Limit theorems for partially hyperbolic systems

Transactions of the American Mathematical Society

Volumn 356, Issue 4, 2004, Pages 1637-1689

  Dolgopyat, Dmitry ^a  

^a University of Maryland   (United States)

Author keywords

Absolute continuity; Central limit theorem; Gibbs measure; Partial hyperbolicity

Indexed keywords

EID: 1642331620     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-03-03335-x     Document Type: Article

References (102)

1. Alves J., Bonatti C. & Viana M. SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Inv. Math. 140 (2000) 35-398. MR 2001j:37063b
2. Anosov D. V. Geodesic flows on closed Riemannian manifolds with negative curvature Proc. Steklov Inst. Math. 90 (1967). MR 39:3527
3. Anosov D. V. & Sinai Ya. G. Certain smooth ergodic systems, Uspekhi Mat. Nauk 22 (1967), no. 5, 107-172; English transl., Russian Math. Surveys 22 (1967), no. 5, 103-167. MR 37:370
4. Anosov D. V. & Sinai Ya. G. Certain smooth ergodic systems, Uspekhi Mat. Nauk 22 (1967), no. 5, 107-172; English transl., Russian Math. Surveys 22 (1967), no. 5, 103-167. MR 37:370
5. Bakhtin, V. I. Random processes generated by a hyperbolic sequence of mappings. I, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 2, 40-72; English transl., Russian Acad. Sci. Izv. Math. 44 (1995) 247-279.
6. Bakhtin, V. I. Random processes generated by a hyperbolic sequence of mappings. I, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 2, 40-72; English transl., Russian Acad. Sci. Izv. Math. 44 (1995) 247-279.
7. Bakhtin, V. I. On the averaging method in a system with fast hyperbolic motions, Proc. Belorussian Math. Inst. 6 (2000) 23-26.
8. Barreira L. & Schmeling J. Sets of non-typical points have full topological entropy and full Haussdorff dimension, Israel J. Math 116 (2000) 29-70. MR 2002d:37040
9. Bonnatti C. & Viana M. SRB measures for partially hyperbolic systems: mostly contracting case, Israel J. Math 115 (2000) 157-193. MR 2001j:37063a
10. Bowen R. 'Equilibrium states and ergodic theory of Anosov diffeomorphisms' Lect. Notes in Math. 470 (1975) Springer New York. MR 56:1364
11. Bowen R. Weak mixing and unique ergodicity on homogeneous spaces Isr. J. Math. 23 (1976) 267-273. MR 53:11016
12. Brezin J. & Shub M. Stable ergodicity in homogeneous spaces Bol. Soc. Brasil. Mat. (N.S.) 28 (1997) 197-210. MR 99a:58103
13. Brin M. Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature Func. An. & Appl. 9 (1975) 8-16. MR 51:6886
14. Brin M. Topology of group extensions of Anosov systems Math. Notes 18 (1975) 858-864. MR 52:15563
15. Brin M. & Pesin Ya. B. Partially hyperbolic dynamical systems Math. USSR-Izvestiya 8 (1974) 177-218. MR 49:8058
16. Burns K., Pugh C. C. & Wilkinson A. Stable ergodicity and Anosov flows, Topology 39 (2000) 149-159. MR 2000i:37031
17. Burns K. & Wilkinson A. Stable ergodicity of skew products, Ann. Sci. Esc. Norm. Sup. 32 (1999) 859-889. MR 2000g:37030
18. Castro A. Backward inducing and statistical properties of some partially hyperbolic attractors, Israel J. Math. 130 (2002) 29-75.
19. Chernov N. I. Limit theorems and Markov approximations for chaotic dynamical systems Prob. Th., Rel. Fields 101 (1995) 321-363. MR 96m:28016
20. Chernov N. I. & Haskell C. Nonuniformly hyperbolic K-systems are Bernoulli Erg. Th. & Dyn. Sys. 16 (1996) 19-44. MR 97k:28031
21. Chernov N. I. & Kleinbock D. Y. Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math. 122 (2001), 1-27. MR 2002h:37003
22. Dani S. G. On orbits of endomorphisms of tori and the Schmidt game, Erg. Th. & Dyn. Sys. 8 (1988) 523-529. MR 90b:58145
23. Denker M. The Central Limit Theorem for dynamical systems, Banach Center Publ. 23 Warsaw, Polish Sci. Publ. (1989) 33-62. MR 92d:28007
24. Denker M. Remarks on weak limit laws for fractal sets, In Fractal geometry and stochastics (C. Bandt, S. Graf & M. Zahle editors) Progr. Prob. 37 (1995) Birkhauser, Basel. MR 97k:28032
25. Denker M. & Philipp W. Approximation by Brownian motion for Gibbs measures and flows under a function, Erg. Th. & Dyn. Sys. 4 (1984) 541-552. MR 86g:28025
26. Dolgopyat D. On decay of correlations in Anosov flows, Ann. Math. 147 (1998) 357-390. MR 99g:58073
27. Dolgopyat D. Prevalence of rapid mixing in hyperbolic flows, Erg. Th. & Dyn. Sys. 18 (1998) 1097-1114. MR 2000a:37014
28. Dolgopyat D. On mixing properties of compact group extensions of hyperbolic systems, Israel Math. J. 130 (2002) 157-205.
29. Dolgopyat D. On dynamics of mostly contracting diffeomorphisms, Comm. Math. Phys. 213 (2000) 181-201. MR 2001h:37056
30. Dolgopyat D. On differentiability of SRB states for partially hyperbolic systems, to appear in Inv. Math.
31. Dolgopyat D., Kaloshin V. & Koralov L. Sample path properties of stochastic flows, to appear in Ann. Prob.
32. Eberlein E. & Taqqu M. S. (editors) Dependence in probability and steatistics. A survey of recent results, Progress in prob. and Stat. 11, Birkhauser, Boston, 1986. MR 88b:60006
33. Ellis R. & Perrizo W. Unique ergodicity of flows on homogeneous spaces, Isr. J. Math. 29 (1978) 276-284. MR 57:12774
34. Gordin M. I. On Central Limit Theorem for stationary processes, Doklany AN SSSR 188 (1969) 739-741; English transl., Soviet Math. Dokl. 10 (1969), 1174-1176. MR 40:5012
35. Gordin M. I. On Central Limit Theorem for stationary processes, Doklany AN SSSR 188 (1969) 739-741; English transl., Soviet Math. Dokl. 10 (1969), 1174-1176. MR 40:5012
36. Gordin M. I. Double extensions of dynamical systems and construction of the mixing filtrations, J. Math. Sci. (New York) 99 (2000), 1053-1060. MR 2000e:37003
37. Gordin M. I. Double extensions of dynamical systems and construction of the mixing filtrations-2: quasi-hyperbolic toral automorphisms, 3. Math. Sci. (New York) 109 (2002), 2103-2114. MR 2001e:37008
38. Grayson M., Pugh C. C. & Shub M. Stably ergodic diffeomorphisms, Ann. Math. 140 (1994) 295-329. MR 95g:58128
39. Guivarch Y. & Hardy J. Theoremes limites pour une classe de chaines de Markov et applications aux diffeomorphisms d'Anosov, Annales de Inst. H. Poincare (Ser. Prob & Stat) 24 (1988) 73-98. MR 89m:60080
40. Guivarch Y. & Le Borgne S. Methode de martingales et flot geodesique sur une surface de courbure constante negative, preprint.
41. Hall P. & Heyde C. C. Martingale limit theory and its application, Academic Press, New York, 1980. MR 83a:60001
42. Hill R. & Velani S. L. The ergodic theory of shrinking targets, Invent. Math. 119 (1995) 175-198. MR 96e:58088
43. Hirata M. Poisson law for Axiom A diffeomorphisms, Erg. Th. & Dyn. Sys. 13 (1993) 533-556. MR 94m:58137
44. Hirata M. Poisson law for the dynamical systems with self-mixing conditions. In Dynamical systems and chaos, Ed. N. Aoki, K. Shiraiwa & Y. Takahashi, World Scientific, Singapore (1995) v. 1, 87-96. MR 98h:81032
45. Hirata M., Saussol B. & Vaienti S. Statistics of return times: a general framework and new applications, Comm. Math. Phys. 206 (1999) 33-55. MR 2001c:37007
46. Hirsch M., Pugh C. C. & Shub M. Invariant manifolds, Lect. Notes in Math. 583 Springer-Verlag, Berlin, 1977. MR 58:18595
47. Ibragimov I. A. & Linnik Yu. V. Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. MR 48:1287
48. Katok, A. Smooth non-Bernoulli K-automorphisms, Invent. Math. 61 (1980) 291-299. MR 84e:58063a, MR 84e:58063b
49. Katok A. & Kononenko A.Cocycle stability for partially hyperbolic systems, Math. Res. Lett. 3 (1996) 191-210. MR 97d:58152
50. Katok A. & Spatzier R. J. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Etudes Sci. Publ. Math. 79 (1994) 131-156. MR 96c:58132
51. Katok A. & Strelcyn J.-M. Invariant manifolds, entropy and billiards; smooth maps with singularities, Lect. Notes Math. 1222 Springer, Berlin, 1986. MR 88k:58075
52. n are Bernoulli shifts, Israel J. Math. 10 (1971) 186-195. MR 45:3672
53. Kifer Yu. Large deviations in dynamical systems and stochastic processes. Transactions AMS 321 (1990) 505-524. MR 91e:60091
54. Kifer Yu. Averaging in dynamical systems and large deviations, Inv. Math. 110 (1992) 337-370. MR 93m:60118
55. Kifer Yu. Limit theorems in averaging for dynamical systems, Erg. Th. & Dyn. Sys. 15 (1995) 1143-1172. MR 97c:58087
56. Kleinbock D. Y. & Margulis G. A. Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Amer. Math. Soc. Transl. 171 (1996) 141-172. MR 96k:22022
57. Kleinbock D. Y. & Margulis G. A. Logarithm laws for flows on homogeneous spaces, Inv. Math. 138 (1999) 451-494. MR 2001i:37046
58. Krzyzewski K. On convergence of certain series related to Axiom A diffeomorphisms, Bull. Acad. Polon. Ser. Math. 30 (1982) 25-30. MR 84b:58086
59. Le Borgne S. Limit theprems for non-hyperbolic automorphisms of the torus, Israel J. Math. 109 (1999) 61-73. MR 2000a:37001
60. Le Borgne S. Principes d'invariance pour les flots diagonaux sur SL(d, ℝ)/SL(d, ℤ), Ann. Inst. H. Poincare Probab. Stat. 38 (2002), 581-612. MR 2003c:37008
61. Liverani C. Central limit theorem for deterministic systems, in International conference on dynamical systems (Eds. F. Ledrappier, J. Lewowics & S. Newhouse), Pitman Res. Notes 363 (1996) 56-75. MR 98k:28025
62. Liverani C. On Contact Anosov flows, preprint.
63. Margulis G. A. Certain measures that are connected with U-flows on compact manifolds, Func. Anal. Appl. 4 (1970) 62-76. MR 42:7865
64. Moore C. C. Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966) 154-178. MR 33:1409
65. Nitica V. & Torok A. An open dense set of stably ergodic diffeomorphisms in a neighbourhood of a non-ergodic one, Topology 40 (2001), 259-278. MR 2001m:37058
66. Ornstein D. & Weiss B. On Bernoulli nature of systems with some hyperbolic structure. Erg. Th. & Dyn. Sys. 18 (1998) 441-456. MR 99c:58096
67. Parry W. & Pollicott M. Zeta Functions and Periodic Orbit Structure of Hyperbolic Dynamics, Asterisque v. 187-188 (1990). MR 92f:58141
68. Pene F. Averaging method for differential equations perturbed by dynamical systems, ESAIM Probab. Stat. 6 (2002), 33-88 (electronic). MR 2003i:34118
69. Pesin Ya. B. Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR 40 (1976) 1332-1379; English tranl., Math. USSR Izv. 10 (1976), 1261-1305. MR 56:16690
70. Pesin Ya. B. Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR 40 (1976) 1332-1379; English tranl., Math. USSR Izv. 10 (1976), 1261-1305. MR 56:16690
71. Pesin Ya. B. Characteristic Lyapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977) 55-112; English transl., Russian Math. Surveys 32 (1977), no. 4, 55-114. MR 57:6687
72. Pesin Ya. B. Characteristic Lyapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977) 55-112; English transl., Russian Math. Surveys 32 (1977), no. 4, 55-114. MR 57:6687
73. Pesin Ya. B. General theory of smooth hyperbolic dynamical systems, In Encyclopedia Math. Sci., Dynamical Systems-2. Ed. Ya. G. Sinai, Springer-Verlag, Berlin, 1989. MR 911:58079
74. Pesin Ya. B. & Sinai Ya. G. Gibbs measures for partially hyperbolic attractors, Erg. Th. & Dyn. Sys. 2 (1982) 417-438. MR 85f:58071
75. Philipp W. Some metric theorems in number theory, Pacific J. Math 20 (1967) 109-127. MR 34:5755
76. Philipp W. & Stout W. Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. AMS 161 (1975). MR 55:6570
77. Pitskel B. S. Poisson Limit Law for Markov chains, Erg. Th. & Dyn. Sys. 11 (1991) 501-513. MR 92j:60081
78. Pugh C. C. & Shub M Ergodicity of Anosov actions, Inv. Math 15 (1972) 1-23. MR 45:4456
79. Pugh C. C. & Shub M. Ergodic attractors, Trans. AMS 312 (1989) 1-54. MR 90h:58057
80. Pugh C. C. & Shub M.Stably ergodic dynamical systems and partial hyperbolicity, 3. Complexity 13 (1997) 125-179. MR 98e:58110
81. Pugh C. C. & Shub M. Stable ergodicity and Julienne quasi-conformality, J. European Math. Soc. 2 (2000) 1-52. MR 2003j:37046
82. Pugh C. C., Shub M. & Wilkinson A. Hölder foliations, Duke Math. J. 86 (1997) 517-546. MR 97m:58155
83. Ratner M. The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel J. Math. 15 (1973) 181-197. MR 48:11446
84. Sevastjanov B. A. A Poisson Limit Law in a scheme of sums of dependant random variables, Teor. Ver., Primenen. 17 (1972) 733-738; English transl. in Theory Probab. Appl. IT (1972). MR 46:10041
85. Sevastjanov B. A. A Poisson Limit Law in a scheme of sums of dependant random variables, Teor. Ver., Primenen. 17 (1972) 733-738; English transl. in Theory Probab. Appl. 17 (1972). MR 46:10041
86. Shub M. & Wilkinson A. Pathological foliations and removable zero exponents, Inv. Math. 139 (2000) 495-508. MR 2001c:37030
87. Sinai Ya. G. Dynamic systems with countably-multiple Lebesgue spectrum. II, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966) 15-68; English transl., Amer. Math. Soc. Transl. (2) 68 (1968), 34-88. MR 33:5847
88. Sinai Ya. G. Dynamic systems with countably-multiple Lebesgue spectrum. II, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966) 15-68; English transl., Amer. Math. Soc. Transl. (2) 68 (1968), 34-88. MR 33:5847
89. Sinai Ya. G. Markov partitions and U-diffeomorphisms, Funkts. Anal. i Prilozhen. 2 (1968) no. 1 64-89; English transl. in Funct. Anal. Appl. 2 (1968). MR 38:1361
90. Sinai Ya. G. Markov partitions and U-diffeomorphisms, Funkts. Anal. i Prilozhen. 2 (1968) no. 1 64-89; English transl. in Funct. Anal. Appl. 2 (1968). MR 38:1361
91. Sinai Ya. G. Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972) 21-64; English transl., Russian Math. Surveys 27 (1972), no. 4, 21-70. MR 53:3265
92. Sinai Ya. G. Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972) 21-64; English transl., Russian Math. Surveys 27 (1972), no. 4, 21-70. MR 53:3265
93. Sinai Ya. G. Stochasticity of dynamical systems, Nonlinear Waves (Materials All-Union School, Gorki, 1977; A. V. Gaponov-Grekhov, editor), "Nauka", Moscow, 1979, pp. 192-212; English transl., Selecta Math. Sovietica 1 (1981), 101-119. MR 83e:58058
94. Sinai Ya. G. Stochasticity of dynamical systems, Nonlinear Waves (Materials All-Union School, Gorki, 1977; A. V. Gaponov-Grekhov, editor), "Nauka", Moscow, 1979, pp. 192-212; English transl., Selecta Math. Sovietica 1 (1981), 101-119. MR 83e:58058
95. Starkov A. N. New progress in the theory of homogeneous flows, Russian Math. Surveys 52 (1997) 721-818. MR 98k:22047
96. Stroock D. W. & Varadhan S. Multidimensional diffusion processes, (1979) Springer, Berlin-New York. MR 91f:60108
97. Sullivan D. Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesies, Acta Math. 140 (1982) 215-237. MR 84j:58097
98. Viana M. Stochastic dynamics of deterministic systems, Lecture Notes XXI Bras. Math. Colloq. IMPA, Rio de Janeiro, 1997.
99. Viana M. Multidimensional nonhyperbolic attractors, Publ. IHES 85 (1997), 63-96. MR 98j:58073
100. Wilkinson A. Stable ergodicity of the time-one map of a geodesic flow, Erg. Th. & Dyn. Sys. 18 (1998) 1545-1587. MR 99m:58129
101. Young L.-S. Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. 147 (1998) 585-650. MR 99h:58140
102. Young L.-S. Recurrence times and rates of mixing, Israel J. Math. 110 (1999) 153-188. MR 2001j:37062