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Cantor spectrum for the almost Mathieu operator
 Commun. Math. Phys
"... In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, (Hb,φx) n = xn+1 + xn−1 + bcos (2πnω + φ) xn on l 2 (Z) and its associated eigenvalue equation to deduce that for b = 0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of th ..."
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In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, (Hb,φx) n = xn+1 + xn−1 + bcos (2πnω + φ) xn on l 2 (Z) and its associated eigenvalue equation to deduce that for b = 0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the socalled “Ten Martini Problem ” for these values of b and ω. Moreover, we prove that for b  = 0 small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open. 1 Introduction. Main
Stability for quasiperiodically perturbed Hill’s equations
 Comm. Math. Phys
"... We consider a perturbed Hill’s equation of the form ¨ φ + (p0(t) + εp1(t))φ = 0, where p0 is real analytic and periodic, p1 is real analytic and quasiperiodic and ε ∈ is “small”. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potenti ..."
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Cited by 7 (4 self)
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We consider a perturbed Hill’s equation of the form ¨ φ + (p0(t) + εp1(t))φ = 0, where p0 is real analytic and periodic, p1 is real analytic and quasiperiodic and ε ∈ is “small”. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials p0 and p1 and the proper frequency of the unperturbed (ε = 0) Hill’s equation, but without making nondegeneracy assumptions on the perturbing potential p1, we prove that quasiperiodic solutions of the unperturbed equation can be continued into quasiperiodic solutions if ε lies in a Cantor set of relatively large measure in [−ε0, ε0] ⊂ , where ε0 is small enough. Our method is based on a resummation procedure of a formal Lindstedt series obtained as a solution of a generalized Riccati equation associated to Hill’s problem. 1
A Nonperturbative Eliasson’s Reducibility Theorem
"... This paper is concerned with discrete, onedimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasiperiodic Bloch wave if the potential is smaller than a certain const ..."
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This paper is concerned with discrete, onedimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasiperiodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated firstorder system, a quasiperiodic skewproduct, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schrödinger operators. Finally we prove that in our setting, Cantor spectrum implies the existence of a Gδset of energies whose Schrödinger cocycle is not reducible to constant coefficients. Keywords: Quasiperiodic Schrödinger operators, Harperlike equations, reducibility, Floquet theory, quasiperiodic cocycles, skewproduct,
KAM THEORY: THE LEGACY OF KOLMOGOROV’S 1954 PAPER
"... Abstract. KolmogorovArnoldMoser (or kam) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and kam theory establishes persistence results for such tori, which carry quasiperiodic moti ..."
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Cited by 3 (1 self)
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Abstract. KolmogorovArnoldMoser (or kam) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and kam theory establishes persistence results for such tori, which carry quasiperiodic motions. We sketch this theory, which begins with Kolmogorov’s pioneering work. 1.
NONGENERIC CUSPS
"... Abstract. We find the order of contact of the boundaries of the cusp for twoparameter families of vector fields on the real line or diffeomorphisms of the real line, for cusp bifurcations of codimensions 1 and 2. Moreover, we create a machinery that can be used for the same problem in higher codimen ..."
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Abstract. We find the order of contact of the boundaries of the cusp for twoparameter families of vector fields on the real line or diffeomorphisms of the real line, for cusp bifurcations of codimensions 1 and 2. Moreover, we create a machinery that can be used for the same problem in higher codimensions and perhaps for other, similar problems. 1.
Stability diagram for 4D linear periodic systems with applications to homographic
"... solutions ..."
QuasiPeriodicity in Dissipative and Conservative Systems
"... Abstract. In the classical perturbation theory of conditionally periodic motions series occur that, due to resonances, diverge on a dense set. In the complement of the resonances, small divisors make convergence problematic. Nonetheless, convergence of the series can be established in a nowhere dens ..."
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Abstract. In the classical perturbation theory of conditionally periodic motions series occur that, due to resonances, diverge on a dense set. In the complement of the resonances, small divisors make convergence problematic. Nonetheless, convergence of the series can be established in a nowhere dense set of positive Hausdorff measure in a suitable dimension. In the product of phase space and parameter space this gives rise to quasiperiodic invariant tori with Diophantine frequency vectors. This kind of result belongs to kam theory, as this developed from Kolmogorov’s 1954 paper [77]. We sketch elements of this development, both in the dissipative and the conservative setting.