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The Set of Smooth Quasi-periodic Schrödinger Cocycles with Positive Lyapunov Exponent is Not Open

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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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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Yiqian Wang

  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Jiangong You

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  1. Yiqian Wang
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  2. Jiangong You
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Correspondence to Jiangong You.

Additional information

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

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