Communications in Mathematical Physics

, Volume 359, Issue 1, pp 347–373 | Cite as

Lyapunov Exponents and Correlation Decay for Random Perturbations of Some Prototypical 2D Maps

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Abstract

To illustrate the more tractable properties of random dynamical systems, we consider a class of 2D maps with strong expansion on large—but non-invariant—subsets of their phase spaces. In the deterministic case, such maps are not precluded from having sinks, as derivative growth on disjoint time intervals can be cancelled when stable and unstable directions are reversed. Our main result is that when randomly perturbed, these maps possess positive Lyapunov exponents commensurate with the amount of expansion in the system. We show also that initial conditions converge exponentially fast to the stationary state, equivalently time correlations decay exponentially fast. These properties depend only on finite-time dynamics, and do not involve parameter selections, which are necessary for deterministic maps with nonuniform derivative growth.

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References

  1. 1.
    Benedicks M., Carleson L.: On iterations of \({1 - ax^2}\) on (−1, 1). Ann. Math. 122(1), 1–25 (1985)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Benedicks M., Carleson L.: The dynamics of the Hénon map. Ann. Math. 133(1), 73–169 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Benedicks, M., Young, L.-S.: Sinai–Bowen–Ruelle measures for certain Hénon maps. In: Hunt, B.R., Kennedy, J.A., Li, T.-Y., Nusse, H.E. (eds.) The Theory of Chaotic Attractors, pp. 364–399. Springer, Berlin (1993)Google Scholar
  4. 4.
    Blumenthal A., Xue J., Young L.-S.: Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. Ann. Math. 185(1), 285–310 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bochi J.: Genericity of zero Lyapunov exponents. Ergod. Theory Dyn. Syst. 22(06), 1667–1696 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bochi J., Viana M.: The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. Math. 161, 1423–1485 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chernov N.: Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122(6), 1061–1094 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chirikov B.V.: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52(5), 263–379 (1979)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dolgopyat D.: Limit theorems for partially hyperbolic systems. Trans. Am. Math. Soc. 356(4), 1637–1689 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Duarte P.: Plenty of elliptic islands for the standard family of area preserving maps. Ann. l’IHP Anal. Non Linéaire 11, 359–409 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gorodetski A.: On stochastic sea of the standard map. Commun. Math. Phys. 309(1), 155–192 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Graczyk J., Swiatek G.: Generic hyperbolicity in the logistic family. Ann. Math. 146, 1–52 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hénon M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50(1), 69–77 (1976)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jakobson M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81(1), 39–88 (1981)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kifer Y.: Ergodic Theory of Random Transformations, Volume 10 of Progress in Probability and Statistics. Springer, Berlin (2012)Google Scholar
  16. 16.
    Lian Z., Stenlund M.: Positive Lyapunov exponent by a random perturbation. Dyn. Syst. 27(2), 239–252 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Liverani C.: Decay of correlations. Ann. Math. 142(2), 239–301 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lyubich M.: Dynamics of quadratic polynomials, I–II. Acta Math. 178(2), 185–297 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lyubich M.: Almost every real quadratic map is either regular or stochastic. Ann. Math. 156, 1–78 (2002)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Masmoudi N., Young L.-S.: Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs. Commun. Math. Phys. 227(3), 461–481 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Newhouse S.E.: The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 50(1), 101–151 (1979)CrossRefMATHGoogle Scholar
  22. 22.
    Pesin Y.B., Sinai Y.G.: Gibbs measures for partially hyperbolic attractors. Ergod. Theory Dyn. Syst. 2(3–4), 417–438 (1982)MathSciNetMATHGoogle Scholar
  23. 23.
    Ruelle D.: A measure associated with Axiom-A attractors. Am. J. Math. 98, 619–654 (1976)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wang Q., Young L.-S.: Strange attractors with one direction of instability. Commun. Math. Phys. 218(1), 1–97 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wang Q., Young L.-S.: From invariant curves to strange attractors. Commun. Math. Phys. 225(2), 275–304 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Wang Q., Young L.-S.: Toward a theory of rank one attractors. Ann. Math. 167(2), 349–480 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Young L.-S.: Decay of correlations for certain quadratic maps. Commun. Math. Phys. 146(1), 123–138 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Young L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147(3), 585–650 (1998)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Young L.-S.: Mathematical theory of Lyapunov exponents. J. Phys. A Math. Theor. 46(25), 254001 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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