Ergodic Theory of Chaotic Dynamical Systems

  • L.-S. Young

Abstract

A dynamical system in this article consists of a map of a manifold to itself or a flow generated by an autonomous system of ordinary differential equations. For definiteness, we shall discuss the discrete-time case, although most things here have their continuous-time versions. We shall not attempt to define “chaos, ” except to mention two of its essential ingredients:
  1. 1.

    exponential divergence of nearby orbits

     
  2. 2.

    lack of predictability.

     

Keywords

Entropy Manifold Mane Shoe 

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References

  1. [A]
    Anosov, D., Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math. No. 90 1967. Amer. Math. Soc. 1969 (translation), pp. 1–235.Google Scholar
  2. [BB]
    Ballman, W. and Brin, M., On the ergodicity of geodesin flows, Ergod.Theory Dynam. Syst. 2 (1982), 311–315.CrossRefGoogle Scholar
  3. [BC1]
    Benedicks, M. and Carleson, L., On iterations of 1-ax2 on (−1, 1)Ann. Math. 122 (1985), 1–25.MathSciNetMATHCrossRefGoogle Scholar
  4. [BC2]
    ———, and ———, The dynamics of the Henon map, Ann. Math. 133 (1991), 73–169.MathSciNetMATHCrossRefGoogle Scholar
  5. [BY1]
    ———, and Young, L.-S., Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps, Erg. Th. & Dynam. Sys. 12 (1992), 13–38.MATHGoogle Scholar
  6. [BY2]
    ——— and ———, SBR measures for certain Henon maps, to appear in Inventiones (1993).Google Scholar
  7. [BK]
    Brin, M. and Katok, A., On local entropy, Geometric Dynamics, Springer Lecture Notes in Mathematics, No. 1007, Springer-Verlag, Berlin, 1983, pp. 30–38.Google Scholar
  8. [BL]
    Blokh, A.M. and Lyubich, M. Yu., Measurable dynamics of S-unimodal maps of the interval, preprint, 1990.Google Scholar
  9. [Bo]
    Bowen, R., Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer Lectures Notes in Mathematics No. 470, Springer-Verlag, Berlin, 1975.Google Scholar
  10. [Bu]
    Bunimovich, L.A., On the ergodic properties of nowhere dispersing billiards, Commun. Math. Phys. 65 (1979), 295–312.MathSciNetMATHCrossRefGoogle Scholar
  11. [C]
    Collet, P., Ergodic properties of some unimodal mappings of the interval, preprint, 1984.Google Scholar
  12. [CL]
    ——— and Levy, Y., Ergodic properties of the Lozi mappings, Commun.Math. Phys. 93 (1984), 461–482.MathSciNetMATHCrossRefGoogle Scholar
  13. [CS]
    Chernov, N.I. and Sinai Ya.G., Ergodic properties of some systems of 2-dimensional discs and 3-dimensional spheres, Russ. Math. Surveys 42 (1987), 181–207.MathSciNetMATHGoogle Scholar
  14. [ER]
    Eckmann, J.-P. and Ruelle, D., Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57 (1985), 617–656.MathSciNetCrossRefGoogle Scholar
  15. [FHY]
    Fathi, A., Herman, M., and Yoccoz, J.-C., A proof of Pesin’s stable manifold theorem, Springer Lecture Notes in Mathematics. No. 1007, Springer-Verlag, Berlin, 1983, pp. 117–215.Google Scholar
  16. [FK]
    Furstenberg, H. and Kifer, Yu., Random matrix products and measures on projective spaces, Israel J. Math. 46 (1–2) (1983), 12–32.MathSciNetMATHCrossRefGoogle Scholar
  17. [FKYY]
    Frederickson, P., Kaplan, K.L., Yorke, E.D. and Yorke, J.A., The Lyapunov dimension of strange attractors, J. Diff. Eq. 49 (1983) 185–207.MathSciNetMATHCrossRefGoogle Scholar
  18. [GH]
    Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.MATHGoogle Scholar
  19. [J]
    Jakobson, M., Absolutely continuous invariant measures for oneparameter families of one-dimensional maps, Commun. Math. Phys. 81 (1981), 39–88.MathSciNetMATHCrossRefGoogle Scholar
  20. [Ka1]
    Katok, A., Bernoulli diffeomorphisms on surfaces, Ann. Math. 110 (1979), 529–547.MathSciNetMATHCrossRefGoogle Scholar
  21. [Ka2]
    ———, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES 51 (1980), 137–174.MathSciNetMATHGoogle Scholar
  22. [KK]
    ——— and Kifer, Yu., Random perturbations of transformations of an interval, J. Analy. Math. 47 (1986), 194–237.MathSciNetGoogle Scholar
  23. [KS]
    ——— and Strelcyn, J.M., Invariant Manifolds, Entropy and Billiards;Smooth Maps with Singularities, Springer Lecture Notes in Mathematics No. 1222, Springer-Verlag, Berlin, 1986.Google Scholar
  24. [Ki1]
    Kifer, Yu., On small random perturbations of some smooth dynamical systems, Math. USSR Ivestjia 8 (1974), 1083–1107.MATHCrossRefGoogle Scholar
  25. [Ki2]
    ———, General random perturbations of hyperbolic and expanding transformations, J. Anal. Math. 47 (1986), 111–150.MathSciNetMATHCrossRefGoogle Scholar
  26. [Ki3]
    ———, Random Perturbations of Dynamical Systems, Birkhauser, Basel, 1986.Google Scholar
  27. [KSS]
    Kramli, A., Simanyi, N., and Szasz, D., The K-property of three billiard balls, Ann. Math. 133 (1991), 37–72.MathSciNetMATHCrossRefGoogle Scholar
  28. [Le1]
    Ledrappier, F., Preprietes ergodiques des mesures de Sinai, Publ Math.IHES 59 (1984), 163–188.MathSciNetMATHGoogle Scholar
  29. [Le2]
    ———, Dimension of invariant measures, Teubner-Texte zur Math. 94 (1987), 116–124.MathSciNetGoogle Scholar
  30. [LS]
    ——— and Strelcyn, J.-M., A proof of the estimation from below in Pesin entropy formula, Ergod. Theory Dynam. Syst. 2 (1982), 203–219.MathSciNetMATHCrossRefGoogle Scholar
  31. [LY1]
    ——— and Young, L.-S., The metric entropy of diffeomorphisms, Ann.Math. 122 (1985), 509–574.MathSciNetMATHCrossRefGoogle Scholar
  32. [LY2]
    ——— and ———, Stability of Lyapunov exponents, Ergod. Th. & Dynam. Sys. 11 (1991), 469–484.MathSciNetMATHGoogle Scholar
  33. [Lo]
    Lorenz, E., Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963), 130–141.CrossRefGoogle Scholar
  34. [M1]
    Mãné, R., A proof of Pesin’s formula, Ergod. Theory Dynam. Syst. 1 (1981), 95–102.MATHCrossRefGoogle Scholar
  35. [M2]
    ———, On the dimension of compact invariant sets of certain nonlinear maps, Springer Lecture Notes in Mathematics No. 898, Springer-Verlag, Berlin, 1981, pp. 230–241 (see erratum).Google Scholar
  36. [M3]
    ———, A proof of the C 1-stability conjecture, Publ. Math. IHES 66 (1988), 160–210.Google Scholar
  37. [N]
    Newhouse, S., Lectures on Dynamical Systems, Birkhauser, Basel, (1980), 1–114.Google 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. [Pa]
    Palis, J. On the C 1-stability conjecture, Publ. Math. IHES 66 (1988), 211–215.MathSciNetMATHGoogle Scholar
  40. [Pe1]
    Pesin, Ya.B., Families if invariant manifolds corresponding to non zero characteristic exponents, Math. USSR Izvestjia 10 (1978), 1261–1305.MATHCrossRefGoogle Scholar
  41. [Pe2]
    ———, Characteristic Lyapunov exponents and smooth ergodic theory, Russ. Math. Surveys 32 (1977), 55–114.MathSciNetCrossRefGoogle Scholar
  42. [PeSi]
    ——— and Sinai, Ya.G., Gibbs measures for partially hyperbolic at tractors, Ergod. Theory Dynam. Syst. 2 (1982), 417–438.MathSciNetMATHCrossRefGoogle Scholar
  43. [PuSh]
    Pugh, C. and Shub, M., Ergodic attractors, Trans. AMS 312 (1) (1989), 1–54.MathSciNetMATHCrossRefGoogle Scholar
  44. [Robb]
    Robbin, J., A structural stability theorem, Ann. Math. 94 (1971), 447–493.MathSciNetMATHCrossRefGoogle Scholar
  45. [Robi]
    Robinson, C., Structural stability of C 1 diffeomorphisms, J. Diff. Eqs. 22 (1976), 28–73.MATHCrossRefGoogle Scholar
  46. [Ru1]
    Ruelle, D., A measure associated with Axiom A attractors, Amer. J.Math. 98 (1976), 619–654.MathSciNetMATHCrossRefGoogle Scholar
  47. [Ru2]
    ———, An inequality of the entropy of differentiate maps, Bol. Sc. Bra. Mat. 9 (1978), 83–87.MathSciNetMATHCrossRefGoogle Scholar
  48. [Ru3]
    ———, Ergodic theory of differentiate systems, Publ. Math. IHES 50 (1979), 27–58.MathSciNetMATHGoogle Scholar
  49. [Ry]
    Rychlik, M., A proof of Jakobson’s theorem, Ergod. Theory Dynam.Syst. 8 (1988), 93–109.MathSciNetMATHCrossRefGoogle Scholar
  50. [Si1]
    Sinai, Ya.G., Markov partitions and C-diffeomorphisms, Funct. Anal. Appl. 2 (1968), 64–89.MathSciNetCrossRefGoogle Scholar
  51. [Si2]
    ———, Dynamical systems with elastic reflections: ergodic properties of dispersing billiards, Russ. Math. Surveys 25(2) (1970), 137–189.MathSciNetMATHCrossRefGoogle Scholar
  52. [Si3]
    ———, Gibbs measures in ergodic theory, Russ. Math. Surveys 27(4) (1972), 21–69.MathSciNetMATHCrossRefGoogle Scholar
  53. [Si4]
    ———(ed.), Dynamicals Systems II, Springer-Verlag, Berlin, 1989.Google Scholar
  54. [Sm]
    Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.MathSciNetMATHCrossRefGoogle Scholar
  55. [VF]
    Ventsel, A.D. and Freidlin, M.I., Small random perturbations of dynamical systems, Russ. Math. Surveys 25 (1970), 1–55.CrossRefGoogle Scholar
  56. [W1]
    Wojtkowski, M., Principles for the design of billiards with nonvanishing Lyapunov exponents, Commun. Math. Phys. 105 (1986), 391–414.MathSciNetMATHCrossRefGoogle Scholar
  57. [W2]
    ———, A system of one-domensional balls with gravity, Commun.Math. Phys. 126 (1990), 507–533.MathSciNetMATHCrossRefGoogle Scholar
  58. [Y1]
    Young, L.-S., Dimension, entropy and Lyapunov exponents, Ergod.Theory Dynam. Syst. 2 (1982), 109–129.MATHCrossRefGoogle Scholar
  59. [Y2]
    ———, Stochastic stability of hyperbolic attractors, Ergod. Theory Dynam. Syst. 6 (1986), 311–319.MATHGoogle Scholar
  60. [Y3]
    ———, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. AMS, 287, No. 1 (1985), 41–48.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

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  • L.-S. Young

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