Limit theorems for partially hyperbolic systems

Dolgopyat, Dmitry

doi:10.1090/S0002-9947-03-03335-X

Author: Dmitry Dolgopyat
Journal: Trans. Amer. Math. Soc. 356 (2004), 1637-1689
MSC (2000): Primary 37D30; Secondary 60Fxx
DOI: https://doi.org/10.1090/S0002-9947-03-03335-X
Published electronically: September 22, 2003
MathSciNet review: 2034323
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Abstract: We consider a large class of partially hyperbolic systems containing, among others, affine maps, frame flows on negatively curved manifolds, and mostly contracting diffeomorphisms. If the rate of mixing is sufficiently high, the system satisfies many classical limit theorems of probability theory.

1. Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193. MR 1749677, https://doi.org/10.1007/BF02810585
2. D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature., Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969. MR 0242194
3. D. V. Anosov and Ja. G. Sinaĭ, Certain smooth ergodic systems, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 107–172 (Russian). MR 0224771
4. 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.
5. Bakhtin, V. I. On the averaging method in a system with fast hyperbolic motions, Proc. Belorussian Math. Inst. 6 (2000) 23-26.
6. Luis Barreira and Jörg Schmeling, Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), 29–70. MR 1759398, https://doi.org/10.1007/BF02773211
7. Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193. MR 1749677, https://doi.org/10.1007/BF02810585
8. Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
9. Rufus Bowen, Weak mixing and unique ergodicity on homogeneous spaces, Israel J. Math. 23 (1976), no. 3-4, 267–273. MR 407233, https://doi.org/10.1007/BF02761804
10. Jonathan Brezin and Michael Shub, Stable ergodicity in homogeneous spaces, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 197–210. MR 1479501, https://doi.org/10.1007/BF01233391
11. M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen. 9 (1975), no. 1, 9–19 (Russian). MR 0370660
12. M. I. Brin, The topology of group extensions of 𝐶-systems, Mat. Zametki 18 (1975), no. 3, 453–465 (Russian). MR 394764
13. M. I. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212 (Russian). MR 0343316
14. Keith Burns, Charles Pugh, and Amie Wilkinson, Stable ergodicity and Anosov flows, Topology 39 (2000), no. 1, 149–159. MR 1710997, https://doi.org/10.1016/S0040-9383(98)00064-0
15. Keith Burns and Amie Wilkinson, Stable ergodicity of skew products, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 6, 859–889 (English, with English and French summaries). MR 1717580, https://doi.org/10.1016/S0012-9593(00)87721-6
16. Castro A. Backward inducing and statistical properties of some partially hyperbolic attractors, Israel J. Math. 130 (2002) 29-75.
17. N. I. Chernov, Limit theorems and Markov approximations for chaotic dynamical systems, Probab. Theory Related Fields 101 (1995), no. 3, 321–362. MR 1324089, https://doi.org/10.1007/BF01200500
18. N. I. Chernov and C. Haskell, Nonuniformly hyperbolic 𝐾-systems are Bernoulli, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 19–44. MR 1375125, https://doi.org/10.1017/S0143385700008695
19. N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math. 122 (2001), 1–27. MR 1826488, https://doi.org/10.1007/BF02809888
20. S. G. Dani, On orbits of endomorphisms of tori and the Schmidt game, Ergodic Theory Dynam. Systems 8 (1988), no. 4, 523–529. MR 980795, https://doi.org/10.1017/S0143385700004673
21. Manfred Denker, The central limit theorem for dynamical systems, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 33–62. MR 1102700
22. Manfred Denker, Remarks on weak limit laws for fractal sets, Fractal geometry and stochastics (Finsterbergen, 1994) Progr. Probab., vol. 37, Birkhäuser, Basel, 1995, pp. 167–178. MR 1391975, https://doi.org/10.1007/978-3-0348-7755-8_8
23. Manfred Denker and Walter Philipp, Approximation by Brownian motion for Gibbs measures and flows under a function, Ergodic Theory Dynam. Systems 4 (1984), no. 4, 541–552. MR 779712, https://doi.org/10.1017/S0143385700002637
24. Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357–390. MR 1626749, https://doi.org/10.2307/121012
25. Dmitry Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1097–1114. MR 1653299, https://doi.org/10.1017/S0143385798117431
26. Dolgopyat D. On mixing properties of compact group extensions of hyperbolic systems, Israel Math. J. 130 (2002) 157-205.
27. Dmitry Dolgopyat, On dynamics of mostly contracting diffeomorphisms, Comm. Math. Phys. 213 (2000), no. 1, 181–201. MR 1782146, https://doi.org/10.1007/s002200000238
28. Dolgopyat D. On differentiability of SRB states for partially hyperbolic systems, to appear in Inv. Math.
29. Dolgopyat D., Kaloshin V. & Koralov L. Sample path properties of stochastic flows, to appear in Ann. Prob.
30. Ernst Eberlein and Murad S. Taqqu (eds.), Dependence in probability and statistics, Progress in Probability and Statistics, vol. 11, Birkhäuser Boston, Inc., Boston, MA, 1986. A survey of recent results; Papers from the conference held at the Mathematical Research Institute Oberwolfach, Oberwolfach, April 1985. MR 899982
31. Robert Ellis and William Perrizo, Unique ergodicity of flows on homogeneous spaces, Israel J. Math. 29 (1978), no. 2-3, 276–284. MR 473095, https://doi.org/10.1007/BF02762015
32. M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739–741 (Russian). MR 0251785
33. M. I. Gordin, Double extensions of dynamical systems and the construction of mixing filtrations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 244 (1997), no. Veroyatn. i Stat. 2, 61–72, 330–331 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 99 (2000), no. 2, 1053–1060. MR 1700380, https://doi.org/10.1007/BF02673626
34. M. I. Gordin, Double extensions of dynamical systems and the construction of mixing filtrations. II. Quasihyperbolic automorphisms of tori, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 260 (1999), no. Veroyatn. i Stat. 3, 103–118, 318–319 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 109 (2002), no. 6, 2103–2114. MR 1759157, https://doi.org/10.1023/A:1014573115086
35. Matthew Grayson, Charles Pugh, and Michael Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2) 140 (1994), no. 2, 295–329. MR 1298715, https://doi.org/10.2307/2118602
36. Y. Guivarc’h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), no. 1, 73–98 (French, with English summary). MR 937957
37. Guivarch Y. & Le Borgne S. Methode de martingales et flot geodesique sur une surface de courbure constante negative, preprint.
38. P. Hall and C. C. Heyde, Martingale limit theory and its application, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Probability and Mathematical Statistics. MR 624435
39. Richard Hill and Sanju L. Velani, The ergodic theory of shrinking targets, Invent. Math. 119 (1995), no. 1, 175–198. MR 1309976, https://doi.org/10.1007/BF01245179
40. Masaki Hirata, Poisson law for Axiom A diffeomorphisms, Ergodic Theory Dynam. Systems 13 (1993), no. 3, 533–556. MR 1245828, https://doi.org/10.1017/S0143385700007513
41. Masaki Hirata, Poisson law for the dynamical systems with the “self-mixing” conditions, Dynamical systems and chaos, Vol. 1 (Hachioji, 1994) World Sci. Publ., River Edge, NJ, 1995, pp. 87–96. MR 1479911
42. Masaki Hirata, Benoît Saussol, and Sandro Vaienti, Statistics of return times: a general framework and new applications, Comm. Math. Phys. 206 (1999), no. 1, 33–55. MR 1736991, https://doi.org/10.1007/s002200050697
43. M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
44. I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR 0322926
45. A. Katok, Smooth non-Bernoulli 𝐾-automorphisms, Invent. Math. 61 (1980), no. 3, 291–299. MR 592695, https://doi.org/10.1007/BF01390069
46. A. Katok and A. Kononenko, Cocycles’ stability for partially hyperbolic systems, Math. Res. Lett. 3 (1996), no. 2, 191–210. MR 1386840, https://doi.org/10.4310/MRL.1996.v3.n2.a6
47. Anatole Katok and Ralf J. 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. MR 1307298
48. Anatole Katok, Jean-Marie Strelcyn, F. Ledrappier, and F. Przytycki, Invariant manifolds, entropy and billiards; smooth maps with singularities, Lecture Notes in Mathematics, vol. 1222, Springer-Verlag, Berlin, 1986. MR 872698
49. Yitzhak Katznelson, Ergodic automorphisms of 𝑇ⁿ are Bernoulli shifts, Israel J. Math. 10 (1971), 186–195. MR 294602, https://doi.org/10.1007/BF02771569
50. Yuri Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc. 321 (1990), no. 2, 505–524. MR 1025756, https://doi.org/10.1090/S0002-9947-1990-1025756-7
51. Yuri Kifer, Averaging in dynamical systems and large deviations, Invent. Math. 110 (1992), no. 2, 337–370. MR 1185587, https://doi.org/10.1007/BF01231336
52. Yuri Kifer, Limit theorems in averaging for dynamical systems, Ergodic Theory Dynam. Systems 15 (1995), no. 6, 1143–1172. MR 1366312, https://doi.org/10.1017/S0143385700009834
53. D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinaĭ’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 141–172. MR 1359098, https://doi.org/10.1090/trans2/171/11
54. D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999), no. 3, 451–494. MR 1719827, https://doi.org/10.1007/s002220050350
55. Karol Krzyżewski, On the convergence of certain series related to Axiom A diffeomorphisms, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), no. 1-2, 25–30 (English, with Russian summary). MR 667382
56. Stéphane Le Borgne, Limit theorems for non-hyperbolic automorphisms of the torus, Israel J. Math. 109 (1999), 61–73. MR 1679589, https://doi.org/10.1007/BF02775027
57. Stéphane Le Borgne, Principes d’invariance pour les flots diagonaux sur 𝑆𝐿(𝑑,ℝ)/𝕊𝕃(𝕕,ℤ), Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 4, 581–612 (French, with English and French summaries). MR 1914940, https://doi.org/10.1016/S0246-0203(01)01105-0
58. Carlangelo Liverani, Central limit theorem for deterministic systems, International Conference on Dynamical Systems (Montevideo, 1995) Pitman Res. Notes Math. Ser., vol. 362, Longman, Harlow, 1996, pp. 56–75. MR 1460797
59. Liverani C. On Contact Anosov flows, preprint.
60. G. A. Margulis, Certain measures that are connected with U-flows on compact manifolds, Funkcional. Anal. i Priložen. 4 (1970), no. 1, 62–76 (Russian). MR 0272984
61. Calvin C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966), 154–178. MR 193188, https://doi.org/10.2307/2373052
62. Viorel Niţică and Andrei Török, An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology 40 (2001), no. 2, 259–278. MR 1808220, https://doi.org/10.1016/S0040-9383(99)00060-9
63. Donald Ornstein and Benjamin Weiss, On the Bernoulli nature of systems with some hyperbolic structure, Ergodic Theory Dynam. Systems 18 (1998), no. 2, 441–456. MR 1619567, https://doi.org/10.1017/S0143385798100354
64. William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 268 (English, with French summary). MR 1085356
65. Françoise Pène, Averaging method for differential equations perturbed by dynamical systems, ESAIM Probab. Statist. 6 (2002), 33–88. MR 1905767, https://doi.org/10.1051/ps:2002003
66. Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1332–1379, 1440 (Russian). MR 0458490
67. A. B. Vasil′eva and A. A. Hasanov, Some singularly perturbed systems in the critical case, Ukrain. Mat. Ž. 29 (1977), no. 3, 291–297, 427 (Russian). MR 0466811
68. L. A. Bunimovich, I. P. Cornfeld, R. L. Dobrushin, M. V. Jakobson, N. B. Maslova, Ya. B. Pesin, Ya. G. Sinaĭ, Yu. M. Sukhov, and A. M. Vershik, Dynamical systems. II, Encyclopaedia of Mathematical Sciences, vol. 2, Springer-Verlag, Berlin, 1989. Ergodic theory with applications to dynamical systems and statistical mechanics; Edited and with a preface by Sinaĭ; Translated from the Russian. MR 1024068
69. Ya. B. Pesin and Ya. G. Sinaĭ, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 417–438 (1983). MR 721733, https://doi.org/10.1017/S014338570000170X
70. Walter Philipp, Some metrical theorems in number theory, Pacific J. Math. 20 (1967), 109–127. MR 205930
71. Walter Philipp and William Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 2 issue 2, 161 (1975), iv+140. MR 0433597, https://doi.org/10.1090/memo/0161
72. B. Pitskel′, Poisson limit law for Markov chains, Ergodic Theory Dynam. Systems 11 (1991), no. 3, 501–513. MR 1125886, https://doi.org/10.1017/S0143385700006301
73. Charles Pugh and Michael Shub, Ergodicity of Anosov actions, Invent. Math. 15 (1972), 1–23. MR 295390, https://doi.org/10.1007/BF01418639
74. Charles Pugh and Michael Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), no. 1, 1–54. MR 983869, https://doi.org/10.1090/S0002-9947-1989-0983869-1
75. Charles Pugh and Michael Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity 13 (1997), no. 1, 125–179. MR 1449765, https://doi.org/10.1006/jcom.1997.0437
76. Charles Pugh and Michael Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS) 2 (2000), no. 1, 1–52. MR 1750453, https://doi.org/10.1007/s100970050013
77. Charles Pugh, Michael Shub, and Amie Wilkinson, Hölder foliations, Duke Math. J. 86 (1997), no. 3, 517–546. MR 1432307, https://doi.org/10.1215/S0012-7094-97-08616-6
78. M. Ratner, The central limit theorem for geodesic flows on 𝑛-dimensional manifolds of negative curvature, Israel J. Math. 16 (1973), 181–197. MR 333121, https://doi.org/10.1007/BF02757869
79. B. A. Sevast′janov, A Poisson limit law in a scheme of sums of dependent random variables, Teor. Verojatnost. i Primenen. 17 (1972), 733–738 (Russian, with English summary). MR 0310943
80. Michael Shub and Amie Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508. MR 1738057, https://doi.org/10.1007/s002229900035
81. Ja. G. Sinaĭ, Classical dynamic systems with countably-multiple Lebesgue spectrum. II, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 15–68 (Russian). MR 0197684
82. Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen 2 (1968), no. 1, 64–89 (Russian). MR 0233038
83. Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
84. Ya. G. Sinaĭ, The stochasticity of dynamical systems, Selecta Math. Soviet. 1 (1981), no. 1, 100–119. Selected translations. MR 645903
85. A. N. Starkov, New progress in the theory of homogeneous flows, Uspekhi Mat. Nauk 52 (1997), no. 4(316), 87–192 (Russian); English transl., Russian Math. Surveys 52 (1997), no. 4, 721–818. MR 1480891, https://doi.org/10.1070/RM1997v052n04ABEH002060
86. G. Yin, On operator-valued mixingales, Stochastic Anal. Appl. 7 (1989), no. 3, 355–366. MR 1010631, https://doi.org/10.1080/07362998908809187
87. Dennis Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math. 149 (1982), no. 3-4, 215–237. MR 688349, https://doi.org/10.1007/BF02392354
88. Viana M. Stochastic dynamics of deterministic systems, Lecture Notes XXI Bras. Math. Colloq. IMPA, Rio de Janeiro, 1997.
89. Marcelo Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math. 85 (1997), 63–96. MR 1471866
90. Amie Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergodic Theory Dynam. Systems 18 (1998), no. 6, 1545–1587. MR 1658611, https://doi.org/10.1017/S0143385798117984
91. Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585–650. MR 1637655, https://doi.org/10.2307/120960
92. Lai-Sang Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153–188. MR 1750438, https://doi.org/10.1007/BF02808180