References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt145",{id:"formSmash:upper:j_idt145",widgetVar:"widget_formSmash_upper_j_idt145",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt146_j_idt148",{id:"formSmash:upper:j_idt146:j_idt148",widgetVar:"widget_formSmash_upper_j_idt146_j_idt148",target:"formSmash:upper:j_idt146:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

The Dynamics of a Class of Quasi-Periodic Schrödinger CocyclesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2015 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 16, no 4, 961-1031 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2015. Vol. 16, no 4, 961-1031 p.
##### National Category

Materials Engineering
##### Identifiers

URN: urn:nbn:se:kth:diva-163944DOI: 10.1007/s00023-014-0330-8ISI: 000350669300002ScopusID: 2-s2.0-84924176441OAI: oai:DiVA.org:kth-163944DiVA: diva2:810440
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt375",{id:"formSmash:j_idt375",widgetVar:"widget_formSmash_j_idt375",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt381",{id:"formSmash:j_idt381",widgetVar:"widget_formSmash_j_idt381",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt387",{id:"formSmash:j_idt387",widgetVar:"widget_formSmash_j_idt387",multiple:true});
##### Funder

Swedish Research Council
##### Note

Let f : T -> R be a Morse function of class C-2 with exactly two critical points, let omega is an element of T be Diopharitine, and let lambda > 0 be sufficiently large (depending on f and omega). For any value of the parameter E is an element of R, we make a careful analysis of the dynamics of the skew-product map Phi(E)(theta, r) = (theta + omega, lambda f(theta) - E - 1/r), acting on the "torus" T x (R) over cap. Here, (R) over cap denotes the projective space R boolean OR {infinity}. The map Phi(E) is intimately related to the quasi-periodic Schrodinger cocycle (omega, A(E)) : T x R-2 -> T x R-2, (theta, x) -> (theta + omega, A(E)(theta) . x), where A(E) : T -> SL(2, R) is given by A(E)(theta) = ((0)(-1) 1(lambda f(theta) - E)), E is an element of R. More precisely, (omega, A(E)) naturally acts on the space T x (R) over cap, and Phi(E) is the map thus obtained. The cocycle (omega, A(E)) arises when investigating the eigenvalue equation H(theta)u = Eu, where H-theta is the quasi-periodic Schrodinger operator (H(theta)u)(n) = -(u(n+1) + u(n-1)) + lambda f (theta + (n - 1)omega)u(n), (1) The (maximal) Lyapunov exponent of the Schrodinger cocycle (omega, A(E)) is greater than or similar to log lambda, uniformly in E is an element of R. This implies that the map PE has exactly two ergodic probability measures for all E is an element of R; (2) If E is on the edge of an open gap in the spectrum sigma(H), then there exist a phase 0 is an element of T and a vector u is an element of l(2)(Z), exponentially decaying at +/-infinity, such that H(theta)u = Eu;acting on the space l(2) (Z). It is well known that the spectrum of H-theta, sigma(H), is independent of the phase theta is an element of T. Under our assumptions on f, omega and lambda, Sinai (in J Stat Phys 46(5-6):861-909, 1987) has shown that sigma(H) is a Cantor set, and the operator H-theta has a pure-point spectrum, with exponentially decaying eig,enfunctions, for a.e. theta is an element of T The analysis of Phi(E) allows us to derive three main results: (3) The map Phi(E) is minimal iff E E is an element of sigma(H)\ {edges of open gaps}. In particular, Phi(E) is minimal for all E is an element of R for which the fibered rotation number alpha(E) associated with (omega, A(E)) is irrational with respect to omega.

QC 20150507

Available from: 2015-05-07 Created: 2015-04-13 Last updated: 2015-05-07Bibliographically approvedReferences$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1080",{id:"formSmash:lower:j_idt1080",widgetVar:"widget_formSmash_lower_j_idt1080",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1081_j_idt1083",{id:"formSmash:lower:j_idt1081:j_idt1083",widgetVar:"widget_formSmash_lower_j_idt1081_j_idt1083",target:"formSmash:lower:j_idt1081:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});