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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
Article
  • 44 Downloads

Abstract

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.

Keywords

Cocycles Lyapunov exponents Irrational rotations Proximal extensions Fast periodic approximation 

Mathematics Subject Classification (2000)

37B55 34A30 58F15 

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityCamdenUSA

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