Advertisement

From Kolmogorov’s Work on entropy of dynamical systems to Non-uniformly hyperbolic dynamics

  • Denis V. Kosygin
  • Yakov G. Sinai

Keywords

Strange Attractor Tangency Point Bernoulli Shift Positive Lebesgue Measure Invariant Circle 
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. [ABS77]
    Afra⌣movič, V.S., Bykov, V.V., Sil″nikov, L.P.: The origin and structure of the Lorenz attractor. Dokl. Akad. Nauk SSSR, 234, 336–339 (1977)MathSciNetGoogle Scholar
  2. [AL83]
    Aubry, S., Le Daeron, P.Y.: The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states. Physica D, 8, 381–422 (1983)CrossRefMathSciNetGoogle Scholar
  3. [Ano67]
    Anosov, D.V.: Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Inst. Math., 90, 235 p. (1967)Google Scholar
  4. [Arn63]
    Arnol′d, V.I.: Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspehi Mat. Nauk, 18, 13–40 (1963)Google Scholar
  5. [BC85]
    Benedicks, M., Carleson, L.: On iterations of 1-ax2 on (-1,1). Ann. of Math., 122, 1–25 (1985)CrossRefMathSciNetGoogle Scholar
  6. [BC91]
    Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. of Math., 133, 73–169 (1991)CrossRefMathSciNetGoogle Scholar
  7. [Bun70]
    Bunimovič, L.A.: On a transformation of the circle. Akad. Nauk SSSR Mat. Zametki, 8, 205–216 (1970)Google Scholar
  8. [CE83]
    Collet, P., Eckmann, J.-P.: Positive Liapunov exponents and absolute continuity for maps of the interval. Ergodic Theory Dynam. Systems, 3, 13–46 (1983)zbMATHMathSciNetGoogle Scholar
  9. [CFS82]
    Cornfeld, I.P., Fomin, S.V., Sina⌣, Ya.G.: Ergodic Theory. Grund. math. Wiss. no 245, Springer (1982)Google Scholar
  10. [Chi79]
    Chirikov, B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep., 52, 264–379 (1979)Google Scholar
  11. [Fom50]
    Fomin, S.: On dynamical systems in a space of functions. Ukrain. Mat. Žurnal, 2, 25–47 (1950)zbMATHGoogle Scholar
  12. [GF52]
    Gel′fand, I.M., Fomin, S.V.: Geodesic flows on manifolds of constant negative curvature. Uspehi Matem. Nauk (N.S.), 7, 118–137 (1952)zbMATHMathSciNetGoogle Scholar
  13. [GF55]
    Gel′fand, I.M., Fomin, S.V.: Geodesic flows on manifolds of constant negative curvature. Amer. Math. Soc. Transl., 1, 49–65 (1955)zbMATHMathSciNetGoogle Scholar
  14. [Gir58]
    Girsanov, I.V.: Spectra of dynamical systems generated by stationary Gaussian processes. Dokl. Akad. Nauk SSSR (N.S.), 119, 851–853 (1958)zbMATHMathSciNetGoogle Scholar
  15. [GS99]
    Graczyk, J., Świątek, G.: Smooth unimodal maps in the 1990s. Ergodic Theory Dynam. Systems, 19, 263–287 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [GW79]
    Guckenheimer, J., Williams, R.F.: Structural stability of Lorenz attractors. Publ. Math. Inst. Hautes Études Sci. Publ. Math., 50, 59–72 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Hén76]
    Hénon, M.: A two-dimensional mapping with a strange attractor. Comm. Math. Phys., 50, 69–77 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [Hop39]
    Hopf, E.: Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91, 261–304 (1939)MathSciNetGoogle Scholar
  19. [Hop40]
    Hopf, E.: Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. II. Math. Ann., 117, 590–608 (1940)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [Jak81]
    Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys., 81, 39–88 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [Kol41a]
    Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers. C. R. (Doklady) Acad. Sci. URSS (N.S.), 30, 301–305 (1941)MathSciNetGoogle Scholar
  22. [Kol41b]
    Kolmogorov, A.N.: Dissipation of energy in the locally isotropic turbulence. C. R. (Doklady) Acad. Sci. URSS (N.S.), 32, 16–18 (1941)MathSciNetGoogle Scholar
  23. [Kol42]
    Kolmogorov, A.N.: Equations of turbulent motion of an incompressible fluid. Bull. Acad. Sci. URSS. Ser. Phys. [Izvestia Akad. Nauk SSSR], 6, 56–58 (1942)Google Scholar
  24. [Kol54a]
    Kolmogorov, A.N.: Théorie générale des systèmes dynamiques et mécanique classique. In: Proceedings of the International Congress of Mathematicians, vol. 1, 315–333, Amsterdam (1954), Erven P. Noordhoff N.V., Groningen (1957)Google Scholar
  25. [Kol54b]
    Kolmogorov, A.N.: On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR (N.S.), 98, 527–530 (1954)zbMATHMathSciNetGoogle Scholar
  26. [Kol58]
    Kolmogorov, A.N.: A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR (N.S.), 119, 861–864 (1958)zbMATHMathSciNetGoogle Scholar
  27. [Kol91a]
    Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. Roy. Soc. London Ser. A, 434, 9–13 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Kol91b]
    Kolmogorov, A.N.: Dissipation of energy in the locally isotropic turbulence. Proc. Roy. Soc. London Ser. A, 434, 15–17 (1991)MathSciNetzbMATHGoogle Scholar
  29. [Koz00]
    Kozlovski, O.S.: Getting rid of the negative Schwarzian derivative condition. Ann. of Math., 152, 743–762 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  30. [Lor63]
    Lorenz, E.N.: Deterministic Nonperiodic Flow. J. Atmospheric Sci., 20, 130–141 (1963)CrossRefGoogle Scholar
  31. [Mar49]
    Maruyama, G.: The harmonic analysis of stationary stochastic processes. Mem. Fac. Sci. Kyūsyū Univ. A, 4, 45–106 (1949)MathSciNetGoogle Scholar
  32. [Mat84]
    Mather, J.N.: Nonexistence of invariant circles. Ergodic Theory Dynam. Systems, 4, 301–309 (1984)Google Scholar
  33. [Mis81]
    Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Publ. Math. Inst. Hautes Études Sci., 53, 17–51 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  34. [Mos62]
    Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962, 1–20 (1962)Google Scholar
  35. [MV93]
    Mora, L., Viana, M.: Abundance of strange attractors. Acta Math., 171, 1–71 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  36. [NV88]
    Nowicki, T., van Strien, S.: Absolutely continuous invariant measures for C2 unimodal maps satisfying the Collet-Eckmann conditions. Invent. Math., 93, 619–635 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  37. [NV90]
    Nowicki, T., van Strien, S.: Hyperbolicity properties of C2 multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions. Trans. Amer. Math. Soc., 321, 793–810 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  38. [Ogn81]
    Ognev, A.I.: Metric properties of a class of mappings of a segment. Akad. Nauk SSSR Mat. Zametki, 30, 723–736, 797 (1981)Google Scholar
  39. [Orn70]
    Ornstein, D.S.: Bernoulli shifts with the same entropy are isomorphic. Adv. in Math., 4, 337–352 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  40. [Orn74]
    Ornstein, D.S.: Ergodic theory, randomness, and dynamical systems. Yale University Press, New Haven, Conn. (1974)zbMATHGoogle Scholar
  41. [Per79]
    Percival, I.C.: Variational principles for invariant tori and cantori. In: Nonlinear dynamics and the beam-beam interaction, AIP Conf. Proc. no 57, 302–310, Sympos. Brookhaven Nat. Lab., New York (1979)Google Scholar
  42. [Pes77]
    Pesin, Ja.B.: Characteristic Ljapunov exponents, and smooth ergodic theory. Uspehi Mat. Nauk, 32, 55–112, 287 (1977)MathSciNetGoogle Scholar
  43. [Rén57]
    Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar, 8, 477–493 (1957)zbMATHCrossRefMathSciNetGoogle Scholar
  44. [RT71]
    Ruelle, D., Takens, F.: On the nature of turbulence. Comm. Math. Phys., 20, 167–192 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  45. [Rue77]
    Ruelle, D.: Applications conservant une mesure absolument continue par rapport à dx sur [0,1]. Comm. Math. Phys., 55, 47–51 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  46. [Sin59]
    Sinaĭ, Ya.G.: On the concept of entropy for a dynamic system. Dokl. Akad. Nauk SSSR, 124, 768–771 (1959)zbMATHMathSciNetGoogle Scholar
  47. [Sin61]
    Sinaĭ, Ya.G.: Dynamical systems with countable Lebesgue spectrum. I. Izv. Akad. Nauk SSSR Ser. Mat., 25, 899–924 (1961)zbMATHMathSciNetGoogle Scholar
  48. [Sin66]
    Sinaĭ, Ya.G.: Classical dynamic systems with countably-multiple Lebesgue spectrum. II. Izv. Akad. Nauk SSSR Ser. Mat., 30, 15–68 (1966)MathSciNetGoogle Scholar
  49. [Sma67]
    Smale, S.: Differentiable dynamical systems. Bull. Amer. Math. Soc., 73, 747–817 (1967)CrossRefMathSciNetGoogle Scholar
  50. [Szá00]
    Szász, D. (ed.): Hard ball systems and the Lorentz gas. Encyclopaedia of Mathematical Sciences no 101, Springer, Berlin (2000)Google Scholar
  51. [Tuc99]
    Tucker, W.: The Lorenz attractor exists. C. R. Acad. Sci. Paris Sér. I Math., 328, 1197–1202 (1999)zbMATHMathSciNetGoogle Scholar
  52. [UV47]
    Ulam, S., von Neumann, J.: On combinations of stochastic and determinisitc processes. Bull. Amer. Math. Soc., 53, 1120 (1947)Google Scholar
  53. [vonN32]
    von Neumann, J.: Zur Operatorenmethode in der klassischen Mechanik. Ann. of Math., 33, 587–642 (1932)CrossRefMathSciNetGoogle Scholar
  54. [WY01]
    Wang, Q., Young, L.S.: Strange attractors with one direction of instability. Comm. Math. Phys., 218, 1–97 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Denis V. Kosygin
    • 1
  • Yakov G. Sinai
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityUSA
  2. 2.Department of MathematicsPrinceton UniversityUSA

Personalised recommendations