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On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows

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  • We show that a generic $SL(2,R)$ valued cocycle in the class of $C^r$, ($0 < r < 1$) cocycles based on a rotation flow on the $d$-torus, is either uniformly hyperbolic or has zero Lyapunov exponents provided that the components of winding vector $\bar \gamma = (\gamma^1,\cdot \cdot \cdot,\gamma^d)$ of the rotation flow are rationally independent and satisfy the following super Liouvillian condition :

    $ |\gamma^i - \frac{p^i_n}{q_n}| \leq Ce^{-q^{1+\delta}_n}, 1\leq i\leq d, n\in N,$

    where $C > 0$ and $\delta > 0$ are some constants and $p^i_n, q_n$ are some sequences of integers with $q_n\to \infty$.

    Mathematics Subject Classification: Primary: 37B55, 34A30; Secondary: 58F15.


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