag, 1982) 9. D. Ruelle, Dynamical systems with turbulent behavior, in Mathematical Problems in Theoretical Physics, Proceedings Rome 1977, Lecture Notes in Physics 80, (Springer-Verlag, 1978) 10. D. Ruelle, Measures describing a turbulent flow, in N.Y. Acad. of Sci., Nonlinear Dynamics 1980, R. Helleman editor 11. V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differenetial Equations, (Springer-Verlag, 1983) 107

2 12. D.V. Anosov, Geodesic Flows on Closed Manifolds with Negative Curvature Proc. Steklov Inst. of Math. 90, 1 (1967) 13. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73, 747 (1967) 14. Ya.B. Pesin, Lyapunov characteristic exponents and ergodic properties of smooth dynamical systems with an invariant measure, Soviet Math. Dokl. 17, 196 (1976) 15. M. Hirsch, C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics 583, (Springer-Verlag, 1977) 16. M. Shub, Global Stability of Dynamical systems, (Springer-Verlag, 1987) 17. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470, (Springer-Verlag, 1975) 18. Ya.B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR Izv. 10, 1261 (1976) 19. D. Ruelle, Ergodic theory of differentiable dynamical systems, Publications IHES 50, 275 (1979) 20. Ya.B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys 32:4, 55 (1977) 21. V.A. Rohlin, On the fundamental ideas of measure theory, Mat. Sbornik (N.S.) 25(67), 107 (1949); English translation, AMS Translations, series 1 10, 1 (1962) 22. D. Ruelle, A measure associated with axiom A attractors, Amer. J. Math. 98, 619 (1976) 23. Ya.G. Sinai, Introduction to ergodic theory, (Princeton University Press, 1977) 24. Ya.G. Sinai, Classical dynamical systems with countably-multiple Lebesgue spectrum, II, Izv. Akad. Nauk SSSR Ser. Mat 30, 15 (1966) 108

3 25. F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms, Part I: Characterization of measures satisfying Pesin s entropy formula, Annals of Math 122, 509 (1985) 26. F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms, Part II: Relations between entropy, exponents, and dimension, Annals of Math 122, 540 (1985) 27. A. Connes, Sur la theorie non commutative de l integration, in Algebres d Operateurs, Seminaire, Les Plans-sur-Bex, Suisse, 1978, Lecture Notes in Mathematics 725, (Springer-Verlag, 1979) 28. D. Kastler, On A. Connes Noncommutative Integration Theory, Comm. Math. Phys. 85, 99 (1982) 29. G. W. Mackey, Ergodic theory, group theory, and differential geometry, Proc. Nat. Acad. Sci. USA 50, 1184 (1963) 30. G. W. Mackey, Ergodic theory and virtual groups, Math. Ann. 166, 187 (1966) 31. A. Ramsey, Virtual groups and Group actions, Adv. Math. 6, 253 (1971) 32. P. Hahn, Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242, 1 (1978) 33. F. Ledrappier and J. M. Strelcyn, Estimation from below in Pesin s entropy formula, Ergod. Th. & Dynam. Sys. 2, 203 (1982) 34. Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis Manifolds and Physics, Revised Edition, (North Holland, 1982) 35. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. I, (Interscience, 1969) 36. Sheldon E. Newhouse, Lectures on Dynamical Systems, in C.I.M.E. Lectures, Bressanone, Italy, June 1978, J. Guckenhiemer, J. Moser, and S. Newhouse editors, (Birkhauser, 1980) 109

4 37. A.N. Lifshits and Ya. G. Sinai, On invariant measures compatible with the smooth structure for transitive U-systems, Soviet Math. Dokl. 13, 1656 (1972) 38. Ya.G. Sinai, Gibbs measures in ergodic theory, Usp. Mat. Nauk 27; English translation, Russ. Math. Surv. 27, 21 (1972) 39. R. de la Llave, J. M. Marco, and R. Moriyon, Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation, Ann. Math. 123, 537 (1986) 40. R. de la Llave, private communication. 41. Ya.B. Pesin and Ya.G. Sinai Hyperbolicity and Stochasticity of Dynamical Systems, in Mathematical Physics Reviews, Soviet Scientific Reviews/ Section C 2, (1981), S. P. Novikov editor 42. C.C. Pugh and M. Shub, Differentiability and continuity of invariant manifolds, N.Y. Acad. of Sci., Nonlinear Dynamics 1980, R. Helleman editor 43. V.I. Oseledec, Multiplicative ergodic theorem, Lyapunov numbers for dynamical systems, Trans. Moscow Math. Soc. 19, 197 (1968) 44. A. Fathi, M. herman, and J.-C. Yocozz, A Proof of Pesin s Stable Manifold Theorem, in Lecture Notes in Mathematics 1007, (Springer-Verlag, 1983) 45. V.A.Rohlin, Lectures on the entropy thoery of measure-preserving transformations, Russian Mathematical Surveys 22:5, 1 (1967) 46. D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat. 9, 83 (1978) 47. J. Milnor, On the concept of an attractor, Comm. Math. Phys. 99, 117 (1985) 48. D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys. 82, 137 (1981) 110

5 49. D. Ruelle, Small random perturbations and the definition of attractor, in Lecture Notes in Mathematics 1007, (Springer-Verlag, 1983) 50. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference, ser. 35, (American Mathematical Society, 1978) 51. Yu.I. Kifer, On small random perturbations of some smooth dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 38, 1091 (1974); English translation, Math. USSR Izv. 8, 1083 (1974) 52. A. Katok, Lyapunov exponents, entropy and peroidic orbits for diffeomorphisms, Publications IHES 51, 137 (1980) 53. Ya.G. Sinai, Markov partitions and C diffeomorphisms Func. Anal. and Appl. 2, 61 (1968) 54. R. Bowen and D. Ruelle, The Ergodic Theory of Axiom A Flows, Inv. Math. 29, 181 (1975) 55. Mane, Ricardo A proof of Pesin s formula Ergod. Th. & Dynam. Sys. 1, 95 (1981) 56. Ya.B. Pesin, Description of π-partition (sic) of a diffeomorphism with invariant measure, Mathematical Notes 22, 506 (1978) 57. G.W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85, 134 (1957) 58. K.R. Parthasarathy, Probability measures on metric spaces, (Academic Press, 1967) 59. K. Kuratowski, Topology, (Academic Press, 1966) 60. Donald L. Cohn, Measure Theory, (Birkhauser, 1980) 61. A. Connes, A Survey of Foliations and Operator Algebras, in Operator Algebras and Applications, Kingston 1980, V. Kadison editor, Proceedings of Symposia in Pure Mathematics, American Mathematical Society (1982) 111

6 62. A. Connes, C Algebres et Geometrie Differentielle, C.R. Acad. Sc. Paris, 290A, 599 (1980) 63. A. Connes, Non-Commutative Differential Geometry, Publications IHES 62, 41 (1985) 64. J. Feldman and C.C. Moore, Ergodic Equivalence Relations, Cohomology, and von Neumann Algebras I, Trans. Amer. Math. Soc. 234, 325 (1977) 65. S. MacLane, Categories for the Working Mathematician, (Springer-Verlag, 1971) 66. C.T.J. Dodson, Categories, Bundles and Spacetime Topology, Shiva Mathematics Series, no. 1, (Shiva, 1980) 67. Paul R. Halmos, Measure Theory, (Springer-Verlag, 1974) 68. A. Katok and J.M. Strelcyn, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, Lecture Notes in Mathematics 1222, (Springer-Verlag, 1986) 69. A. Connes, The von Neumann Algebra of a Foliation, in Mathematical Problems in Theoretical Physics, Proceedings Rome 1977, Lecture Notes in Physics 80, (Springer-Verlag, 1978) 70. A. Connes, Feuilletages et Algebres d Operateurs, Seminaire Bourbaki 32e Annee, no. 551, 1979/80, Lectures Notes in Mathematics 842, (Springer- Verlag, 1981) 71. R. Bowen, Anosov Foliations are Hyperfinite, Ann. Math. 106, 161 (1977) 72. J. Plante, Foliations with measure preserving holonomy, Ann. Math. 102, 327 (1975) 73. D. Ruelle and D. Sullivan, Currents, flows, and diffeomorphisms, Topology 14, 319 (1975) 74. D. Ruelle, Integral representation of measures associated with a foliation, 112

7 Publications IHES 48, 127 (19??); Invariant measures for a diffeomorphism which expands the leaves of a foliation, Publications IHES 48, 133 (19??) 75. S. Hurder and A. Katok, Secondary classes and transverse measure theory of a foliation, Bull. Amer. Math. Soc. 11, 347 (1984) 76. S. Hurder and A. Katok, Differentiability, Rigidity and Godbillon- Vey Classes for Anosov Flows, preprint 77. S. Hurder and A. Katok, Ergodic Theory and Weil Measures for Foliations, preprint 78. C.C. Moore, Ergodic Theory and von Neumann Algebras, in Operator Algebras and Applications, Kingston 1980, V. Kadison editor, Proceedings of Symposia in Pure Mathematics, American Mathematical Society (1982) 79. A. Connes, J. Feldman, and B. Weiss, An amenable relation is generated by a single transformation, Ergod. Th. & Dynam. Sys. 1, 431 (1981) 113