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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Camden, NJ, 08102, USA

    Mahesh Nerurkar

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  1. Mahesh Nerurkar
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Correspondence to Mahesh Nerurkar.

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Dedicated to Professor Russell Johnson on the occasion of his birthday.

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Nerurkar, M. On \({SL(2, \mathbb R)}\) Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows. J Dyn Diff Equat 23, 451–473 (2011). https://doi.org/10.1007/s10884-011-9215-4

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  • DOI: https://doi.org/10.1007/s10884-011-9215-4

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