holder continuity of the rotation number for quasi periodic co cycles in sl 2 mathbb r we prove two results on the rotation number of the skew product system omega a theta y in mathbb t d times mathbb r 2 mapsto theta omega a theta y in mathbb t d times mathbb r 2 where is diophantine and a theta in sl 2 mathbb r is homotopic to the identity on the one hand we prove that this function has the behavior of a frac 1 2 holder function on the other we show that the length of the gaps has a sub exponential estimate which depends on its label given by the gap labeling theorem we give also an estimate of the complement of the spectrum these results are obtained by studying the reducibility of the quasi periodic co cycle a

on the reducibility of linear quasi periodic systems with liouvillean basic frequencies and multiple eigenvalues abstract in this paper we consider the linear quasi periodic system x a p t x x r d where a is a d d constant matrix of elliptic type and has multiple eigenvalues p t is analytic quasi periodic with respect to t with basic frequencies 1 where is irrational and is a small perturbation parameter under suitable non resonant condition non degeneracy condition and 0 r where lim sup n ln q n 1 q n q n is the sequence of denominations of the best rational approximations for r q 0 r r r is the initial radius of analytic strip it is proved that for most sufficiently small this system can be reduced to a constant system x a x where a is a constant matrix close to a as some applications we apply our results to quasi periodic hill s equations three dimensional skew symmetric systems and n coupled schrodinger equations to study the lyapunov stability of the equilibrium and the existence of quasi periodic solutions

stability for quasi periodically perturbed hill s equations we consider a perturbed hill s equation of the form open image in new window p0 t p1 t 0 where p0 is real analytic and periodic p1 is real analytic and quasi periodic and 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 0 hill s equation but without making any assumptions on the perturbing potential p1 other than analyticity we prove that quasi periodic solutions of the unperturbed equation can be continued into quasi periodic solutions if lies in a cantor set of relatively large measure in open image in new window where 0 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

almost reducibility and non perturbative reducibility of quasi periodic linear systems in this paper we prove that a quasi periodic linear differential equation in sl 2 with two frequencies 1 is almost reducible provided that the coefficients are analytic and close to a constant in the case that is diophantine we get the non perturbative reducibility we also obtain the reducibility and the rotations reducibility for an arbitrary irrational under some assumption on the rotation number and give some applications for schrodinger operators our proof is a generalized kam type iteration adapted to all irrational

positive lyapunov exponents for continuous quasiperiodic schrodinger equations we prove that the continuous one dimensional schrodinger equation with an analytic quasi periodic potential has positive lyapunov exponents in the bottom of the spectrum for large couplings

ballistic transport in one dimensional quasi periodic continuous schrodinger equation abstract for the solution q t to the one dimensional continuous schrodinger equation i t q x t x 2 q x t v x q x t x r with r d satisfying a diophantine condition and v a real analytic function on t d we consider the growth rate of the diffusion norm q t d r x 2 q x t 2 d x 1 2 for any non zero initial condition q 0 h 1 r with q 0 d we prove that q t d grows linearly with t if v is sufficiently small

on topological genericity of the mode locking phenomenon we study circle homeomorphisms extensions over a strictly ergodic homeomorphism under a very mild restriction we show that the fibered rotation number is locally constant on an open and dense subset of all circle homeomorphisms extensions homotopic to the trivial extension in the complement of this set we find a dense subset in which every map is conjugate to a direct product our result provides a generalisation of avila bochi damanik s result on mathrm sl 2 mathbb r cocycles and jager wang zhou s result on quasi periodically forced maps to a broader setting

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

quasi periodic solutions for two level systems we consider the schrodinger equation for a class of two level atoms in a quasi periodic external field in the case in which the spacing 2 between the two unperturbed energy levels is small and we study the problem of finding quasi periodic solutions of a related generalized riccati equation we prove the existence of quasi periodic solutions of the latter equation for a cantor set of values of around the origin which is of positive lebesgue measure such solutions can be obtained from the formal power series by a suitable resummation procedure the set can be characterized by requesting infinitely many diophantine conditions of mel nikov type

holder continuity of the spectral measures for one dimensional schrodinger operator in exponential regime avila and jitomirskaya prove that the spectral measure v xf of quasi periodic schrodinger operator is 1 2 holder continuous with appropriate initial vector f if satisfies diophantine condition and is small in the present paper the conclusion is extended to that for all with the spectral measure v xf is 1 2 holder continuous with small if v is real analytic in a neighbor of x c where c is a large absolute constant in particular the spectral measure xf of almost mathieu operator is 1 2 holder continuous if e c with c a large absolute constant

purely absolutely continuous spectrum for almost mathieu operators using a recent result of sinai we prove that the almost mathieu operators acting onl2 ly n l 1 l cos n n have a purely absolutely continuous spectrum for almost all a provided that is a good irrational and is sufficiently small furthermore the generalized eigen functions are quasiperiodic

dispersive estimate for quasi periodic schrodinger operators on 1 d lattices abstract consider the one dimensional discrete schrodinger operator h h q n q n 1 q n 1 v n q n n z with r d diophantine and v a real analytic function on t d r 2 z d for v sufficiently small we prove the dispersive estimate for every l 1 z 1 e i t h l k 0 ln e 0 a ln ln 2 t 2 d t 1 3 l 1 t 1 t 2 with a and k 0 two absolute constants and e 0 an analytic norm of v the estimate holds for every t d

on reducibility for a class of n dimensional quasi periodic systems with a small parameter in this paper we consider the reducibility of a class of n dimensional real analytic quasi periodic systems with a small parameter x a q t x x r n we prove that if the basic frequencies of q and the eigenvalues of a satisfy some non resonance conditions then for most of the sufficiently small parameters in the sense of lebesgue measure the system is reducible without any non degeneracy assumption with respect to the parameter moreover under some assumptions we obtain a similar result for nonlinear quasi periodic systems

on perturbations of quasiperiodic schrodinger operators abstract using relative oscillation theory and the reducibility result of eliasson we study perturbations of quasiperiodic schrodinger operators in particular we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

lower dimensional invariant tori for quasiperiodically forced circle diffeomorphisms abstract for any analytic quasiperiodically forced circle diffeomorphisms p q e f where f is fixed and e is small we show that if is diophantine and the fibred rotation number of the diffeomorphism remains constant in a unilateral neighborhood of e 0 i e there is a unilateral phase locking at e 0 then the diffeomorphism has at least one analytic q invariant torus provided e is small enough

embedding of analytic quasi periodic cocycles into analytic quasi periodic linear systems and its applications in this paper we prove that any analytic quasi periodic cocycle close to constant is the poincare map of an analytic quasi periodic linear system close to constant which bridges both methods and results in quasi periodic linear systems and cocycles we also show that the almost reducibility of an analytic quasi periodic linear system is equivalent to the almost reducibility of its corresponding poincare cocycle by the local embedding theorem and the equivalence we transfer the recent local almost reducibility results of quasi periodic linear systems hou and you in invent math 190 209 260 2012 to quasi periodic cocycles and the global reducibility results of quasi periodic cocycles avila in almost reducibility and absolute continuity 2010 avila et al in geom funct anal 21 1001 1019 2011 to quasi periodic linear systems finally we give a positive answer to a question of avila et al geom funct anal 21 1001 1019 2011 and use it to study point spectrum of long range quasi periodic operator with liouvillean frequency the embedding also holds for some nonlinear systems

on the absolutely continuous spectrum of one dimensional quasi periodic schrodinger operators in the adiabatic limit cet article est consacre a l etude du spectre de certaines familles d equations de schrodinger quasi periodiques sur l axe reel lorsque les varietes iso energetiques adiabatiques sont etendues dans la direction des positions nous demontrons que dans un intervalle d energie ou ceci est le cas le spectre est dans sa majeure partie purement absolument continu et que les fonctions propres generalisees correspondantes sont des fonctions de bloch floquet

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

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

local rigidity of reducibility of analytic quasi periodic cocycles on gl n c abstract in this paper we consider the reducibility problem of an analytic d dimensional quasi periodic cocycle on g l n c we prove that for any diophantine vector and diagonal matrix c satisfying some arithmetic conditions if a cocycle a can be conjugated to the constant cocycle c by some 0 b with b l 2 t d g l n c and a is sufficiently close to some diagonal matrix it is analytically reducible moreover b is actually analytic if it is continuous

sharp holder continuity of the lyapunov exponent of finitely differentiable quasi periodic cocycles we show that if the base frequency is diophantine then the lyapunov exponent of a c k quasi periodic sl 2 mathbb r cocycle is 1 2 holder continuous in the almost reducible regime as a consequence we show that if the frequency is diophantine and the potential is small then the integrated density of states of the corresponding quasi periodic schrodinger operator is 1 2 holder continuous

uniqueness of solutions of the kdv hierarchy via dubrovin type flows abstract we consider the cauchy problem for the kdv hierarchy a family of integrable pdes with a lax pair representation involving one dimensional schrodinger operators under a local in time boundedness assumption on the solution for reflectionless initial data we prove that the solution stays reflectionless for almost periodic initial data with absolutely continuous spectrum we prove that under craig type conditions on the spectrum dirichlet data evolve according to a lipschitz dubrovin type flow so the solution is uniquely recovered by a trace formula this applies to algebro geometric finite gap solutions more notably we prove that it applies to small quasiperiodic initial data with analytic sampling functions and diophantine frequency this also gives a uniqueness result for the cauchy problem on the line for periodic initial data even in the absence of craig type conditions

persistence of lower dimensional tori of general types in hamiltonian systems this work is a generalization to a result of j you 1999 we study the persistence of lower dimensional tori of general type in hamiltonian systems of general normal forms by introducing a modified linear kam iterative scheme to deal with small divisors we shall prove a persistence result under a melnikov type of non resonance condition which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori

dynamics of the quasi periodic schrodinger cocycle at the lowest energy in the spectrum in this paper we consider the quasi periodic schrodinger cocycle over mathbb t d d 1 and in particular its projectivization in the regime of large coupling constants and diophantine frequencies we give an affirmative answer to a question posed by m herman 21 p 482 concerning the geometric structure of certain strange non chaotic attractors which appear in the projective dynamical system we also show that for some phase the lowest energy in the spectrum of the associated schrodinger operator is an eigenvalue with an exponentially decaying eigenfunction this generalizes 39 to the multi frequency case d 1

a new approach to nonlinear partial differential equations abstract the author s decomposition method for the solution of operator equations which may be nonlinear and or stochastic is generalized to multidimensional cases

absolutely continuous spectrum of multifrequency quasiperiodic schrodinger operator abstract in this paper we prove that for any d frequency analytic quasiperiodic schrodinger operator if the frequency is weak liouvillean and the potential is small enough then the corresponding operator has absolutely continuous spectrum moreover in the case d 2 we even establish the existence of ac spectrum under small potential and some super liouvillean frequency and this result is optimal due to a recent counterexample of avila and jitomirskaya 6

floquet solutions for the 1 dimensional quasi periodic schrodinger equation we show that the 1 dimensional schrodinger equation with a quasiperiodic potential which is analytic on its hull admits a floquet representation for almost every energye in the upper part of the spectrum we prove that the upper part of the spectrum is purely absolutely continuous and that for a generic potential it is a cantor set we also show that for a small potential these results extend to the whole spectrum

counterexamples to the kotani last conjecture for continuum schrodinger operators via character automorphic hardy spaces abstract the kotani last conjecture states that every ergodic operator in one space dimension with non empty absolutely continuous spectrum must have almost periodic coefficients this statement makes sense in a variety of settings for example discrete schrodinger operators jacobi matrices cmv matrices and continuum schrodinger operators in the main body of this paper we show how to construct counterexamples to the kotani last conjecture for continuum schrodinger operators by adapting the approach developed by volberg and yuditskii to construct counterexamples to the kotani last conjecture for jacobi matrices this approach relates the reflectionless operators associated with the prescribed spectrum to a family of character automorphic hardy spaces and then relates the shift action on the space of operators to the resulting action on the associated characters the key to our approach is an explicit correspondence between the space of continuum reflectionless schrodinger operators associated with a given set and the space of reflectionless jacobi matrices associated with a derived set once this correspondence is established we can rely to a large extent on the previous work of volberg and yuditskii to produce the resulting action on the space of characters we analyze this action and identify situations where we can observe absolute continuity without almost periodicity in the appendix we show how to implement this strategy and obtain analogous results for extended cmv matrices

almost periodicity in time of solutions of the kdv equation we study the cauchy problem for the kdv equation tu 6u xu xu 0 with almost periodic initial data u x 0 v x we consider initial data v for which the associated schrodinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify and show the existence uniqueness and almost periodicity in time of solutions this establishes a conjecture of percy deift for this class of initial data the result is shown to apply to all small analytic quasiperiodic initial data with diophantine frequency vector

quantum averaging for driven systems with resonances abstract we discuss the effects of resonances in driven quantum systems within the context of quantum averaging techniques in the floquet representation we consider in particular iterative methods of kam type and the extensions needed to take into account resonances the approach consists in separating the coupling terms into resonant and nonresonant components at a given scale of time and intensity the nonresonant part can be treated with perturbative techniques which we formulate in terms of kam type unitary transformations that are close to the identity these can be interpreted as averaging procedures with respect to the dynamics defined by effective uncoupled hamiltonians the resonant parts are treated with a different kind of unitary transfomations that are not close to the identity and are adapted to the structure of the resonances they can be interpreted as a renormalization of the uncoupled hamiltonian that yields an effective dressed hamiltonian around which a perturbation expansion can be developed the combination of these two ingredients provides a strongly improved approximation technique

reducibility for wave equations of finitely smooth potential with periodic boundary conditions abstract in the present paper the reducibility is derived for the wave equations with finitely smooth and time quasi periodic potential subject to periodic boundary conditions more exactly the linear wave equation u t t u x x m u e v 0 t u x x v t x u 0 x r 2 z can be reduced to a linear hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set where v is finitely smooth in t x quasi periodic in time t with diophantine frequency r n and v 0 is finitely smooth and quasi periodic in time t with diophantine frequency r n moreover it is proved that the corresponding wave operator possesses the property of pure point spectra and zero lyapunov exponent

resummation of perturbation series and reducibility for bryuno skew product flows we consider skew product systems on d l sl 2 for bryuno base flows close to constant coefficients depending on a parameter in any dimension d and we prove reducibility for a large measure set of values of the parameter the proof is based on a resummation procedure of the formal power series for the conjugation and uses techniques of renormalisation group in quantum field theory

sharp phase transitions for the almost mathieu operator it is known that the spectral type of the almost mathieu operator amo depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency we study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place which solves jitomirskaya s conjecture together with a previous work by avila this gives the sharp description of phase transitions for the amo for the a e phase

on product nonlinearities in stochastic differential equations the author s methods for solution of stochastic differential equations shown elsewhere to be applicable for nonlinear cases including polynomial exponential and trigonometric terms is shown to be applicable to product nonlinearities as well

the isospectral torus of quasi periodic schrodinger operators via periodic approximations we study the quasi periodic schrodinger operator begin aligned psi x v x psi x e psi x quad x in mathbb r end aligned in the regime of small v x sum m in mathbb z nu c m exp 2 pi i m omega x omega omega 1 dots omega nu in mathbb r nu c m le varepsilon exp kappa 0 m we show that the set of reflectionless potentials isospectral with v is homeomorphic to a torus moreover we prove that any reflectionless potential q isospectral with v has the form q x sum m in mathbb z nu d m exp 2 pi i m omega x with the same omega and with d m le sqrt 2 varepsilon exp frac kappa 0 2 m our derivation relies on the study of the approximation via hill operators with potentials tilde v x sum m in mathbb z nu c m exp 2 pi i m tilde omega x where tilde omega is a rational approximation of omega it turns out that the multi scale analysis method of damanik and goldstein publ math inst hautes etudes sci 119 217 401 2014 applies to these hill operators namely in damanik et al trans am math soc to appear arxiv 1409 2147 2016 we developed the multi scale analysis for the operators dual to the hill operators in question the main estimates obtained in damanik et al trans am math soc to appear arxiv 1409 2147 2016 allow us here to establish estimates for the gap lengths and the fourier coefficients in a form that is considerably stronger than the estimates known in the theory of hill operators with analytic potentials in the general setting due to these estimates the approximation procedure for the quasi periodic potentials is effective despite the fact that the rate of approximation omega tilde omega thicksim tilde t delta 0 delta 1 2 is slow on the scale of the period tilde t of the hill operator

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

linearization of quasiperiodically forced circle flows beyond brjuno condition we prove that an analytic quasiperiodically forced circle flow with a not super liouvillean base frequency and which is close enough to some constant rotation is c rotations reducible provided its fibered rotation number is diophantine with respect to the base frequency as a corollary we obtain that among such systems the linearizable ones and those displaying mode locking are locally dense for the c topology

phase transition and semi global reducibility for 1d continuous schodinger operators with large analytic quasi periodic potentials of two frequencies one knows that the spectral measure is singular at the bottom of the spectrum and purely absolutely continuous in the upper part of the spectrum so there is a phase transition when energy increases in this paper we obtain the exact power law for the phase transition in energy by the semi global reducibility theory of analytic quasi periodic linear systems

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

anderson transitions for a family of almost periodic schrodinger equations in the adiabatic case this work is devoted to the study of a family of almost periodic one dimensional schrodinger equations using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit we prove the existence of an asymptotically sharp anderson transition in the low energy region more explicitly we prove the existence of energy intervals containing only singular spectrum and of other energy intervals containing absolutely continuous spectrum the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions the analysis may be viewed as utilizing a complex wkb method for adiabatic perturbations of periodic schrodinger equations the transition energies are interpreted in terms of phase space tunneling

on the reducibility of a class of finitely differentiable quasi periodic linear systems abstract in this paper we consider the following system x a e q t x where a is a constant matrix with different eigenvalues and q t is quasi periodic with frequencies 1 2 r moreover q q t q t has continuous partial derivatives b q j b for j 1 2 r where b 9 4 r 1 z and the moduli of continuity of b q j b satisfy a condition of finiteness condition on an integral which is more general than a holder condition under suitable hypothesis of non resonance conditions and non degeneracy conditions we prove that for most sufficiently small e the system can be reducible to a constant coefficient differentiable equation by means of a quasi periodic homeomorphism

spectral theory of extended harper s model and a question by erdős and szekeres the extended harper s model proposed by d j thouless in 1983 generalizes the famous almost mathieu operator allowing for a wider range of lattice geometries parametrized by three coupling parameters by permitting 2d electrons to hop to both nearest and next nearest neighboring nnn lattice sites while still exhibiting its characteristic symmetry aubry andre duality previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space one of which is characterized by the positivity of the lyapunov exponent in this paper we complete the picture with a description of the spectral measures over the entire remaining self dual region for all irrational values of the frequency parameter the magnetic flux in the model most notably we prove that in the entire interior of this regime the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the nnn interaction becomes symmetric in physics literature extensive numerical analysis had indicated such spectral collapse however so far not even a heuristic argument for this phenomenon could be provided on the other hand in the remaining part of the self dual region the spectral measures are singular continuous irrespective of such symmetry the analysis requires some rather delicate number theoretic estimates which ultimately depend on the solution of a problem posed by erd s and szekeres on the product prod k 1 n 1 z a k publ de l institut mathematique paris 1950

genericity of mode locking for quasiperiodically forced circle maps abstract we show that a generic quasiperiodically forced circle homeomorphism is mode locked the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system as a consequence this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition the graph of the rotation number as a function of the parameter is a devil s staircase

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

inversion of nonlinear stochastic operators abstract the operator theoretic method adomian and malakian j math anal appl 76 1 1980 183 201 recently extended adomian s solutions of nonlinear stochastic differential equations g adomian stochastic systems analysis in applied stochastic processes nonlinear stochastic differential equations j math anal appl 55 1 1976 441 452 on the modeling and analysis of nonlinear stochastic systems in proceeding international conf on mathematical modeling vol 1 pp 29 40 to provide an efficient computational procedure for differential equations containing polynomial exponential and trigonometric nonlinear terms n y the procedure depends on the calculation of certain quantities an and bn this paper generalizes the calculation of the an and bn to much wider classes of nonlinearities of the form n y y essentially the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance this procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities an and bn for any order n in a convenient form this paper also demonstrates the significance of the iterative series decomposition proposed by adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators since both the nonlinear and stochastic behavior is quite general the results are extremely significant for applications processes need not for example be limited to either gaussian processes white noise or small fluctuations

holder continuity of absolutely continuous spectral measure for multi frequency schrodinger operators abstract we establish sharp results on the modulus of continuity of the distribution of the spectral measure for multi frequency schrodinger operators with diphantine frequencies and small analytic potentials

a normal form theorem for brjuno skew systems through renormalization abstract we develop a dynamical renormalization method for the problem of local reducibility of analytic linear quasi periodic skew product flows on t 2 sl 2 r this approach is based on the continued fraction expansion of the linear base flow and deals with small divisors by turning them into large divisors we use this method to give a new proof of a normal form theorem for a brjuno base flow

a learning approach to the two person decentralized team problem with incomplete information a learning approach to the two person decentralized team problem with incomplete information and with a 2x2 payoff matrix is considered it is shown that if the payoff matrix is unimodal there exists a proper choice of parameters of the learning algorithm that will ensure asymptotically an expected payoff as close to maximum payoff as desired it is shown that a team problem with multimodal payoff matrix gives rise to interesting class of open problems

on the spectrum of odd order self adjoint ordinary differential operators on the real line with quasi periodic coefficients in this paper we prove that an odd order self adjoint differential operator on the real line with quasi periodic coefficients has only absolute continuous simple spectrum for large enough energies ordinary differential operators with quasi periodic coefficients have been studied a lot during the last 20 years these operators arise naturally from physical considerations people have mainly been interested in the one dimensional schro dinger operator with a quasi periodic potential see be te din sin eli ji si mo mo po su and so sp such hamiltonians come up naturally in solid state physics see pa fi in this paper we study ordinary differential operators of higher order with quasi periodic coefficients such operators also come up in physics for example in the study of boussinesq s equation article no de963225