Abstract
One knows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is open and dense in analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be arbitrarily approximated by ones with zero Lyapunov exponent in the space of \({\mathcal{C}^ l (1 \le l \le \infty)}\) smooth quasi-periodic cocycles, which shows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not open in smooth topology.
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Acknowledgements
We are grateful to the referee for the useful suggestions. We are also in debt to S. Jitomirskaya for drawing our attention to this question.
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Communicated by J. Marklof
This work was supported by NSFC of China (Grants 11471155, 11771205), Simons Foundation.
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Wang, Y., You, J. The Set of Smooth Quasi-periodic Schrödinger Cocycles with Positive Lyapunov Exponent is Not Open. Commun. Math. Phys. 362, 801–826 (2018). https://doi.org/10.1007/s00220-018-3223-8
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DOI: https://doi.org/10.1007/s00220-018-3223-8