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Positive Lyapunov exponents for a class of deterministic potentials

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Abstract

LetV(θ) be a smooth, non-constant function on the torus and letT be a hyperbolic toral automorphism. Consider a discrete one dimensional Schrödinger operatorH, whose potential at sitej is given bygV j =gV(T j θ). We prove that wheng≧0 is small andg 1/2 ≦|E|≦2−g 1/2, the Lyapunov exponent for the cocycle generated byH-E is proportional tog 2. The proof relies on a formula of Pastur and Figotin and on symbolic dynamics.

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Authors and Affiliations

  1. Institute for Mathematical Problems in Biology, Pushchino, Russia

    Victor Chulaevsky

  2. Institute for Advanced Study, 08540, Princeton, New Jersey, USA

    Thomas Spencer

Authors
  1. Victor Chulaevsky
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  2. Thomas Spencer
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Communicated By B. Simon

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Chulaevsky, V., Spencer, T. Positive Lyapunov exponents for a class of deterministic potentials. Commun.Math. Phys. 168, 455–466 (1995). https://doi.org/10.1007/BF02101838

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  • DOI: https://doi.org/10.1007/BF02101838

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