Cantor Spectrum for the Almost Mathieu Operator

By Joaquim Puig. 2004

156 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/86507cfb1e1016a01d366a52c7a634a2516d816a

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Almost localization and almost reducibility (2008, background): “i Problem enjoyed a signiﬁcant attention since its ﬁrst formulation in the early 80s. An aﬃrmative answer was obtained for Liouville α[CEY], as well as for a set of (λ,α) of positive Lebesgue measure [P1]. Here we are able to deal with almost every (λ,α). Let DC(κ,τ) be the set of α∈ Rsuch that α− p”
Lyapunov exponents for random perturbations of some area-preserving maps including the standard map (2017, methodology): “We mention as well the formula of Herman [Her83], [Kni92] and the re-lated work [AB02], which use subharmonicity to estimate Lyapunov exponents, and the substantial body of work on 1D Schr¨odinger operators (e.g., [Kot84], [Bou13], [Pui04], [AJ09]).”
The Ten Martini Problem (2009, background): “Especially important for us are [CEY], [J], [ P ], whose methods we improve, but several other ingredients are needed (such as Kotani Theory [Sim2], the recent estimates on Lyapunov exponents of [BJ]).”
Schrödinger operators with dynamically defined potentials (2014, methodology): “This era was ushered in by Puig’s proof that localization implies Cantor spectrum for the almost Mathieu operator via Aubry duality and reducibility [182], and by the proof of zero-measure spectrum for the critical almost Mathieu operator with recurrent Diophantine frequencies by Avila and Krikorian”
KAM theory: The legacy of Kolmogorov’s 1954 paper (2004, background): “[29]), as well as for quasi-periodic potentials in one dimension, describing electrons in quasicrystals; see e.g. [27], [19], [ 60 ].”