arXiv subjects

Silvius Klein

Publications and source records attributed to Silvius Klein.

23 records · Page 2Linked to original sources

arXiv2015

Continuity of the Oseledets Decomposition

We consider an abstract space of measurable linear cocycles and we assume the availability in this space of some appropriate uniform large deviation type estimates. Under these hypotheses we establish the continuity of the Oseledets filtration and decomposition as functions of the cocycle. The same assumptions lead in [6] to a general continuity theorem for the Lyapunov exponents. This result and other technical estimates derived in [6], along with the inductive scheme based on the Avalanche Principle, are the main ingredients of the arguments in this paper. We also give a new proof of the classical Multiplicative Ergodic Theorem of V. Oseledets, using the Avalanche Principle (AP). This is a draft of a chapter in our forthcoming research monograph [6].

math.DS↗

arXiv2013

Continuity of the Lyapunov exponents for quasiperiodic cocycles

Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Holder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.

math.DS↗

arXiv2013

Positive Lyapunov exponents for higher dimensional quasiperiodic cocycles

We consider an m-dimensional analytic cocycle with underlying dynamics given by an irrational translation on the circle. Assuming that the d-dimensional upper left corner of the cocycle is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets-Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schroedinger operators.

math.DS↗

arXiv2013

Localization for quasiperiodic Schrodinger operators with multivariable Gevrey potential functions

We consider an integer lattice quasiperiodic Schrodinger operator. The underlying dynamics is either the skew-shift or the multi-frequency shift by a Diophantine frequency. We assume that the potential function belongs to a Gevrey class on the multi-dimensional torus. Moreover, we assume that the potential function satisfies a generic transversality condition, which we show to imply a Lojasiewicz type inequality for smooth functions of several variables. Under these assumptions and for large coupling constant, we prove that the associated Lyapunov exponent is positive for all energies, and continuous as a function of energy, with a certain modulus of continuity. Moreover, in the large coupling constant regime and for an asymptotically large frequency - phase set, we prove that the operator satisfies Anderson localization.

math-ph↗

arXiv2003

Anderson localization for the discrete one-dimensional quasi-periodic Schroedinger operator with potential defined by a Gevrey-class function

In this paper we consider the discrete one-dimensional Schroedinger operator with quasi-periodic potential v_n = λv (x + n ω). We assume that the frequency ωsatisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove - in the perturbative regime - that for large disorder λand for most frequencies ωthe operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity. We also prove a partial nonperturbative result assuming that the function v belongs to some particular Gevrey classes.

math-ph↗