Positive Lyapunov Exponents for Higher Dimensional Quasiperiodic Cocycles

Duarte, Pedro; Klein, Silvius

doi:10.1007/s00220-014-2082-1

Cited by 23 publications

(30 citation statements)

References 10 publications

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“…To that end, we use the method introduced in P. Duarte and S. Klein [7] for obtaining lower bounds on Lyapunov exponents of quasi-periodic cocycles.…”

Section: Introduction and Statementmentioning

confidence: 99%

Anderson localization for one-frequency quasi-periodic block Jacobi operators

Klein

2017

Journal of Functional Analysis

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We consider a one-frequency, quasi-periodic, block Jacobi operator, whose blocks are generic matrix-valued analytic functions. We establish Anderson localization for this type of operator under the assumption that the coupling constant is large enough but independent of the frequency. This generalizes a result of J. Bourgain and S. Jitomirskaya on localization for band lattice, quasi-periodic Schrödinger operators.

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“…To that end, we use the method introduced in P. Duarte and S. Klein [7] for obtaining lower bounds on Lyapunov exponents of quasi-periodic cocycles.…”

Section: Introduction and Statementmentioning

confidence: 99%

Anderson localization for one-frequency quasi-periodic block Jacobi operators

Klein

2017

Journal of Functional Analysis

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“…We note the fact that the lower bounds on the top LE obtained in [28,5,12] in the one-frequency (d = 1) setting are uniform in the frequency. Furthermore, Z. Zhang [29] obtained a sharp (and uniform) lower bound for the top LE of the Schrödinger cocycle (1.2), while very recently, R. Han and C. Marx [20] obtained a precise (and uniform) asymptotic formula for the top LE of the same cocycles.…”

Section: Introduction and Statementsmentioning

confidence: 84%

Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles

Duarte

Klein

2019

J. Eur. Math. Soc.

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An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We prove large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we establish continuity properties of the Lyapunov exponents regarded as functions on this space of cocycles. This result builds upon our previous work on this topic and its proof uses an abstract continuity theorem of the Lyapunov exponents which we derived in a recent monograph. The new feature of this paper is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible anywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.

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“…It is shown in [4] that N (f − µ) is upper semi-continuous and β (f − µ) is lower semi-continuous in µ, in particularN (f ) < ∞ andβ (f ) > 0. The quantitative version of Observation 2.1 forms our first key lemma:…”

Section: Key Lemmasmentioning

confidence: 99%

“…In particular, (1.4) rigorously establishes the heuristics that for sufficiently large λ, the potential in (1.1) should dominate the discrete Laplacian. We mention that simplified proofs of the Sorets-Spencer bound (1.4) were obtained in [2], [14], and [4]. The question of capturing the large coupling behavior of the LE has a rich history which influenced greatly the development of many aspects of the theory of quasi-periodic operators.…”

Section: Introductionmentioning

confidence: 99%