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

strange non chaotic attractors in quasiperiodically forced circle maps the occurrence of strange non chaotic attractors sna in quasiperiodically forced systems has attracted considerable interest over the last two decades in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos their existence was first described by millionscikov and later by vinograd and also herman for quasiperiodic rm sl 2 mathbb r cocycles and by grebogi et al and later keller for so called pinched skew products however except for these two particular classes there are still hardly any rigorous results on the topic despite a large number of numerical studies confirming the widespread existence of sna in quasiperiodically forced systems here we prove the existence of sna in quasiperiodically forced circle maps under rather general conditions which can be stated in terms of mathcal c 1 estimates as a consequence we obtain the existence of sna for parameter sets of positive measure in suitable parameter families these sna carry the unique physical measure of the system which determines the behaviour of lebesgue almost all initial conditions finally we show that the dynamics are minimal in the considered situations the results apply in particular to a forced version of the arnold circle map for this example we also describe how the first arnold tongue collapses and looses its regularity due to the presence of strange non chaotic attractors and a related unbounded mean motion property

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

finite monoids and the fine structure of nc 1 recently a new connection was discovered between the parallel complexity class nc 1 and the theory of finite automata in the work of barrington on bounded width branching programs there nonuniform nc 1 was characterized as those languages recognized by a certain nonuniform version of a dfa here we extend this characterization to show that the internal structures of nc 1 and the class of automata are closely related in particular using therien s classification of finite monoids we give new characterizations of the classes ac 0 depth k ac 0 and acc the last being the ac 0 closure of the mod q functions for all constant q we settle some of the open questions in 3 give a new proof that the dot depth hierarchy of algebraic automata theory is infinite 8 and offer a new framework for understanding the internal structure of nc 1

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

cantor and band spectra for periodic quantum graphs with magnetic fields we provide an exhaustive spectral analysis of the two dimensional periodic square graph lattice with a magnetic field we show that the spectrum consists of the dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant lyapunov function of a suitable kronig penney hamiltonian in particular between any two dirichlet eigenvalues the spectrum is a cantor set for an irrational flux and is absolutely continuous and has a band structure for a rational flux the dirichlet eigenvalues can be isolated or embedded subject to the choice of parameters conditions for both possibilities are given we show that generically there are infinitely many gaps in the spectrum and the bethe sommerfeld conjecture fails in this case

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

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

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

a lower bound on the lyapunov exponent for the generalized harper s model we obtain a lower bound for the lyapunov exponent of a family of discrete schrodinger operators hu n u n 1 u n 1 2a 1 cos 2 pi theta n alpha u n 2a 2 cos 4 pi theta n alpha u n that incorporates both a 1 and a 2 thus going beyond the herman s bound

nondeterministic space is closed under complementation in this paper we show that nondeterministic space s n is closed under complementation for s n greater than or equal to log n it immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by kuroda inform and control 7 1964 pp 207 233

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

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

almost every real quadratic map is either regular or stochastic in this paper we complete a program to study measurable dynamics in the real quadratic family our goal was to prove that almost any real quadratic map pc z t x2 c c c 2 1 4 has either an attracting cycle or an absolutely continuous invariant measure the final step completed here is to prove that the set of infinitely renormalizable parametric values c c 2 1 4 has zero lebesgue measure we derive this from a renormalization theorem which asserts uniform hyperbolicity of the full renormalization operator this theorem gives the most general real version of the feigenbaum coullet tresser universality simultanuously for all combinatorial types

what are lyapunov exponents and why are they interesting taking as inspiration the fields medal work of artur avila i d like to introduce you to lyapunov exponents my plan is to show how lyapunov exponents play a key role in three areas in which avila s results lie smooth ergodic theory billiards and translation surfaces and the spectral theory of 1 dimensional schrodinger operators but first what are lyapunov exponents let s begin by viewing them in one of their natural habitats the barycentric subdivision of a triangle is a collection of 6 smaller triangles obtained by joining the midpoints of the sides to opposite vertices here s what happens when you start with an equilateral triangle and repeatedly barycentrically subdivide

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

second phase transition line we study the phase transion line of the almost mathieu operator that separates arithmetic regions corresponding to singular continuous and a e pure point regimes and prove that both purely singular continuous and a e pure point spectrum occur for dense sets of frequencies

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

the xi function despite the fact that spectral and inverse spectral properties of one dimensional schrsdinger operators h d2 dx2 have been extensively studied for seventy five years there remain large areas where our knowledge is limited for example while the inverse theory for operators on l2 co co is well understood in case v is periodic 12 24 25 35 39 42 49 it is not understood in case limjx o v x co and h has discrete spectrum our goal here is to introduce a special function x a on r associated to h which we believe will be a valuable tool in the spectral and inverse spectral theory in a sense we will make precise it complements the weyl m functions m x a a main application of which we will make here concerns a generalization of the trace formula for schrsdinger operators to general v s recall the well known trace formula for periodic potentials let v x v x t 1 then by floquet theory see e g 10 37 44

spectral characteristics of the unitary critical almost mathieu operator we discuss spectral characteristics of a one dimensional quantum walk whose coins are distributed quasi periodically the unitary update rule of this quantum walk shares many spectral characteristics with the critical almost mathieu operator however it possesses a feature not present in the almost mathieu operator namely singularity of the associated cocycles this feature is however present in the so called extended harper s model we show that this operator has empty absolutely continuous spectrum and that the lyapunov exponent vanishes on the spectrum hence this model exhibits cantor spectrum of zero lebesgue measure for all irrational frequencies and arbitrary phase which in physics is known as hofstadter s butterfly in fact we will show something stronger namely that all spectral parameters in the spectrum are of critical type in the language of avila s global theory of analytic quasiperiodic cocycles we further prove that it has empty point spectrum for each irrational frequency and away from a frequency dependent set of phases having lebesgue measure zero the key ingredients in our proofs are an adaptation of avila s global theory to the present setting self duality via the fourier transform and a johnson type theorem for singular dynamically defined cmv matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting

examples of discontinuity of lyapunov exponent in smooth quasiperiodic cocycles we study the regularity of the lyapunov exponent for quasiperiodic cocycles t a where t is an irrational rotation x x 2 on s1 and a cl s1 sl 2 r 0 l for any fixed l 0 1 2 and any fixed of bounded type we construct dl cl s1 sl 2 r such that the lyapunov exponent is not continuous at dl in cl topology we also construct such examples in a smaller schrodinger class

on the hausdorff dimension of the spectrum of the thue morse hamiltonian we show that the hausdorff dimension of the spectrum of the thue morse hamiltonian has a common positive lower bound for all coupling

anderson localization for the almost mathieu equation a nonperturbative proof we prove that for any diophantine rotation angle and a e phase the almost mathieu operator h n n 1 n 1 cos 2 n n has pure point spectrum with exponentially decaying eigenfunctions for 15 we also prove the existence of some pure point spectrum for any 5 4

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

recurrence plot analysis of nonstationary data the understanding of curved patterns recurrence plots of the calls of the nomascus concolor western black crested gibbon and hylobates lar white handed gibbon show characteristic circular curved and hyperbolic patterns superimposed to the main temporal scale of the signal it is shown that these patterns are related to particular nonstationarities in the signal some of them can be reproduced by artificial signals like frequency modulated sinusoids and sinusoids with time divergent frequency these modulations are too faint to be resolved by conventional time frequency analysis with similar precision therefore recurrence plots act as a magnifying glass for the detection of multiple temporal scales in slightly modulated signals the detected phenomena in these acoustic signals can be explained in the biomechanical context by taking in account the role of the muscles controlling the vocal folds

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

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

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

optimizing of recurrence plots for noise reduction we propose a way to automatically detect the best neighborhood size for a local projective noise reduction filter where a typical problem is the proper identification of the noise level here we make use of concepts from the recurrence quantification analysis in order to adaptively tune the filter along the incoming time series we define an index to be computed via recurrence plots whose minimum gives a clear indication of the best size of the neighborhood in the embedding space comparison of the local projective noise reduction filter using this optimization scheme with the state of the art is also provided

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

estimating the shannon entropy recurrence plots versus symbolic dynamics recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics a few years later the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis among the different measures introduced a shannon entropy was found to be correlated with the inverse of the largest lyapunov exponent the discrepancy between this and the usual interpretation of a shannon entropy is solved here by using a new definition still based on the recurrence plots and it is verified that this new definition is correlated with the largest lyapunov exponent as expected from the pesin conjecture a comparison with a shannon entropy computed from symbolic dynamics is also provided

anderson localization for the almost mathieu equation ii point spectrum for lambda 2 we prove that for any 2 and a e the pure point spectrum of the almost mathieu operator h n n 1 n 1 cos 2 n n contains the essential closure hat sigma of the spectrum corresponding eigenfunctions decay exponentially the singular continuous component if it exists is concentrated on a set of zero measure which is nowhere dense in hat sigma

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

distinguishing quasiperiodic dynamics from chaos in short time series we propose a procedure to distinguish quasiperiodic from chaotic orbits in short time series which is based on the recurrence properties in phase space the histogram of the return times in a recurrence plot is introduced to disclose the recurrence property consisting of only three peaks imposed by slater s theorem noise effects on the statistics are studied our approach is demonstrated to be efficient in recognizing regular and chaotic trajectories of a hamiltonian system with mixed phase space

recurrence plot based measures of complexity and their application to heart rate variability data the knowledge of transitions between regular laminar or chaotic behaviors is essential to understand the underlying mechanisms behind complex systems while several linear approaches are often insufficient to describe such processes there are several nonlinear methods that however require rather long time observations to overcome these difficulties we propose measures of complexity based on vertical structures in recurrence plots and apply them to the logistic map as well as to heart rate variability data for the logistic map these measures enable us not only to detect transitions between chaotic and periodic states but also to identify laminar states i e chaos chaos transitions the traditional recurrence quantification analysis fails to detect the latter transitions applying our measures to the heart rate variability data we are able to detect and quantify the laminar phases before a life threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event our findings could be of importance for the therapy of malignant cardiac arrhythmias

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

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

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

arithmetic spectral transitions for the maryland model we give a precise description of spectra of the maryland model h u n un 1 un 1 tan n un for all values of parameters we introduce an arithmetically defined index and show that for sc h e e and pp h e e since ac h this gives a complete description of the spectral decomposition for all values of parameters and making it the first case of a family where arithmetic spectral transition is described without any parameter exclusion the set of eigenvalues can be explicitly identified for all parameters using the quantization condition we also establish for the first time for this or any other model a quantization condition for singular continuous spectrum an arithmetically defined measure zero set that supports singular continuous measures for all parameters 2017 wiley periodicals inc

denjoy constructions for fibered homeomorphisms of the torus we construct different types of quasiperiodically forced circle homeomorphisms with transitive but non minimal dynamics concerning the recent poincare like classification by jager and stark for this class of maps we demonstrate that transitive but non minimal behaviour can occur in each of the different cases this closes one of the last gaps in the topological classification actually we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets for example we show that in some of the examples we construct the unique minimal set is a cantor set and its intersection with each vertical fibre is uncountable and nowhere dense but may contain isolated points we also prove that minimal sets of the latter kind cannot occur when the dynamics are given by the projective action of a quasiperiodic sl 2 r cocycle more precisely we show that for a quasiperiodic sl 2 cocycle any minimal proper subset of the torus either is a union of finitely many continuous curves or contains at most two points on generic fibres

singular continuous spectrum for a class of almost periodic jacobi matrices we consider operators parametrized by a o on 2 with matrix ojy i o i fl o y with an cos 2ttcw 6 if ce is a liouville number and 2 we prove that for a e 0 the operator s spectral measures are all singular continuous we consider the operator on l2 z depending upon three parameters x a 0 1 x a 0 u n u n 1 u n 1 x cos 2iran 6 u n in this note we will sketch the proof of the following result whose detailed proof will appear elsewhere 3 theorem 1 fix a an irrational number obeying

the avalanche principle and negative curvature we use the geometric structure of the hyperbolic upper half plane to provide a new proof of the avalanche principle introduced by m goldstein and w schlag in the context of mathrm sl 2 mathbb r matrices this approach allows to interpret and extend this result to arbitrary text cat 1 metric spaces through the proof we deduce a polygonal schur theorem for these spaces

oracle branching programs and logspace versus p abstract we define the notion of an oracle branching program in order to investigate space bounded computation within this new framework we examine the p complete problem gen which consists of determining membership in a subalgebra of a general not necessarily associative binary algebra input as a multiplication table our work begins with the statement of a conceptually simple conjecture highlighting the combinatorics which underlie the relationship between logspace and p we show that natural subclasses of p can be expressed as natural subproblems for gen finally we prove optimal lower bounds on the size of branching programs for gen with certain natural oracles

singular spectrum of lebesgue measure zero for one dimensional quasicrystals the spectrum of one dimensional discrete schrodinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the lyapunov exponent if and only if the lyapunov exponent exists uniformly this is used to obtain the cantor spectrum of zero lebesgue measure for all aperiodic subshifts with uniform positive weights this covers in particular all aperiodic subshifts arising from primitive substitutions including new examples such as e g the rudin shapiro substitution our investigation is not based on trace maps instead it relies on an oseledec type theorem due to a furman and a uniform ergodic theorem due to the author

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

a time space tradeoff for sorting on a general sequential model of computation in a general sequential model of computation no restrictions are placed on the way in which the computation may proceed except that parallel operations are not allowed we show that in such an unrestricted environment text time cdot text space omega n 2 log n in order to sort n integers each in the range 1 n 2

global density of reducible quasi periodic cocycles on bf t 1 times roman su 2 we prove that given a in a set of total haar measure in t 1 r z the set of a c t 1 su 2 for which the skew product system a t 1 su 2 t 1 su 2 a y a y is reducible that is a b a 0 b 1 for some a 0 su 2 b c t 1 su 2 is dense for the c topology

multidimensional continued fractions dynamical renormalization and kam theory the disadvantage of traditional multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors following ideas of dani lagarias and kleinbock margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space sl d mathbb z backslash sl d mathbb r the space of lattices of covolume one that indeed yields best possible approximations to any irrational vector the algorithm is ideally suited for a number of dynamical applications that involve small divisor problems as an example we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension

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

operators with singular continuous spectrum iii almost periodic schrodinger operators we prove that one dimensional schrodinger operators with even almost periodic potential have no point spectrum for a dense g in the hull this implies purely singular continuous spectrum for the almost mathieu equation for coupling larger than 2 and a dense g in even if the frequency is an irrational with good diophantine properties

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

curved structures in recurrence plots the role of the sampling time we give a theoretical explanation of the formation of the curved macropatterns in the recurrence plots of sinusoidal signals with nonstationarity in the phase or in the frequency we show that the large time scales observed and the curved structures are the artificial product of the discretization of the signal recurrence plots are highly sensitive to the phase error introduced by the sampling and we show that this characteristic can be used to detect very small approximately 0 5 phase or frequency shifts of the carrier frequency

positive hausdorff dimensional spectrum for the critical almost mathieu operator we show that there exists a dense set of frequencies with positive hausdorff dimension for which the hausdorff dimension of the spectrum of the critical almost mathieu operator is positive

full measure reducibility and localization for quasiperiodic jacobi operators a topological criterion abstract we establish a topological criterion for connection between reducibility to constant rotations and dual localization for the general family of analytic quasiperiodic jacobi operators as a corollary we obtain the sharp arithmetic phase transition for the extended harper s model in the positive lyapunov exponent region

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

generalized recurrence plots for the analysis of images from spatially distributed systems abstract we propose a new method for the analysis of images showing patterns emerging from the evolution of spatially distributed systems the generalized recurrence plot grp and the generalized recurrence quantification analysis grqa are exploited for the investigation of such patterns we focus on snapshots of spatio temporal processes such as the formation of turing structures and traveling waves in the belousov zhabotinsky reaction satellite images of spatial chlorophyll distribution in seas and oceans similar to turbulent flows colonies of dyctiostelium discoideum fractals and noise the method is based on the grp and grqa and particularly on the measures determinism det and entropy ent providing a new criterion for the assessment and classification of images based on the simultaneous evaluation of their global and local structure the det ent diagram is introduced and compared with the classical image analysis entropy defined on the pixels values the method proposed provides appealing performances in the case of images showing complex spatial patterns

spectral transition line for the extended harper s model in the positive lyapunov exponent regime abstract we study the spectral transition line of the extended harper s model in the positive lyapunov exponent regime we show that both pure point spectrum and purely singular continuous spectrum occur for dense subsets of frequencies on the transition line

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

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

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

absolute continuity of the integrated density of states for the almost mathieu operator with non critical coupling we show that the integrated density of states of the almost mathieu operator is absolutely continuous if and only if the coupling is non critical we deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase settling the measure theoretical case of problem 6 of barry simon s list of schrodinger operator problems for the twenty first century

on the persistence of invariant curves for fibered holomorphic transformations we consider the problem of the persistence of invariant curves for analytical fibered holomorphic transformations we define a fibered rotation number associated to an invariant curve we show that an invariant curve with a prescribed fibered rotation number persists under small perturbations on the dynamics provided that the pair of rotation numbers verifies a brjuno type arithmetical condition nevertheless an extra complex parameter is added to the problem and the persistence becomes a one complex codimension property

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

the algebraic geometry of harper operators following an approach developed by gieseker kn rrer and trubowitz for discretized schr dinger operators we study the spectral theory of harper operators in dimensions 2 and 1 as a discretized model of magnetic laplacians from the point of view of algebraic geometry we describe the geometry of an associated family of bloch varieties and compute their density of states finally we also compute some spectral functions based on the density of states we discuss the difference between the cases with rational or irrational parameters for the two dimensional harper operator the compactification of the bloch variety is an ordinary variety in the rational case and an ind pro variety in the irrational case this gives rise at the algebro geometric level of bloch varieties to a phenomenon similar to the hofstadter butterfly in the spectral theory in dimension 2 the density of states can be expressed in terms of period integrals over fermi curves where the resulting elliptic integrals are independent of the parameters in dimension 1 for the almost mathieu operator with a similar argument we find the usual dependence of the spectral density on the parameter which gives rise to the well known hofstadter butterfly picture

a condition of boshernitzan and uniform convergence in the multiplicative ergodic theorem this paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts if such a subshift satisfies a certain condition originally introduced by boshernitzan every locally constant sl 2 r valued cocycle is uniform as a consequence the corresponding schrodinger operators exhibit cantor spectrum of lebesgue measure zero an investigation of boshernitzan s condition then shows that these results cover all earlier results of this type and moreover provide various new ones in particular boshernitzan s condition is shown to hold for almost all circle maps and almost all arnoux rauzy subshifts

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

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

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

a new method to estimate the kolmogorov entropy from recurrence plots its application to neuronal signals recurrence plots are a useful graphic tool for studying dynamical systems a new variable is proposed to quantify their deterministic structure it allows an easy computation of the k2 entropy since the distance between points needs to be calculated only once instead of every time for each new embedding dimension and it bypasses some of the limitations inherent to the classical grassberger and procaccia method it is valuable for biological data as illustrated here by tests with various dynamical models and its application to experimental signals recorded from neurons

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