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Bounds on the Lyapunov Exponent via Crude Estimates on the Density of States

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Abstract

We study the Chirikov (standard) map at large coupling λ ≫ 1, and prove that the Lyapounov exponent of the associated Schrödinger operator is of order log λ except for a set of energies of measure exp(−c λ β) for some 1 < β < 2. We also prove a similar (sharp) lower bound on the Lyapunov exponent (outside a small exceptional set of energies) for a large family of ergodic Schrödinger operators, the prime example being the d-dimensional skew shift.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Mira Shamis

  2. School of Mathematics, Institute for Advanced Study, Einstein Dr., Princeton, NJ, 08540, USA

    Mira Shamis & Thomas Spencer

Authors
  1. Mira Shamis
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  2. Thomas Spencer
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Correspondence to Mira Shamis.

Additional information

Communicated by W. Schlag

M. Shamis was supported in part by NSF grants PHY-1104596 and DMS-1128155.

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Shamis, M., Spencer, T. Bounds on the Lyapunov Exponent via Crude Estimates on the Density of States. Commun. Math. Phys. 338, 705–720 (2015). https://doi.org/10.1007/s00220-015-2324-x

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  • DOI: https://doi.org/10.1007/s00220-015-2324-x

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