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Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation

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Abstract

A rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl 2(ℤ), defined byHψ(n)=ψ(n+1)+ψ(n−1)+Vf(nσ)ψ(n), whereV is a real parameter,f is a certain discontinuous period-1 function, and\(\sigma = {{\left( { - 1 + \sqrt 5 } \right)} \mathord{\left/ {\vphantom {{\left( { - 1 + \sqrt 5 } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}\) is the golden mean. The renormalization map forH is a diffeomorphism,T, of ℝ3, preserving a cubic surfaceS V . ForV≧8 we prove that the non-wandering set of the restriction ofT toS v is a hyperbolic set, on whichT is conjugate to a subshift on six symbols. It follows from results in dynamical systems theory that the optimally approximating periodic operators toH have spectra which obey a global scaling law. We also define a set which we call the pseudospectrum” of the operatorH. We prove it to be a Cantor set of measure zero, and obtain bounds on its Hausdorff dimension. It is an open question whether the pseudospectrum coincides with the spectrum ofH.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, CV4 7AL, Coventry, United Kingdom

    Martin Casdagli

  2. Center for Studies of Nonlinear Dynamics, La Jolla Institute, 3252 Holiday Court, Suite 208, 92037, La Jolla, CA, USA

    Martin Casdagli

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Communicated by B. Simon

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Casdagli, M. Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Commun.Math. Phys. 107, 295–318 (1986). https://doi.org/10.1007/BF01209396

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  • DOI: https://doi.org/10.1007/BF01209396

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