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Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators

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Abstract

Simon’s subshift conjecture states that for every aperiodic minimal subshift of Verblunsky coefficients, the common essential support of the associated measures has zero Lebesgue measure. We disprove this conjecture in this paper, both in the form stated and in the analogous formulation of it for discrete Schrödinger operators. In addition we prove a weak version of the conjecture in the Schrödinger setting. Namely, under some additional assumptions on the subshift, we show that the density of states measure, a natural measure associated with the operator family and whose topological support is equal to the spectrum, is singular. We also consider one-frequency quasi-periodic Schrödinger operators with continuous sampling functions and show that generically, the density of states measure is singular as well.

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Author information

Authors and Affiliations

  1. CNRS UMR 7586, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Bâtiment Sophie Germain, Case 7012, 75205, Paris Cedex 13, France

    Artur Avila

  2. IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil

    Artur Avila

  3. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    David Damanik

  4. Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Zhenghe Zhang

Authors
  1. Artur Avila
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  2. David Damanik
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  3. Zhenghe Zhang
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Corresponding author

Correspondence to David Damanik.

Additional information

Communicated by B. Simon

A. A. was supported by the ERC Starting Grant “Quasiperiodic” and by the Balzan project of Jacob Palis.

D. D. was supported in part by a Simons Fellowship and NSF Grants DMS–0800100 and DMS–1067988.

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Avila, A., Damanik, D. & Zhang, Z. Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators. Commun. Math. Phys. 330, 469–498 (2014). https://doi.org/10.1007/s00220-014-1968-2

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  • DOI: https://doi.org/10.1007/s00220-014-1968-2

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