Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators

Tao, Kai

doi:10.24033/bsmf.2675

Tao, Kai

Bulletin de la Société Mathématique de France, Volume 142 (2014) no. 4, pp. 635-671

Abstract

We consider the quasi-periodic Jacobi operator $H_{x, ω}$ in $l^{2} (ℤ)$ : $(H_{x, ω} φ) (n) = - b (x + (n + 1) ω) φ (n + 1) - b (x + n ω) φ (n - 1) + a (x + n ω) φ (n) = E φ (n)$ , $n \in ℤ$ , where $a (x), b (x)$ are analytic functions on $𝕋$ , $b$ is not identically zero, and $ω$ obeys some strong Diophantine condition. We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent $L (E)$ of the cocycle is positive for some $E = E_{0}$ , then there exist $ρ_{0} = ρ_{0} (a, b, ω, E_{0})$ , $β = β (a, b, ω)$ such that $| L (E) - L (E^{'}) | < | E - E^{'} |^{β}$ for any $E, E^{'} \in (E_{0} - ρ_{0}, E_{0} + ρ_{0})$ . If $L (E) > 0$ for all $E$ in some compact interval $I$ , then $L (E)$ is Hölder continuous on $I$ with Hölder exponent $β = β (a, b, ω, I)$ . In our derivation we follow the refined version of the Goldstein-Schlag method [3] developed by Bourgain and Jitomirskaya [2].

Published online: 2014-07-21

MR Zbl

DOI: 10.24033/bsmf.2675

@article{BSMF_2014__142_4_635_0,
     author = {Tao, Kai},
     title = {H\"older continuity of {Lyapunov} exponent for quasi-periodic {Jacobi} operators},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {635--671},
     year = {2014},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {142},
     number = {4},
     doi = {10.24033/bsmf.2675},
     mrnumber = {3306872},
     zbl = {1309.47033},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2675/}
}

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Tao, Kai. Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators. Bulletin de la Société Mathématique de France, Volume 142 (2014) no. 4, pp. 635-671. doi: 10.24033/bsmf.2675

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Cited by

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