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Mathematische Annalen

, Volume 370, Issue 1–2, pp 271–285 | Cite as

Second phase transition line

  • Artur Avila
  • Svetlana Jitomirskaya
  • Qi ZhouEmail author
Article

Abstract

We study the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum occur for dense sets of frequencies.

Notes

Acknowledgements

A.A. and Q.Z. were partially supported by the ERC Starting Grant “Quasiperiodic”. S.J. was a 2014–15 Simons Fellow, and was partially supported by NSF DMS-1401204. Q.Z. was also supported by “Deng Feng Scholar Program B” of Nanjing University and NNSF of China (11671192), he would like to thank the hospitality of the UCI where this work was started. S.J. and Q.Z. are grateful to the Isaac Newton Institute for Mathematical Sciences, Cambridge, for its hospitality supported by EPSRC Grant No. EP/K032208/1, during the programme “Periodic and Ergodic Spectral Problems” where they worked on this paper.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.CNRS UMR 7586, Institut de Mathématiques de Jussieu - Paris Rive GaucheBâtiment Sophie GermainParis Cedex 13France
  2. 2.IMPARio de JaneiroBrazil
  3. 3.Department of MathematicsUniversity of CaliforniaIrvineUSA
  4. 4.Department of MathematicsNanjing UniversityNanjingChina

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