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Communications in Mathematical Physics

, Volume 362, Issue 3, pp 801–826 | Cite as

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

  • Yiqian Wang
  • Jiangong You
Article
  • 89 Downloads

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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Notes

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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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina

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