the set of smooth quasi periodic schrodinger cocycles with positive lyapunov exponent is not open one knows that the set of quasi periodic schrodinger cocycles with positive lyapunov exponent is open and dense in analytic topology in this paper we construct cocycles with positive lyapunov exponent which can be arbitrarily approximated by ones with zero lyapunov exponent in the space of mathcal c l 1 le l le infty smooth quasi periodic cocycles which shows that the set of quasi periodic schrodinger cocycles with positive lyapunov exponent is not open in smooth topology

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

global theory of one frequency schrodinger operators we study schrodinger operators with a one frequency analytic potential focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials from the dynamical point of view the transition signals the emergence of non uniform hyperbolicity so the dependence of the lyapunov exponent with respect to parameters plays a central role in the analysis though often ill behaved by conventional measures we show that the lyapunov exponent is in fact remarkably regular in a stratified sense which we define the irregularity comes from the matching of nice analytic or smooth functions along sets with complicated geometry this result allows us to establish that the critical set for the transition lies within countably many codimension one subvarieties of the infinite dimensional parameter space a more refined renormalization based analysis shows that the critical set is rather thin within those subvarieties and allows us to conclude that a typical potential has no critical energies such acritical potentials also form an open set and have several interesting properties only finitely many phase transitions may happen but never at any specific point in the spectrum and the lyapunov exponent is minorated in the region of the spectrum where it is positive on the other hand we do show that the number of phase transitions can be arbitrarily large

the lyapunov exponents of generic volume preserving and symplectic maps we show that the integrated lyapunov exponents of c1 volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding oseledets splitting is trivial all lyapunov exponents are equal to zero or else dominated uniform hyperbolicity in the projective bundle almost everywhere we deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold almost every orbit either is projectively hyperbolic or has all lyapunov exponents equal to zero similarly for a residual subset of all c1 symplectic diffeomorphisms on any compact manifold either the diffeomorphism is anosov or almost every point has zero as a lyapunov exponent with multiplicity at least 2 finally given any set s gl d satisfying an accessibility condition for a residual subset of all continuous s valued cocycles over any measure preserving homeomorphism of a compact space the oseledets splitting is either dominated or trivial the condition on s is satisfied for most common matrix groups and also for matrices that arise from discrete schr nodinger operators

loi des grands nombres et perturbations pour des produits reductibles de matrices aleatoires independantes soit x x n n 1 une suite de matrices aleatoires reelles d d independantes de meme loi on etudie le comportement asymptotique de la suite 1 n log x n x 1 x n

anderson localization for the discrete one dimensional quasi periodic schrodinger operator with potential defined by a gevrey class function in this paper we consider the discrete one dimensional schrodinger operator with quasi periodic potential vn v x n we assume that the frequency satisfies 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 and for most frequencies the 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

characterization of half radial matrices abstract numerical radius r a is the radius of the smallest ball with the center at zero containing the field of values of a given square matrix a it is well known that r a a 2 r a where is the matrix 2 norm matrices attaining the lower bound are called radial and have been analyzed thoroughly this is not the case for matrices attaining the upper bound where only partial results are available in this paper we consider matrices satisfying r a a 2 and call them half radial we summarize the existing results and formulate new ones in particular we investigate their singular value decomposition and algebraic structure and provide other necessary and sufficient conditions for a matrix to be half radial based on that we study the extreme case of the attainable constant 2 in crouzeix s conjecture the presented results support the conjecture of greenbaum and overton that the crabb choi crouzeix matrix always plays an important role in this extreme case

analytic quasi perodic cocycles with singularities and the lyapunov exponent of extended harper s model we show how to extend and with what limitations avila s global theory of analytic sl 2 c cocycles to families of cocycles with singularities this allows us to develop a strategy to determine the lyapunov exponent for the extended harper s model for all values of parameters and all irrational frequencies in particular this includes the self dual regime for which even heuristic results did not previously exist in physics literature the extension of avila s global theory is also shown to imply continuous behavior of the le on the space of analytic m 2 mathbb c cocycles this includes rational approximation of the frequency which so far has not been available

continuity of the measure of the spectrum for quasiperiodic schrodinger operators with rough potentials we study discrete quasiperiodic schrodinger operators on ell 2 mathbb z with potentials defined by holder functions we prove a general statement that for 1 2 and under the condition of positive lyapunov exponents measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants an important ingredient in our analysis is a general result on uniformity of convergence from above in the subadditive ergodic theorem for strictly ergodic cocycles

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

positive lyapunov exponents for schrodinger operators with quasi periodic potentials we present a new simple way to estimate the rate of exponential growth lyapunov exponent of solutions of the finite difference schrodinger equation h e psi n mathop limits def psi n 1 psi n 1 lambda f alpha n theta psi n heref is a non constant real analytic function of period 1 and is irrational for large we prove that the lyapunov exponent is positive for every energye in the spectrum ofh and a e in particular the absolutely continuous spectrum ofh is empty in the continuum we study the quasi periodic operator onl2 r h frac d 2 dx 2 k 2 cos x cos alpha x theta for largek and show that for wide intervals of low energies the lyapunov exponent is positive the main idea which originated from m herman s subharmonic argument 11 is to deform the phase to the complex plane this enables us to avoid small denominator problems by moving them off the axis making estimates much easier to perform we recover the information for real using an elementary extension of jensen s formula subharmonicity

uniform positivity and continuity of lyapunov exponents for a class of c 2 quasiperiodic schrodinger cocycles abstract we show that for a class of c 2 quasiperiodic potentials and for any fixed diophantine frequency the lyapunov exponent of the corresponding schrodinger cocycles as a function of energies are uniformly positive and weakly holder continuous as a corollary we obtain that the corresponding integrated density of states is weakly holder continuous as well our approach is of purely dynamical systems which depends on a detailed analysis of asymptotic stable and unstable directions we also apply it to more general sl 2 r cocycles which in turn can be applied to get uniform positivity and continuity of lyapunov exponents around unique nondegenerate extremal points of any smooth potential and to a certain class of c 2 szeg cocycles

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

continuity of the lyapunov exponent for quasiperiodic operators with analytic potential we study regularity properties of the lyapunov exponent l of one frequency quasiperiodic operators with analytic potential under no assumptions on the diophantine class of the frequency we prove joint continuity of l in frequency and energy at every irrational frequency

almost all cocycles over any hyperbolic system have nonvanishing lyapunov exponents we prove that for any s 0 the majority of c s linear cocycles over any hyperbolic uniformly or not ergodic transformation exhibit some nonzero lyapunov exponent this is true for an open dense subset of cocycles and actually vanishing lyapunov exponents correspond to codimension 1 this open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation

the ten martini problem we prove the conjecture known as the ten martini problem after kac and simon that the spectrum of the almost mathieu operator is a cantor set for all nonzero values of the coupling and all irrational frequencies

extremal lyapunov exponents an invariance principle and applications we propose a new approach to analyzing dynamical systems that combine hyperbolic and non hyperbolic center behavior e g partially hyperbolic diffeomorphisms a number of applications illustrate its power