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Positive Lyapunov exponent for a class of quasi-periodic cocycles

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  • Young [17] proved the positivity of Lyapunov exponent in a large set of the energies for some quasi-periodic cocycles. Her result is also proved to be applicable for some quasi-periodic Schrödinger cocycles by Zhang [18]. However, her result cannot be applied to the Schrödinger cocycles with the potential $ v = \cos(4\pi x)+w( x) $, where $ w\in C^2(\mathbb R/\mathbb Z,\mathbb R) $ is a small perturbation. In this paper, we will improve her result such that it can be applied to more cocycles.

    Mathematics Subject Classification: Primary: 37A30.


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  • Figure 1.  graph of the function in $ \mathcal F $

    Figure 2.  Graphs of the angle functions

    Figure 3.  Bifurcation of Type Ⅲ functions with $ f'_1(c_1)f'_2(c_2)<0 $

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