Advertisement

Ergodic Theory of Differentiable Dynamical Systems

Chapter
Part of the NATO ASI Series book series (ASIC, volume 464)

Abstract

These notes are about the dynamics of systems with hyperbolic properties. The setting for the first half consists of a pair (f, µ), where f is a diffeomorphism of a Riemannian manifold and µ is an f-invariant Borel probability measure. After a brief review of abstract ergodic theory, Lyapunov exponents are introduced, and families of stable and unstable manifolds are constructed. Some relations between metric entropy, Lyapunov exponents and Hausdorff dimension are discussed. In the second half we address the following question: given a differentiable mapping, what are its natural invariant measures? We examine the relationship between the expanding properties of a map and its invariant measures in the Lebesgue measure class. These ideas are then applied to the construction of Sinai-Ruelle-Bowen measures for Axiom A attractors. The nonuniform case is discussed briefly, but its details are beyond the scope of these notes.

Keywords

Lyapunov Exponent Invariant Measure Ergodic Theory Unstable Manifold Borel Probability Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    Azencott, R., Diffeomorphismes d’Anosov et schemas de Bernoulli, CRAS Paris A270 (1970) 1105–1107.MathSciNetzbMATHGoogle Scholar
  2. [BB]
    Ballman, W. and Brin, M., On the ergodicity of geodesic flows, Ergod. Th. and Dyn. Sys. 2 (1982) 311–315. (Note: gap in proof of ergodicity).Google Scholar
  3. [BC1]
    Benedicks, M. and Carleson, L., On iterations of 1-axe on (-1,1), Annals of Math. 122 (1985) 1–25.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [BC2]
    Benedicks, M. and Carleson, L., The dynamics of the Henon map, Ann. Math. 133 (1991) 73–169.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [BY]
    Benedicks, M. and Young, L.-S., SBR measures for certain Henon maps, Inventiones Math. 112 (1993) 541–576.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Bi]
    Billingsley, P., Ergodic theory and information, John Wiley and Sons, Inc., New York-London-Sydney (1965).Google Scholar
  7. [Bo]
    Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lectures Notes in Math. 470 (1975).Google Scholar
  8. [BR]
    Bowan, R. and Ruelle, D., The ergodic theory of Axiom A flows, Invent. Math. 29 (1975) 181–202.Google Scholar
  9. [BrK]
    Brin, M. and Katok, A., On local entropy, Geometric Dynamics, Springer Lecture Notes 1007 (1983) 30–38.MathSciNetGoogle Scholar
  10. [BuK]
    Burns, K. and Katok, A., Infinitesimal Lyapunov functions, invariant cone families, and stochastic properties of smooth dynamical systems, to appear in Erg. Th. and Dynam. Sys.Google Scholar
  11. [CE]
    P. Collet and J. P. Eckmann, Positive Lyapunov exponents and absolutely continuity, Ergod. Th. and Dynam. Syst. 3 (1983), 13–46.MathSciNetzbMATHGoogle Scholar
  12. [ER]
    Eckmann, J.-P. and Ruelle, D., Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57 (1985) 617–656.MathSciNetCrossRefGoogle Scholar
  13. [FHY]
    Fathi, A., Herman, M. and Yoccoz, J.-C., A proof of Pesin’s stable manifold theorem, Springer Lecture Notes in Math. 1007 (1983) 117–215.MathSciNetGoogle Scholar
  14. [Fa]
    Falconer, K., Fractal Geometry, mathematical foundations and applications, Wiley (1990).Google Scholar
  15. [Fu]
    Furstenberg, H., Noncommuting random products, Trans. AMS 108 (1963) 377–428.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [G]
    M. de Guzmán, Differentiation of Integrals in R“, Springer Lecture Notes in Math. 481 (1975), 2.Google Scholar
  17. [GM]
    Goldsheid, I. and Margulis, G., Lyapunov indices of a product of random matrices, Russ. Math. Surveys 44: 5 (1989) 11–71.MathSciNetCrossRefGoogle Scholar
  18. [GR]
    Guivarc’h, Y. and Raugi, A., Frontière de Furstenberg, propriétés de contraction et theorèmes de convergence, Z. Wahrscheinlichkeitstheorie verw. Geb. 69 (1985), 187–242.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [HP]
    M. W. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. Sym. in Pure Math. 14, A.M.S., Providence, RI (1970).Google Scholar
  20. [HY]
    Hu, H. and Young, L.-S., Nonexistence of SBR measures for some systems that are “almost Anosov”, to appear in Erg. Th. and Dyn. Sys.Google Scholar
  21. [J]
    Jakobson, M., Absolutely continuous invariant measures for one-parameter families of onedimensional maps, Commun. Math. Phys. 81 (1981) 39–88.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [Ka]
    Katok, A., Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES 51 (1980) 137–174.zbMATHGoogle Scholar
  23. [KaS]
    Katok, A., and Strelcyn, J.M., Invariant manifolds, entropy and billiards; smooth maps with singularities, Springer Lecture Notes in Math. 1222 (1986).Google Scholar
  24. [KrS]
    Krzyzewski, K. and Szlenk, W., On invariant measures for expanding differentiable mappings, Studia Math. 33 (1969), 83–92.MathSciNetzbMATHGoogle Scholar
  25. [Ki]
    Kingman, C., Subadditive processes, Springer Lecture Notes in Math. 539 (1976).Google Scholar
  26. [L1]
    Ledrappier, F., Preprietes ergodiques des mesures de Sinai, Publ. Math. IHES 59 (1984) 163–188.MathSciNetzbMATHGoogle Scholar
  27. [L2]
    Ledrappier, F., Quelques propriétés des exposants caractéristiques, Springer Lecture Notes in Math 1097 (1984), 305–396.MathSciNetCrossRefGoogle Scholar
  28. [L3]
    Ledrappier, F., Dimension of invariant measures, Teubner-Texte zur Math. 94 (1987) 116–124.MathSciNetGoogle Scholar
  29. [LS]
    Ledrappier, F., and Strelcyn, J.-M., A proof of the estimation from below in Pesin entropy formula, Ergod. Th. and Dynam. Sys. 2 (1982) 203–219.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [LY]
    Ledrappier, F., and Young, L.-S., The metric entropy of diffeomorphisms, Annals of Math. 122 (1985) 509–574.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [M1]
    Mané, R., A proof of Pesin’s formula, Ergod. Th. and Dynam. Sys. 1 (1981) 95–102.zbMATHCrossRefGoogle Scholar
  32. [M2]
    Mané, R., On the dimension of compact invariant sets of certain non-linear maps, Springer Lecture Notes in Math. 898 (1981) 230–241 (see erratum).Google Scholar
  33. [M3]
    Mané, R., Ergodic Theory and Differentiable Dynamics,Springer-Verlag (1987).Google Scholar
  34. [Mi]
    M. Misiurewicz, Absolutely continuous measures for certain maps of the interval, Publ. Math. IHES 53 (1981), 17–51.MathSciNetzbMATHGoogle Scholar
  35. [MV]
    Mora, L. and Viana, M., Abundance of strange attractors, to appear in Acta Math.Google Scholar
  36. [N]
    Newhouse, S., Lectures on Dynamical Systems, Progress in Math. 8, Birkhauser (1980) 1–114.Google Scholar
  37. [NS]
    Nowicki, T. and van Strien, S., Absolutely continuous invariant measures under a summability condition, Invent. Math 105 (1991) 123–136.zbMATHGoogle Scholar
  38. [O]
    Oseledec, V. I., A multiplicative ergodic theorem: Liapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968) 197–231.MathSciNetGoogle Scholar
  39. [OW]
    Ornstein, D. and Weiss, B., Geodesic flows are Bernoullian, Israel J. Math 14 (1973), 184–198.MathSciNetzbMATHGoogle Scholar
  40. [P1]
    Pesin, Ya. B., Families if invariant manifolds corresponding to non-zero characteristic ex- ponents, Math. of the USSR, Izvestjia 10 (1978) 1261–1305.Google Scholar
  41. [P2]
    Pesin, Ya. B., Characteristic Lyapunov exponents and smooth ergodic theory, Russ. Math. Surveys 32 (1977) 55–114.MathSciNetCrossRefGoogle Scholar
  42. [PS]
    Pesin, Ya. B., and Sinai, Ya. G., Gibbs measures for partially hyperbolic attractors, Ergod. Th. and Dynam. Sys. 2 (1982) 417–438.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [PS1]
    Pugh, C. and Shub, M., Ergodicity of Anosov actions, Inventiones Math. 15 (1972) 1–23.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [PS2]
    Pugh, C. and Shub, M., Ergodic attractors, Trans. AMS, Vol. 312, No. 1 (1989) 1–54.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [Re]
    Rees, M., Positive measure sets of ergodic rational maps, Ann. Scient. Ec. Norm. Sup. 40 série 19 (1986) 93–109.MathSciNetGoogle Scholar
  46. [RY]
    Robinson, C. and Young, L.-S., Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math. 61 (1980) 159–176.MathSciNetzbMATHGoogle Scholar
  47. [Rol]
    V. A. Rohlin, On the fundamental ideas of measure theory, A.M.S. Transl. (1) 10 (1962) 1–52.Google Scholar
  48. [Ro2]
    V. A. Rohlin, Lectures on the theory of entropy of transformations with invariant measures, Russ. Math. Surveys 22: 5 (1967) 1–54.MathSciNetCrossRefGoogle Scholar
  49. [Rul]
    Ruelle, D., A measure associated with Axiom A attractors, Amer. J. Math. 98 (1976) 619–654.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [Ru2]
    Ruelle, D., Applications conservant une measure absolument continue par raport a dx sur [0, 1], Commun. Math. Phys. 55 (1987) (47–52).Google Scholar
  51. [Ru3]
    Ruelle, D., An inequality of the entropy of differentiable maps, Bol. Sc. Bra. Mat. 9 (1978) 83–87.MathSciNetzbMATHCrossRefGoogle Scholar
  52. [Ru4]
    Ruelle, D., Ergodic theory of differentiable systems, Publ. Math. IHES 50 (1979) 27–58.MathSciNetzbMATHGoogle Scholar
  53. [Ru5]
    Ruelle, D., The thermodynamics formalism for expanding maps, Commun. Math. Phys., Vol. 125 (1989) 239–262.MathSciNetzbMATHCrossRefGoogle Scholar
  54. [Ry]
    Rychlik, M., A proof of Jakobson’s theorem, Ergod. Th. and Dyn. Sys. 8 (1988) 93–109.MathSciNetzbMATHGoogle Scholar
  55. [Sh]
    Shub, M., Global Stability of Dynamical Systems, Springer (1987).Google Scholar
  56. [51]
    Sinai, Ya. G., Markov partitions and C-diffeomorphisms, Func. Anal. and its Appl. 2 (1968) 64–89.Google Scholar
  57. [S2]
    Sinai, Ya. G., Dynamical systems with elastic reflections: ergodic properties of dispersing bil- liards, Russ. Math. Surveys 25, No. 2 (1970) 137–189.MathSciNetzbMATHCrossRefGoogle Scholar
  58. [S3]
    Sinai, Ya. G., Gibbs measures in ergodic theory, Russ. Math. Surveys 27 No. 4 (1972) 21–69.CrossRefGoogle Scholar
  59. [S4]
    Sinai, Ya. G., Introduction to Ergodic Theory, Princeton Univ. Press (1976).Google Scholar
  60. [S5]
    Sinai, Ya. G., (ed.), Dynamicals systems II, Encyclopaedia of Math. Sc. Vol. 2, Springer-Verlag (1989).Google Scholar
  61. [Sm]
    Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747–817.MathSciNetzbMATHCrossRefGoogle Scholar
  62. [T]
    Tsujii, M., Positive Lyapunov exponents in families of 1-dimensional systems, Invent. Math. 11 (1993) 113–117.MathSciNetGoogle Scholar
  63. [TTY]
    Thieullen, P., Tresser, C. and Young, L.-S., Positive Lyapunov exponent for generic 1-parameter families of unimodal maps, C.R. Acad. Sci. Paris, t. 315 Série I (1992) 69–72; longer version to appear in J. d’Analyse.Google Scholar
  64. [Wa]
    Walters, P., An introduction to Ergodic Theory, Grad. Texts in Math., Springer-Verlag (1981).Google Scholar
  65. [Wl]
    Wojtkowski, M., Invariant families of cones and Lyapunov exponents, Erg. Th. and Dyn. Sys. 5 (1985) 145–161.MathSciNetzbMATHGoogle Scholar
  66. [W2]
    Wojtkowski, M., Principles for the design of billiards with nonvanishing Lyapunov exponents, Commun. Math. Phys. 105 (1986) 391–414.MathSciNetzbMATHCrossRefGoogle Scholar
  67. [W3]
    Wojtkowski, M., Systems of classical interacting particles with nonvanishing Lyapunov exponents, Springer Lecture Notes in Math 1486 (1991).Google Scholar
  68. [Y]
    Young, L.-S., Dimension, entropy and Lyapunov exponents, Ergod. Th. and Dynam. Sys. 2 (1982) 109–129.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations