Chapter 4 Invariant measures for hyperbolic dynamical systems

Handbook of Dynamical Systems

Volumn 1, Issue PART A, 2002, Pages 321-407

  Chernov, N ^a  

^a UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

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EID: 1042287164     PISSN: 1874575X     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-575X(02)80006-6     Document Type: Article

References (81)

1. Franks J., and Misiurewicz M. Topological methods in dynamics. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1 A (2002), Elsevier, Amsterdam 547-598
2. Hasselblatt B. Hyperbolic dynamical systems. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 239-319
3. Katok A., and Hasselblatt B. Principal structures. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1 A (2002), Elsevier, Amsterdam 1-203
4. Pollicott M. Periodic orbits and zeta functions. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 407-452
5. Thouvenot J.-P. Ergodic theory: Entropy, isomorphisms and Kakutani equivalence. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 205-238
6. Abramov L.M. On the entropy of a flow. Amer. Math. Soc. Transi., Ser.2 Vol. 49 (1966) 167-170
7. Anosov D.V. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967)
8. Anosov D.V. On tangent fields of transversal foliations in Y-systems. Mat. Zametki 2 (1967) 539-548
9. Anosov D.V., and Sinai Ya.G. Some smooth ergodic systems. Russian Math. Surv. 22 (1967) 103-167
10. Adler R., and Weiss B. Entropy a complete metric invariant for automorphisms of the torus. Proc. Nat. Acad. Sci. USA 57 (1967) 1573-1576
11. Arnold V.I. Some remarks on flows of line elements and frames. Soviet Math. Dokl. 2 (1961) 562-564
12. (in Russian). Arnold V.I. Remarks on winding numbers. Sibirsk. Mat. Zh. 2 (1961) 807-813
13. Barreira L., Pesin Y., and Schmeling J. Dimension and product structure of hyperbolic measures. Ann. of Math. 149 (1999) 755-783
14. Benedicks M., and Young L.-S. SBR measures for certain Henon maps. Invent. Math. 112 (1993) 541-576
15. Bowen R. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973) 429-459
16. Bowen R. Equilibrium States and the Ergodic Theory of Anosov Dijfeomorphisms. Lect. Notes Math. Vol. 470 (1975), Springer-Verlag, Berlin
17. Bowen R. Markov partitions are not smooth. Proc. Amer. Math. Soc. 71 (1978) 130-132
18. Bowen R., and Ruelle D. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975) 181-202
19. Bunimovich L.A., and Sinai Ya.G. Markov partitions for dispersed billiards. Comm. Math. Phys. 73 (1980) 247-280
20. Bunimovich L.A., Sinai Ya.G., and Chernov N.I. Markov partitions for two-dimensional billiards. Russian Math. Surv. 45 (1990) 105-152
21. Cencova N.N. Statistical properties of smooth Smale horseshoes. In: Dobrushin R.L. (Ed). Mathematical Problems of Statistical Mechanics and Dynamics (1986), Reidel, Dordrecht 199-256
22. Chernov N.I. Markov approximations and decay of correlations for Anosov flows. Ann. Math. 147 (1998) 269-324
23. Chernov N., and Markarian R. Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28 (1997) 271-314
24. Chernov N., and Markarian R. Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat. 28 (1997) 315-342
25. Chernov N., Markarian R., and Troubetzkoy S. Conditionally invariant measures for Anosov maps with small holes. Ergodic Theory Dynamical Systems 18 (1998) 1049-1073
26. Chernov N., Markarian R., and Troubetzkoy S. Invariant measures for Anosov maps with small holes. Ergodic Theory Dynamical Systems 20 (2000) 1007-1044
27. Denker M. The central limit theorem for dynamical systems. Dyn. Syst. Ergod. Theory, Banach Center Publ. Vol. 23 (1989), PWN-Polish Sci. Publ., Warsaw
28. Dobrushin R.L. The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13 (1968) 197-224
29. Dobrushin R.L. Gibbsian random fields for lattice systems with pairwise interaction. Funct. Anal. Appl. 2 (1968) 292-301
30. Dobrushin R.L. The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Funct. Anal. Appl. 2 (1968) 302-312
31. Dolgopyat D. On decay of correlations in Anosov flows. Ann. Math. 147 (1998) 357-390
32. Dunford N., and Schwartz J. Linear Operators: General Theory: Part I (1988), Wiley
33. Franks J., and Williams R. Anomalous Anosov flows. Global Theory of Dynamical Systems, Lecture Notes Math. Vol. 819 (1980), Springer-Verlag, Berlin 158-174
34. Friedman N., and Ornstein D. On the isomorphism of weak Bernoulli transformations. Adv. Math. 5 (1970) 365-394
35. Gallavotti G. Reversible Anosov dijfeomorphisms and large deviations. Math. Phys. Electronic J. 1 (1995) 1-12
36. Gallavotti G., and Cohen E.D.G. Dynamical ensembles in stationary states. J. Stat. Phys. 80 (1995) 931-970
37. Gantmacher F. Applications of the Theory of Matrices (1959), Interscience Publishers, New York
38. Gaspard P., and Dorfman J.R. haotic scattering theory, thermodynamic formalism, and transport coefficients. Phys. Rev. E 52 (1995) 3525-3552
39. Gaspard P., and Nicolis G. Transport properties, Lyapunov exponents, and entropy per unit time. Phys. Rev. Lett. 65 (1990) 1693-1696
40. Guivarc'h Y., and Hardi J. Théorèmes limites pour une classe de chaines de Markov et applications aux dijféomorphismes d'Anosov. Ann. Inst. H. Poincaré, Probab. Statist. 24 (1988) 73-98
41. Gurevič B.M. Some conditions for the existence of K-partitions for special flows. Trans. Moscow Math. Soc. 17 (1967) 99-128
42. Haydn N.T. Canonical product structure of equilibrium states. Random Comp. Dynam. 2 (1994) 79-96
43. Hirsch M., and Pugh C. Smoothness of horocycle foliations. J. Differential Geom. 10 (1975) 225-238
44. Hopf E. Statistik der Lösungen geodätischer Probleme vom unstabilen Typus, II. Math. Annalen 117 (1940) 590-608
45. Hu H., and Young L.-S. Nonexistence of SRB measures for some systems that are "almost Anosov ". Ergodic Theory Dynamical Systems 15 (1995) 67-76
46. Katok A., and Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems (1995), Cambridge Univ. Press
47. Krüger T., and Troubetzkoy S. Markov partitions and shadowing for nommiformly hyperbolic systems with singularities. Ergodic Theory Dynamical Systems 12 (1992) 487-508
48. T. Krüger and S. Troubetzkoy, Markov partitions via shadowing for diffeomorphisms with no zero exponents, Manuscript.
49. Lanford O.E., and Ruelle D. Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Phys. 13 (1969) 194-215
50. Ledrappier F. Proprietes ergodiques des mesures de Sinai. Inst. Hantes Études Sci. Publ. Math. 59 (1984) 163-188
51. Ledrappier F., and Strelcyn J.-M. A proof of the estimation from below in Pesin's entropy formula. Ergodic Theory Dynamical Systems 2 (1982) 203-219
52. Ledrappier F., and Young L.-S. The metric entropy of diffeomorphisms: I. Characterization of measures satisfying Pesin's formula, II. Relations between entropy, exponents and dimension. Ann. of Math. 122 (1985) 509-539
53. Ledrappier F., and Young L.-S. The metric entropy of diffeomorphisms: I. Characterization of measures satisfying Pesin's formula, II. Relations between entropy, exponents and dimension. Ann. of Math. 122 (1985) 540-574
54. Livshitz A.N. Cohomology of dynamical systems. Math. USSR Izvestija 6 (1972) 1276-1301
55. Livshitz A.N., and Sinai Ya.G. On invariant measures compatible with the smooth structure for transitive U-systems. Soviet Math. Dokl. 13 (1972) 1656-1659
56. Ornstein D.S., and Weiss B. Geodesic flows are Bernoullian. Israel J. Math. 14 (1973) 184-198
57. Oseledec V.I. A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968) 197-231
58. Parry W., and Pollicott M. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque (1990) 187-188
59. Pesin Ya.B. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. USSR Izv. 10 (1976) 1261-1305
60. Pesin Ya.B. Lyapunov characteristic exponents and smooth ergodic theory. Russian Math. Surv. 32 (1977) 55-114
61. Pesin Ya.B. Dynamical systems with generalized hyperbolic attractors. Ergodic Theory Dynamical Systems 12 (1992) 123-151
62. Philipp W., and Stout W. Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 161 (1975)
63. Plante J. Anosov flows. Amer. J. Math. 94 (1972) 729-754
64. Pugh C., and Shub M. Ergodic attractors. Trans. Amer. Math. Soc. 312 (1989) 1-54
65. Pollicott M. On the rate of mixing of Axiom A flows. Invent. Math. 81 (1985) 413-426
66. Ratner M. The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16 (1973) 181-197
67. Ratner M. Anosov flows with Gibbs measures are also Bernoullian. Israel J. Math. 17 (1974) 380-391
68. Rudin W. Real and Complex Analysis (1987), McGraw-Hill
69. Ruelle D. A measure associated with Axiom-A attractors. Amer. J. Math. 98 (1976) 619-654
70. Ruelle D. Thermodynamic Formalism (1978), Addison-Wesley, Reading, MA
71. Ruelle D. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9 (1978) 83-87
72. Ruelle D. Flots qui ne mélangent pas exponentiellement. C. R. Acad. Sci. Paris 296 (1983) 191-193
73. Ruelle D. Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys. 85 (1996) 1-23
74. 393-168. Ruelle D. Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 94 (1999)
75. Seneta E. Non-Negative Matrices and Markov Chains (1981), Springer-Verlag, New York
76. Sinai Ya.G. Markov partitions and C-dijfeomorphisms. Funct. Anal. Appl. 2 (1968) 61-82
77. Sinai Ya.G. Construction of Markov partitions. Funct. Anal. Appl. 2 (1968) 245-253
78. Sinai Ya.G. Gibbs measures in ergodic theory. Russian Math. Surv. 27 (1972) 21-69
79. Smale S. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967) 747-817
80. Young L.-S. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147 (1998) 585-650
81. Young L.-S. Ergodic theory of chaotic dynamical systems. XII International Congress of Mathematical Physics (ICMP'97) (Brisbane) (1999), Internat. Press, Cambridge, MA 131-143