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Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model

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An Erratum to this article was published on 02 December 2012

Abstract

We show how to extend (and with what limitations) Avila’s global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for the extended Harper’s model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila’s global theory is also shown to imply continuous behavior of the LE on the space of analytic \({M_2(\mathbb{C})}\)-cocycles. This includes rational approximation of the frequency, which so far has not been available.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92717, USA

    S. Jitomirskaya & C. A. Marx

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  1. S. Jitomirskaya
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  2. C. A. Marx
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Correspondence to C. A. Marx.

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Communicated by B. Simon

The work was supported by the NSF Grants DMS - 0601081 and DMS 1101578, and the BSF grant 2006483.

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Jitomirskaya, S., Marx, C.A. Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model. Commun. Math. Phys. 316, 237–267 (2012). https://doi.org/10.1007/s00220-012-1465-4

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  • DOI: https://doi.org/10.1007/s00220-012-1465-4

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