A review of linear response theory for general differentiable dynamical systems

Nonlinearity

Volumn 22, Issue 4, 2009, Pages 855-870

  Ruelle, David ^a  

^a INSTITUT DES HAUTES ÉTUDES SCIENTIFIQUES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67650708843     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/22/4/009     Document Type: Article

References (60)

1. Andrey L 1985 The rate of entropy change in non-Hamiltonian systems Phys. Lett. A 11 45-6
2. Baladi V 2000 Positive Transfer Operators and Decay of Correlations (Singapore: World Scientific)
3. Baladi V and Smania D 2008 Linear response formula for piecewise expanding unimodal maps Nonlinearity 21 677-711
4. Baladi V and Smania D in preparation
5. Billingsley P 1965 Ergodic Theory and Information (New York: Wiley)
6. Bonatti C, Diaz L and Viana M 2005 Dynamics Beyond Uniform Hyperbolicity (Berlin: Springer)
7. Bowen R 1970 Markov partitions for Axiom A diffeomorphisms Am. J. Math. 92 725-47
8. Bowen R 1973 Symbolic dynamics for hyperbolic flows Am. J. Math. 95 429-60
9. Bowen R 1975 Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics vol 470) 2nd edn (Berlin: Springer) 2008
10. Bowen R and Ruelle D 1975 The ergodic theory of Axiom A flows Inventiones Math. 29 181-202
11. Butterley O and Liverani C 2007 Smooth Anosov flows: correlation spectra and stability J. Mod. Dyn. 1 301-22
12. Cessac B 2007 Does the complex susceptibility of the Hénon map have a pole in the upper half plane? A numerical investigation Nonlinearity 20 2883-95
13. Chernov N 1998 Markov approximations and decay of correlations for Anosov flows Ann. Math. 147 269-324
14. Dolgopyat D 1998 Decay of correlations in Anosov flows Ann. Math. 147 357-90
15. Dolgopyat D 1998 Prevalence of rapid mixing in hyperbolic flows Ergod. Theory Dyn. Syst. 18 1097-114
16. Dolgopyat D 2000 Prevalence of rapid mixing-II: topological prevalence Ergod. Theory Dyn. Syst. 20 1045-59
17. Dolgopyat D 2004 On differentiability of SRB states for partially hyperbolic systems Inventiones Math. 155 389-449
18. Evans D J and Morriss G P 1990 Statistical Mechanics of Nonequilibrium Fluids (New York: Academic)
19. Field M, Melbourne I and Tóròk A 2007 Stability of mixing for hyperbolic flows Ann. Math. 166 269-91
20. Gallavotti G and Cohen E G D 1995 Dynamical ensembles in stationary states J. Stat. Phys. 80 931-70
21. Hoover W G 1986 Molecular Dynamics (Lecture Notes in Physics vol 258) (Heidelberg: Springer)
22. Jacobs K 1962/63 Lecture Notes on Ergodic Theory (Denmark: Aarhus Matematisk Institut)
23. Jakobson M V 1981 Absolutely continuous invariant measures for one-parameter families of one-dimensional maps Commun. Math. Phys. 81 39-88
24. Jiang Y and Ruelle D 2005 Analyticity of the susceptibility function for unimodal Markovian maps of the interval Nonlinearity 18 2447-53
25. Katok A, Knieper G, Pollicott M and Weiss H 1989 Differentiability and analyticity of topological entropy for Anosov and geodesic flows Inventiones Math. 98 581-97
26. Ledrappier F and Strelcyn J-M 1982 A proof of the estimation from below in Pesin's entropy formula Ergod. Theory Dyn. Syst. 2 203-19
27. Ledrappier F and Young L S 1985 The metric entropy of diffeomorphisms: I. Characterization of measures satisfying Pesin's formula Ann. Math. 122 509-39
28. Ledrappier F and Young L S 1985 The metric entropy of diffeomorphisms: II. Relations between entropy, exponents and dimension Ann. Math. 122 540-74
29. Liverani C 2004 On contact Anosov flows Ann. Math. 159 1275-312
30. Lucarini V 2008 Response theory for equilibrium and nonequilibrium statistical mechanics: causality and generalized Kramers-Kronig relations J. Stat. Phys. 131 543-58
31. Lucarini V Evidence of dispersion relations for the nonlinear response of the Lorenz 63 system arXiv:0809.0251v1
32. Misiurewicz M 1981 Absolutely continuous measures for certain maps of an interval Publ. Math. IHES 53 17-52
33. Oseledec V I 1968 A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems Tr. Mosk. Mat. Ob. 19 179-210
34. Oseledec V I 1968 Trans. Moscow Math. Soc. 19 197-221 (Engl. Transl.)
35. Parry W and Pollicott M 1990 Zeta functions and the periodic orbit structure of hyperbolic dynamics Astérisque 187-188 1-268
36. Pesin Ya B 1976 Invariant manifold families which correspond to non-vanishing characteristic exponents Izv. Akad. Nauk SSSR Ser. Mat. 40 1332-79
37. Pesin Ya B 1976 Math. USSR Izv. 10 1261-305 (Engl. Transl.)
38. Pesin Ya B 1977 Lyapunov characteristic exponents and smooth ergodic theory Uspehi Mat. Nauk 32 55-112
39. Pesin Ya B 1977 Russian Math. Surveys. 32 55-114 (Engl. Transl.)
40. Ruelle D 1976 A measure associated with Axiom A attractors Am. J. Math. 98 619-54
41. Ruelle D 1979 Ergodic theory of differentiable dynamical systems Publ. Math. IHES 50 27-58
42. Ruelle D 2004 Thermodynamic formalism 2nd edn (Cambridge: Cambridge University Press)
43. Ruelle D 1997 Differentiation of SRB states Commun. Math. Phys. 187 227-41
44. Ruelle D 2003 Correction and complements Commun. Math. Phys. 234 185-90
45. Ruelle D 1998 General linear response formula in statistical mechanics and the fluctuation-dissipation theorem far from equilibrium Phys. Lett. A 245 220-4
46. Ruelle D 2004 Application of hyperbolic dynamics to physics: some problems and conjectures Bull. Am. Math. Soc. (N.S.) 41 275-8
47. Ruelle D 2005 Differentiating the absolutely continuous invariant measure of an interval map f with respect to f Commun. Math. Phys. 258 445-53
48. Ruelle D 2008 Differentiation of SRB states for hyperbolic flows Ergod. Theory Dyn. Syst. 28 613-31
49. Ruelle D 2009 Structure and f-dependence of the a.c.i.m. for a unimodal map f of Misiurewicz type Commun. Math. Phys. at press
50. Sinai Ya G 1968 Markov partitions and C-diffeomorphisms Funkts. Analiz i Ego Pril. 2 64-89
51. Sinai Ya G 1968 Funct. Anal. Appl. 2 61-82 (Engl. Transl.)
52. Sinai Ya G 1968 Constuction of Markov partitions Funkts. Analiz i Ego Pril. 2 70-80
53. Sinai Ya G 1968 Funct. Anal. Appl. 2 245-53
54. Sinai Ya G 1972 Gibbsian measures in ergodic theory Uspehi Mat. Nauk 27 21-64
55. Sinai Ya G 1972 Russian Math. Surveys 27 21-69 (Engl. Transl.)
56. Smale S 1967 Differentiable dynamical systems Bull. Am. Math. Soc. 73 747-817
57. Spohn H and Lebowitz J L 1977 Stationary non-equilibrium states of infinite harmonic systems Commun. Math. Phys. 54 97-120
58. Toll J 1956 Causality and the dispersion relation: logical foundations Phys. Rev. 104 1760-70
59. Whitney H 1934 Analytic expansions of differentiable functions defined in closed sets Trans. Am. Math. Soc. 36 63-89
60. Young L-S 2002 What are SRB measures and which dynamical systems have them? J. Stat. Phys. 108 733-54