On differentiability of SRB states for partially hyperbolic systems

Inventiones Mathematicae

Volumn 155, Issue 2, 2004, Pages 389-449

  Dolgopyat, Dmitry ^a  

^a Bldg 142   (United States)

Author keywords

[No Author keywords available]

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EID: 0842346767     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-003-0324-5     Document Type: Article

References (61)

1. Alves, J., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140, 351-398 (2000)
2. Anosov, D. V.: Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90, 1-209 (1967)
3. Bakhtin, V.I.: A direct method for constructing an invariant measure on a hyperbolic attractor. Russ. Acad. Sci. Izv. Math. 41, 207-227 (1993)
4. Bakhtin, V.I.: Random processes generated by a hyperbolic sequence of mappings I-II. Russ. Acad. Sci. Izv. Math. 44, 247-279, 617-627 (1995)
5. Bonnatti, C., Viana, M.: SRB measures for partially hyperbolic systems: mostly contracting case, Isr. J. Math 115, 157-193 (2000)
6. Bonetto, F., Daems, D., Lebowitz, J.: Properties of stationary nonequilibrium states in the thermostated periodic Lorenz gas I: the one particle system. J. Stat. Phys. 101, 35-60 (2000)
7. Bonetto, F., Gentile, G., Mastropietro, V.: Electric fields on a surface of constant negative curvature. Ergodic Theory Dyn. Syst. 20, 681-696 (2000)
8. Bonetto, F., Lebowitz, J., Rey-Bellet, L.: Fourier Law: a challenge to theorists. In: Mathematical Physics-2000. London: Imperial College Press 2000
9. Bowen, R.: Equilibrium states and ergodic theory of Anosov diffeomorphisms. Lect. Notes Math. 470. New York: Springer 1975
10. Bowen, R.: Mixing Anosov flows. Topology 15, 77-79 (1976)
11. Bowen, R., Marcus, B.: Unique ergodicity for horocycle foliations. Isr. J. Math. 26, 43-67 (1977)
12. Brin, M.: Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature. Funct. Anal. Appl. 9, 8-16 (1975)
13. Brin, M.: The topology of group extensions of C-systems. Mat. Zametki 18, 453-465 (1975)
14. Brin, M., Pesin, Ya.B.: Partially hyperbolic dynamical systems. Math. USSR-Izvestiya 8, 177-218 (1974)
15. Burns, K., Puch, C., Shub, M., Wilkinson, A.: Recent advances in stable ergodicity. Proc. Symp. Pure Math. 69, 327-366 (2001)
16. Burns, K., Wilkinson, A.: Stable ergodicity of skew products. Ann. Sci. ÉC. Norm. Supér., IV Sér. 32, 859-889 (1999)
17. Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Steady-state electrical conduction in the periodic Lorentz gas. Commun. Math. Phys. 154, 569-601 (1993)
18. Contreras, G.: Regularity of topological and metric entropy for hyperbolic flows. Math. Z. 210, 97-111 (1992)
19. Dolgopyat, D.: On decay of correlations in Anosov flows. Ann. Math. 147, 357-390 (1998)
20. Dolgopyat, D.: On mixing properties of compact group extensions of hyperbolic systems. Isr. J. Math. 130, 157-205 (2002)
21. Dolgopyat, D.: On dynamics of mostly contracting diffeomorphisms. Commun. Math. Phys. 213, 181-201 (2000)
22. Dolgopyat, D.: Limit Theorems for partially hyperbolic systems. To appear in Trans. Am. Math. Soc.
23. Dolgopyat, D.: Lectures on u-Gibbs states. Evanston, May 28-June 2. Available at www.math.psu.edu/dolgop/papers.html
24. Dolgopyat, D., Pesin, Ya.B.: Every manifold carries a completely hyperbolic diffeomorphism. Ergodic Theory Dyn. Syst. 22, 409-437 (2002)
25. Graczyk, J., Swiatek, G.: Smooth unimodal maps in the 1990s. Ergodic Theory Dyn. Syst. 19, 263-287 (1999)
26. Grayson, M. Pugh, C.C., Shub, M.: Stably ergodic diffeomorphisms. Ann. Math. (2) 140, 295-329 (1994)
27. Hirsh, M., Pugh, C.C.: Smoothness of horocycle foliations. J. Differ. Geom. 10, 225-238 (1975)
28. Hirsh, M., Pugh, C.C., Shub, M.: Invariant manifolds. Lect. Notes Math. 583. Berlin: Springer 1977
29. Hurder, S., Katok, A.: Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. Publ. Math., Inst. Hautes Étud. Sci. 72, 5-61 (1990)
30. Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39-88 (1981)
31. Katok, A., Knieper, G., Pollicott, M., Weiss, H.: Differentiability of entropy for Anosov and geodesic flows. Bull. Am. Math. Soc. 22, 285-293 (1990)
32. Katok, A., Knieper, G., Pollicott, M., Weiss, H.: Differentiability and analyticity of topological entropy for Anosov and geodesic flows. Invent. Math. 98, 581-597 (1989)
33. Katok, A., Spatzier, R.J.: First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ. Math., Inst. Hautes Étud. Sci. 79, 131-156 (1994)
34. Katok, A., Spatzier, R.J.: Invariant measures for higher-rank hyperbolic abelian actions. Ergodic Theory Dyn. Syst. 16, 751-778 (1996)
35. Kifer, Yu.: Averaging in difference equations driven by dynamical systems. Preprint. To appear in Astérisque
36. Kleinbock, D., Margulis, G.A.: Bounded orbits of nonquasiunipotent flows on homogeneous spaces. In: Sinai's Moscow Seminar on Dynamical Systems, ed. by L.A. Bunimovich, B.M. Gurevich, Ya.B. Pesin. AMS Transl. 171, 141-172 (1996)
37. Lyubich, M.: The quadratic family as a qualitatively solvable model of chaos. Notices Am. Math. Soc. 47, 1042-1052 (2000)
38. Margulis, G.A.: On some problems in the theory of U-systems. PhD thesis, Moscow State University, 1970 (in Russian, English transl. to appear)
39. Palis, J.: A global view of dynamics and a conjecture on the denseness of finitude of attractors. Geometrie complexe et systemes dynamiques (Orsay, 1995). Astérisque 261, 335-347 (2000)
40. Palis, J., Takens, F.: Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Fractal dimensions and infinitely many attractors. Camb. Stud. Adv. Math. 35. Cambridge: Cambridge University Press 1993
41. Pesin, Ya.B.: Families of invariant manifolds that correspond to nonzero characteristic exponents. Izv. Akad. Nauk SSSR 40, 1332-1379 (1976)
42. Pesin, Ya.B.: Characteristic Lyapunov exponents, and smooth ergodic theory. Usp. Mat. Nauk 32, 55-112 (1977)
43. Pesin, Ya.B., Sinai, Ya.G.: Gibbs measures for partially hyperbolic attractors. Ergodic Theory Dyn. Syst. 2, 417-438 (1982)
44. Pugh, C.C., Shub, M., Wilkinson, A.: Holder foliations. Duke Math. J. 86, 517-546 (1997)
45. Pugh, C.C., Shub, M., Wilkinson, A.: Partial differentiability of invariant foliations. Preprint
46. Ruelle, D.: Integral representation of measures associated with a foliation. Publ. Math., Inst. Hautes Étud. Sci. 48, 127-132 (1978)
47. Ruelle, D.: Invariant measures for a diffeomorphism which expands the leaves of a foliation. Publ. Math., Inst. Hautes Étud. Sci. 48, 133-135 (1978)
48. Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227-241 (1997)
49. Ruelle, D.: Differentiation of SRB states: corrections and complements. Commun. Math. Phys. 234, 185-190 (2003)
50. Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95, 393-468 (1999)
51. Ruelle, D.: Nonequilibrium statistical mechanics near equilibrium: computing higher-order terms. Nonlinearity 11, 5-18 (1998)
52. Ruelle, D.: Perturbation theory for Lyapunov exponents of a toral map: extension of a result of Shub and Wilkinson. Isr. J. Math. 134, 345-362 (2003)
53. Ruelle, D., Wilkinson, A.: Absolutely singular dynamical foliations. Commun. Math. Phys. 219, 481-487 (2001)
54. Shub, M., Wilkinson, A.: Pathological foliations and removable zero exponents. Invent. Math. 139, 495-508 (2000)
55. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747-817 (1967)
56. Tsujii, M.: On continuity of Bowen-Ruelle-Sinai measures in families of one-dimensional maps. Commun. Math. Phys. 177, 1-11 (1996)
57. Viana, M.: Stochastic dynamics of deterministic systems. Lecture Notes XXI Bras. Math. Colloq. IMPA, Rio de Janeiro 1997
58. Viana, M.: Multidimensional nonhyperbolic attractors. Publ. Math., Inst. Hautes Étud. Sci. 85, 63-96 (1997)
59. Wilkinson, A.: Stable ergodicity of time-one map of a geodesic flow. Ergodic Theory Dyn. Syst. 18, 1545-1587 (1998)
60. Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. 147, 585-650 (1998)
61. Young, L.-S.: Geometric and ergodic theory of hyperbolic dynamical systems. In: Current developments in mathematics, 1998 (Cambridge, MA), 237-278. Somerville, MA: Int. Press 1999