periodically forced double homoclinic loops to a dissipative saddle abstract in this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point we prove the existence of many complicated dynamical objects for a large class of non autonomous second order equations ranging from attractive quasi periodic torus to newhouse sinks and henon like attractors and to rank one attractors with srb measures and full stochastic behavior this theory enables us to apply rigorously many profound dynamics theories on non uniformly hyperbolic maps developed in the last forty years including the newhouse theory the theory of srb measures the theory of henon like attractors and the theory of rank one attractors to the analysis of the strange attractors in a periodically perturbed duffing equation

quantum ergodicity for graphs related to interval maps we prove quantum ergodicity for a family of graphs that are obtained from ergodic one dimensional maps of an interval using a procedure introduced by pakonski et al j phys a 34 9303 9317 2001 as observables we take the l 2 functions on the interval the proof is based on the periodic orbit expansion of a majorant of the quantum variance specifically given a one dimensional lebesgue measure preserving map of an interval we consider an increasingly refined sequence of partitions of the interval to this sequence we associate a sequence of graphs whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the perron frobenius operator corresponding to the map we show that except possibly for subsequences of density 0 the eigenstates of the quantum graphs equidistribute in the limit of large graphs

quantum variance and ergodicity for the baker s map we prove an egorov theorem or quantum classical correspondence for the quantised baker s map valid up to the ehrenfest time this yields a logarithmic upper bound for the decay of the quantum variance and as a corollary a quantum ergodic theorem for this map

semi continuity of the entropy of sinai probability measures for partially hyperbolic diffeomorphisms abstract we establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

nonlinear statistical coupling abstract by considering a nonlinear combination of the probabilities of a system a physical interpretation of tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed the escort probability is interpreted as the coupled probability with q 1 q defined as the degree of nonlinear coupling between the statistical states positive values of q have coupled statistical states a larger entropy metric and a maximum coupled entropy distribution of compact support coupled gaussians negative values of q have decoupled statistical states and for 2 q 0 a maximum coupled entropy distribution of heavy tail coupled gaussians the conjugate transformation between the heavy tail and compact support domains is shown to be q 2 q 2 q for coupled gaussian distributions this conjugate relationship has been used to extend the generalized fourier transform to the compact support domain and to define a scale invariant correlation structure with heavy tail limit distribution in the present paper we show that the conjugate is a mapping between the source of nonlinearity in non stationary stochastic processes and the nonlinear coupling which defines the coupled gaussian limit distribution the effects of additive and multiplicative noise are shown to be separable into the coupled variance and the coupling parameter q providing further evidence of the importance of the generalized moments

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

recent progress on the quantum unique ergodicity conjecture we report on some recent striking advances on the quantum unique ergodicity or que conjecture concerning the distribution of large frequency eigenfunctions of the laplacian on a negatively curved manifold the account falls naturally into two categories the first concerns the general conjecture where the tools are more or less limited to micro local analysis and the dynamics of the geodesic flow the second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue the recent account by zelditch zel4 covers some of the same material and his and our discussion complement each other nicely as he goes into more detail with the first category and we with the second this note is not meant to be a survey of these topics and the discussion is not chronological our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context recent progress on que 2 1 the general que conjecture in figure 1 the domains e s and b are an ellipse a stadium and a barnett billiard table respectively superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions states modes of the laplacian that is they are solutions to 4 j j j 0 in 0 dirichlet boundary conditions j dxdy 1 0 here 4 divgrad 2 x2 2 y2 1 2 3 are the eigenvalues and the eigenfunctions are normalized to have unit l norm the sequences are of 12 consecutive modes around the 5600 eigenvalue they are ordered from left to right and then down and the grayscale represents the probability density with zero white and larger values darker

scarred eigenstates for quantum cat maps of minimal periods in this paper we construct a sequence of eigenfunctions of the quantum arnold s cat map that in the semiclassical limit shows a strong scarring phenomenon on the periodic orbits of the dynamics more precisely those states have a semiclassical limit measure that is the sum of 1 2 the normalized lebesgue measure on the torus plus 1 2 the normalized dirac measure concentrated on any a priori given periodic orbit of the dynamics it is known the schnirelman theorem that most sequences of eigenfunctions equidistribute on the torus the sequences we construct therefore provide an example of an exception to this general rule our method of construction and proof exploits the existence of special values of h for which the quantum period of the map is relatively short and a sharp control on the evolution of coherent states up to this time scale we also provide a pointwise description of these states in phase space which uncovers their hyperbolic structure in the vicinity of the fixed points and yields more precise localization estimates

large supremum norms and small shannon entropy for hecke eigenfunctions of quantized cat maps this paper concerns the behavior of eigenfunctions of quantized cat maps and in particular their supremum norm we observe that for composite integer values of n the inverse of planck s constant some of the desymmetrized eigenfunctions have very small support and hence very large supremum norm we also prove an entropy estimate and show that our functions satisfy equality in this estimate in the case when n is a prime power with even exponent we calculate the supremum norm for a large proportion of all desymmetrized eigenfunctions and we find that for a given n there is essentially at most four different values these assume

a lower bound for periods of matrices for a nonsingular integer matrix a we study the growth of the order of a modulo n we say that a matrix is exceptional if it is diagonalizable and a power of the matrix has all eigenvalues equal to powers of a single rational integer or all eigenvalues are powers of a single unit in a real quadratic field for exceptional matrices it is easily seen that there are arbitrarily large values of n for which the order of a modulo n is logarithmically small in contrast we show that if the matrix is not exceptional then the order of a modulo n goes to infinity faster than any constant multiple of log n

nodal domain distributions for quantum maps the statistics of the nodal lines and nodal domains of the eigenfunctions of quantum billiards have recently been observed to be fingerprints of the chaoticity of the underlying classical motion by blum et al 2002 phys rev lett 88 114101 and by bogomolny and schmit 2002 phys rev lett 88 114102 these statistics were shown to be computable from the random wave model of the eigenfunctions we here study the analogous problem for chaotic maps whose phase space is the two torus we show that the distributions of the numbers of nodal points and nodal domains of the eigenvectors of the corresponding quantum maps can be computed straightforwardly and exactly using random matrix theory we compare the predictions with the results of numerical computations involving quantum perturbed cat maps

entropy of semiclassical measures in dimension 2 we study the high energy asymptotic properties of eigenfunctions of the laplacian in the case of a compact riemannian surfacem of anosov type to do this we look at families of distributions associated to them on the cotangent bundle t m and we derive entropic properties on their accumulation points in the high energy limit the so called semiclassical measures we show that the kolmogorov sinai entropy of a semiclassical measure for the geodesic ow gt is bounded from below by half of the ruelle upper bound i e hks g 1 2 s m d where is the upper lyapunov exponent at point

pathological foliations and removable zero exponents the ergodic theory of uniformly hyperbolic or axiom a diffeomorphisms has been studied extensively beginning with the pioneering work of anosov sinai ruelle and bowen an si ru bo while uniformly hyperbolic systems enjoy strong mixing properties they are not dense among c diffeomorphisms sm as using the concept of lyapunov exponents pesin introduced a weaker form of hyperbolicity which he termed nonuniform hyperbolicity nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms as a curious by product of our construction we prove that a pathological feature of central foliations the complete failure of absolute continuity can exist in a c open set of volume preserving diffeomorphisms we recall the definition of a nonuniformly hyperbolic diffeomorphism a real number is a lyapunov exponent of the diffeomorphism g m m if there exists a nonzero vector v tm such that

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

on herman s positive entropy conjecture abstract we show that any area preserving c r diffeomorphism of a two dimensional surface displaying an elliptic fixed point can be c r perturbed to one exhibiting a chaotic island whose metric entropy is positive for every 1 r this proves a conjecture of herman stating that the identity map of the disk can be c perturbed to a conservative diffeomorphism with positive metric entropy this implies also that the chirikov standard map for large and small parameter values can be c approximated by a conservative diffeomorphisms displaying a positive metric entropy a weak version of sinai s positive metric entropy conjecture finally this sheds light onto a herman s question on the density of c r conservative diffeomorphisms displaying a positive metric entropy we show the existence of a dense set formed by conservative diffeomorphisms which either are weakly stable so conjecturally uniformly hyperbolic or display a chaotic island of positive metric entropy

random products of standard maps we develop a general geometric method to establish the existence of positive lyapunov exponents for a class of skew products the technique is applied to show non uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps notably for skew products whose fiber dynamics is given by a continuum of parameters in the froeschle family these types of coupled systems appear as some induced maps in models for the study of arnold diffusion consequently we are able to present new examples of partially hyperbolic diffeomorphisms having rich high dimensional center dynamics the methods are also suitable for studying cocycles over shift spaces and do not demand any low dimensionality condition on the fiber

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

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

nested cantor sets we give sufficient conditions for two cantor sets of the line to be nested for a positive set of translation parameters this problem occurs in diophantine approximations it also occurs as a toy model of the parameter selection for non uniformly hyperbolic attractors of the plane for natural cantors sets we show that this condition is optimal

fluctuations correlations and the nonextensivity abstract examples of joint probability distributions are studied in terms of tsallis nonextensive statistics both for correlated and uncorrelated variables in particular it is explicitely shown how correlations in the system can make tsallis entropy additive and that the effective nonextensivity parameter q n decreases towards unity when the number of variables n increases we demonstrate that tsallis distribution of energies of particles in a system leads in a natural way to the negative binomial multiplicity distribution in this system

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

hecke theory and equidistribution for the quantization of linear maps of the torus 1 1 background one of the key issues of quantum chaos is the nature of the semiclassical limit of eigenstates of classically chaotic systems when the classical system is given by the geodesic flow on a compact riemannian manifoldm or rather on its cotangent bundle one can formulate the problem as follows the quantum hamiltonian is in suitable units represented by the positive laplacian on m to measure the distribution of its eigenstates we start with a smooth classical observable that is a smooth function on the unit cotangent bundle s m via some choice of quantization from symbols to pseudodifferential operators we form its quantization op f this is a zero order pseudodifferential operator with principal symbol f the expectation value of op f in the eigenstate is op f let j be a sequence of normalized eigenfunctions j j j 0 m j 2 1 the problem then is to understand the possible limits as j of the distributions f c s m op f j j 1 1

central limit theorem for anomalous scaling due to correlations we derive a central limit theorem for the probability distribution of the sum of many critically correlated random variables the theorem characterizes a variety of different processes sharing the same asymptotic form of anomalous scaling and is based on a correspondence with the levy gnedenko uncorrelated case in particular correlated anomalous diffusion is mapped onto levy diffusion under suitable assumptions the nonstandard multiplicative structure used for constructing the characteristic function of the total sum allows us to determine correlations of partial sums exclusively on the basis of the global anomalous scaling

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

on the distribution of matrix elements for the quantum cat map for many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed that is the matrix elements of smooth observables tend to the phase space average of the observable in this paper we study the fluctuations of the matrix elements for the desymmetrized quantum cat map we present a conjecture for the distribution of the normalized matrix elements namely that their distribution is that of a certain weighted sum of traces of independent matrices in su 2 this is in contrast to generic chaotic systems where the distribution is expected to be gaussian we compute the second and fourth moment of the normalized matrix elements and obtain agreement with our conjecture

quantisations of piecewise parabolic maps on the torus and their quantum limits for general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand the wignerfunctions of eigenstates converge weakly to invariant measures of the classical system the so called quantum limits and one would like to understand which invariant measures can occur that way thereby classifying the semiclassical behaviour of eigenfunctions

heteroclinic tangles in time periodic equations abstract in this paper we prove that when a heteroclinic loop is periodically perturbed three types of heteroclinic tangles are created and they compete in the space of where is a parameter representing the magnitude of the perturbations the three types are a transient heteroclinic tangles containing no gibbs measures b heteroclinic tangles dominated by sinks representing stable dynamical behavior and c heteroclinic tangles with strange attractors admitting srb measures representing chaos we also prove that as 0 the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections the theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical birkhoff melnikov smale method

arithmetical properties of strongly chaotic motions abstract the orbits of the generalized arnold sinai cat maps or hyperbolic automorphisms of the two dimensional torus typify purely chaotic anosov motion we transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers classify all periodic orbits and show how to determine their periods and initial conditions the methods are based on ideal theory in quadratic fields which is reviewed it is shown that the structure of orbits rests upon some basic arithmetical notions such as unique factorization into prime ideals

dissipative homoclinic loops of two dimensional maps and strange attractors with one direction of instability we prove that when subjected to periodic forcing of the form p t d h x y c sin t certain two dimensional vector fields with dissipative homoclinic loops generate strange attractors with sinai ruelle bowen measures for a set of forcing parameters of positive lebesgue measure the proof extends ideas of afraimovich and shilnikov and applies the recent theory of rank 1 maps developed by wang and young we prove a general theorem and then apply this theorem to an explicit model a forced duffing equation of the form d 2 q

ergodic properties of quantized toral automorphisms we study the ergodic properties for a class of quantized toral automorphisms namely the cat and kronecker maps the present work uses and extends the results of klimek and le niewski ann phys 244 173 198 1996 we show that quantized cat maps are strongly mixing while kronecker maps are ergodic and nonmixing we also study the structure of these quantum maps and show that they are effected by unitary endomorphisms of a suitable vector bundle over a torus this allows us to exhibit explicit relations between our toeplitz quantization and the semiclassical quantization of cat maps proposed by hannay and berry physica d 1 267 290 1980

asymptotically scale invariant occupancy of phase space makes the entropy sq extensive phase space can be constructed for n equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated if they are locally correlated we expect the boltzmann gibbs entropy s bg k i p i ln p i to be extensive i e s bg n n for n in particular if they are independent s bg is strictly additive i e s bg n ns bg 1 n however if the subsystems are globally correlated we expect for a vast class of systems the entropy s q k 1 i p q i q 1 with s 1 s bg for some special value of q 1 to be the one which is extensive i e s q n n for n another concept which is relevant is strict or asymptotic scale freedom or scale invariance defined as the situation for which all marginal probabilities of the n system coincide or asymptotically approach for n the joint probabilities of the n 1 system if each subsystem is a binary one scale freedom is guaranteed by what we hereafter refer to as the leibnitz rule i e the sum of two successive joint probabilities of the n system coincides or asymptotically approaches the corresponding joint probability of the n 1 system the kinds of interplay of these various concepts are illustrated in several examples one of them justifies the title of this paper we conjecture that these mechanisms are deeply related to the very frequent emergence in natural and artificial complex systems of scale free structures and to their connections with nonextensive statistical mechanics summarizing we have shown that for asymptotically scale invariant systems it is s q with q 1 and not s bg the entropy which matches standard clausius like prescriptions of classical thermodynamics

classical limit of the quantized hyperbolic toral automorphisms the canonical quantization of any hyperbolic symplectomorphisma of the 2 torus yields a periodic unitary operator on an dimenional hilbert space n 1 h we prove that this quantum system becomes ergodic and mixing at the classical limit n n prime which can be interchanged with the time average limit the recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the lebesgue measure the wigner functions of the eigenstates supported on the classical periodic orbits are indeed proved to become uniformly speread in phase space

markov processes hurst exponents and nonlinear diffusion equations with application to finance abstract we show by explicit closed form calculations that a hurst exponent h 1 2 does not necessarily imply long time correlations like those found in fractional brownian motion fbm we construct a large set of scaling solutions of fokker planck partial differential equations pdes where h 1 2 thus markov processes which by construction have no long time correlations can have h 1 2 if a markov process scales with hurst exponent h 1 2 then it simply means that the process has nonstationary increments for the scaling solutions we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient d x t is specified as an example we generate a class of student t like densities from the class of quadratic diffusion coefficients notably the tsallis density is one member of that large class the tsallis density is usually thought to result from a nonlinear diffusion equation but instead we explicitly show that it follows from a markov process generated by a linear fokker planck equation and therefore from a corresponding langevin equation having a tsallis density with h 1 2 therefore does not imply dynamics with correlated signals e g like those of fbm a short review of the requirements for fbm is given for clarity and we explain why the usual simple argument that h 1 2 implies correlations fails for markov processes with scaling solutions finally we discuss the question of scaling of the full green function g x t x t of the fokker planck pdes

on stochastic sea of the standard map consider a generic one parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism we show that for many parameters residual subset in an open set approaching the critical value the corresponding diffeomorphism has a transitive invariant set of full hausdorff dimension the set is a topological limit of hyperbolic sets and is accumulated by elliptic islands

bowen ruelle measures for certain piecewise hyperbolic maps we consider a class of piecewise c 2 lozi like maps and prove the existence of invariant measures with absolutely continuous conditional mea sures on unstable manifolds

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

eigenvalue spacings for quantized cat maps according to one of the basic conjectures in quantum chaos the eigenvalues of a quantized chaotic hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices we study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms cat maps our numerical experiments indicate that for generic choices of cat maps the unfolded consecutive spacing distribution in the irreducible components of the nth quantization given by the n dimensional weil representation approaches the goe gse law of random matrix theory for certain special arithmetic transformations related to the ramanujan graphs of lubotzky phillips and sarnak the experiments indicate that the unfolded consecutive spacing distribution follows poisson statistics we provide a sharp estimate in that direction

on the ergodicity of partially hyperbolic systems pugh and shub have conjectured that essential accessibility implies ergodicity for a c partially hyperbolic volume preserving diffeomorphism we prove this conjecture under a mild center bunching assumption which is satisfied in particular by all partially hyperbolic systems with 1 dimensional center bundle we also obtain ergodicity results for c 1ci partially hyperbolic systems

on quantum ergodicity for linear maps of the torus we prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus cat maps we show that there is a density one sequence of integers so that as n tends to infinity along this sequence all eigenfunctions of the quantum propagator at inverse planck constant n are uniformly distributed

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus we look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribu tion in the semiclassical limit we construct super scars that are stable under the arithmetic symmetries of the system and localize on invariant manifolds we show that these super scars exist only when there are isotropic rational subspaces invariant under the linear map in the case where there are no such scars we com pute the variance of the fluctuations of the matrix elements for the desymmetrized system and present a conjecture for their limiting distributions

scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms we exhibit scarring for the quantization of certain nonlinear ergodic maps on the torus we consider perturbations of hyperbolic toral automorphisms preserving certain co isotropic submanifolds the classical dynamics is ergodic hence in the semiclassical limit almost all quantum eigenstates converge to the volume measure of the torus nevertheless we show that for each of the invariant submanifolds there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

on the quantisation of arnold s cat characterises the quantisation ua of the classical map a in sl 2 z using the heisenberg group constructs the eigenstates for n perfect square where 2 pi n and shows that the fourier components of the wigner functions of a complete set of eigenstates go to zero for n p2 p prime p to infinity a hyperbolic

using the hadamard and related transforms for simplifying the spectrum of the quantum baker s map we rationalize the somewhat surprising efficacy of the hadamard transform in simplifying the eigenstates of the quantum baker s map a paradigmatic model of quantum chaos this allows us to construct closely related but new transforms that do significantly better thus nearly solving many states of the quantum baker s map these transforms which combine the standard fourier and hadamard transforms in an interesting manner are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit flip parity and possess bit reversal time reversal symmetry

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

the quantum perturbed cat map and symmetry we investigate a class of quantum symmetries of the perturbed cat map which exist only for a subset of possible values of planck s constant the effect of these symmetries is to change the spectral statistics along this positive density subset the symmetries are shown to be related to some simple classical symmetries of the map