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.
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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