Journal of Dynamics and Differential Equations

, Volume 23, Issue 3, pp 451–473 | Cite as

On \({SL(2, \mathbb R)}\) Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

  • Mahesh Nerurkar


Consider the class of C r -smooth \({SL(2, \mathbb R)}\) valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: \(\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n})\), where C >  0 and \({0 < \kappa <1 }\) are some constants and \({\frac{P_n}{q_n}}\) is some sequence of irreducible fractions.


Cocycles Lyapunov exponents Irrational rotations Proximal extensions Fast periodic approximation 

Mathematics Subject Classification (2000)

37B55 34A30 58F15 


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  1. 1.
    Avila A.: Density of positive Lyapunov exponents for quasiperiodic \({SL(2, \mathbb R)}\) cocycles in arbitrary dimension. J. Mod. Dyn. 3, 629–634 (2009)Google Scholar
  2. 2.
    Bochi J.: Genericity of zero Lyapunov exponents. Ergod. Theory Dyn. Syst. 22, 1667–1696 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    De Concini C., Johnson R.: The algebraic-geometric AKNS potentials. Ergod. Theory Dyn. Syst. 7, 1–24 (1987)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ellis R.: Lectures in Topological Dynamics. Benjamin, New York (1969)Google Scholar
  5. 5.
    Fabbri, R.: Genericita dell’iperbolicita nei sistemi differenziali lineari di dimensione due, Ph.D. Thesis, Universita di Firenze (1997)Google Scholar
  6. 6.
    Fabbri R., Johnson R.: On the Lyapunov exponent of certain \({SL(2, \mathbb R)}\) valued cocycles. Differ. Equ. Dyn. Syst. 7, 349–370 (1999)MathSciNetMATHGoogle Scholar
  7. 7.
    Fabbri R., Johnson R., Pavani R.: On the nature of the spectrum of the quasi-periodic Schrodinger operator. Nonlin. Anal. 3 37, 37–59 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Furman A.: On the multiplicative ergodic theorem for uniquely ergodic systems. Ann. Inst. H. Poincare Probab. Stat. 33, 797–815 (1997)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fayad B., Krikorian R.: Rigidity results for quasiperiodic \({SL(2, \mathbb R)}\) cocycles. J. Mod. Dyn. 3, 479–510 (2009)MathSciNetMATHGoogle Scholar
  10. 10.
    Johnson R.: Two dimensional almost periodic linear system with proximal and recurrent behaviour. Proc. AMS 82(3), 417–422 (1981)MATHCrossRefGoogle Scholar
  11. 11.
    Johnson R.: Exponential dichotomy, rotation number and linear differential operators with bounded coefficients. J. Differ. Equ. 61, 54–78 (1986)MATHCrossRefGoogle Scholar
  12. 12.
    Johnson R.: Hopf bifurcation from non-periodic solutions of differential equations I-Linear Theory. J. Dyn. Differ. Equ. 1(2), 179–198 (1989)MATHCrossRefGoogle Scholar
  13. 13.
    Johnson R.: Cantor spectrum for the quasi periodic Schrödinger equation. J. Differ. Equ. 91, 88–110 (1991)MATHCrossRefGoogle Scholar
  14. 14.
    Johnson R., Moser J.: The rotation number for almost periodic potentials. Comm. Math. Phys. 84, 403–438 (1982)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Johnson, R., Nerurkar, M.: On \({SL(2, \mathbb R)}\) valued cocycles of Hölder class with zero exponents over Kronecker flows, Commun. Pure Appl. Anal. 10(3) (2011)Google Scholar
  16. 16.
    Johnson R., Palmer K., Sell G.: Ergodic properties of linear dynamical systems. Siam J. Math. Anal. 18, 1–33 (1987)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Krikorian, R.: Reducibility, Differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on \({\mathbb{T} \times SL(2, \mathbb R)}\), (2002), preprint (
  18. 18.
    Knill O.: Positive Lyapunov exponents for a dense set of bounded measurable \({SL(2, \mathbb R)}\) cocycles. Ergod. Theory Dyn. Syst. 12, 319–331 (1992)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kotani, S.: Lyapunov indices determine absolutely continuous spectrum of stationary random Schrodinger operators, Proc. Taniguchi Symposium SA, Kataka, 225–247 (1982)Google Scholar
  20. 20.
    Magnus W., Winkler S.: Hill’s Equation. Dover Publication, New York (1979)Google Scholar
  21. 21.
    Nerurkar M.: Recurrent proximal linear differential systems with almost periodic coefficients’. Proc. Am. Math. Soc. 100(4), 739–743 (1987)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Nerurkar M.: Positive exponents for a dense set of continuous cocycles which arise as solutions to strongly accessible linear differential systems. Contemp. Math. Ser. AMS 215, 265–278 (1998)MathSciNetGoogle Scholar
  23. 23.
    Viana M.: Almost all cocycles over a hyperbolic system have non-vanishing Lyapunov exponents. Ann. Math. 107, 643–680 (2008)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityCamdenUSA

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