entropy of quantum limits in this paper we show that any measure arising as a weak limit of microlocal lifts of eigenfunctions of the laplacian on certain arithmetic manifolds have dimension at least 11 9 and in particular all ergodic components of this measure with respect to the geodesic flow have positive entropy

entropic bounds on semiclassical measures for quantized one dimensional maps quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic hamiltonian systems are equidistributed in phase space this however does not prohibit existence of exceptional sequences which might converge to different non liouville classical invariant measures it has been recently shown by n anantharaman and s nonnenmacher in 20 21 with h koch that for anosov geodesic flows the metric entropy of any semiclassical measure must satisfy a certain bound this remarkable result seems to be optimal for manifolds of constant negative curvature but not in the general case where it might become even trivial if the negative curvature of the riemannian manifold varies a lot it has been conjectured by the same authors that in fact a stronger bound valid in the general case should hold

the eigenvector moment flow and local quantum unique ergodicity we prove that the distribution of eigenvectors of generalized wigner matrices is universal both in the bulk and at the edge this includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries the proof relies on analyzing the eigenvector flow under the dyson brownian motion the key new ideas are 1 the introduction of the eigenvector moment flow a multi particle random walk in a random environment 2 an effective estimate on the regularity of this flow based on maximum principle and 3 optimal finite speed of propagation holds for the eigenvector moment flow with very high probability

concentration of symmetric eigenfunctions abstract in this article we examine the concentration and oscillation effects developed by high frequency eigenfunctions of the laplace operator in a compact riemannian manifold more precisely we are interested in the structure of the possible invariant semiclassical measures obtained as limits of wigner measures corresponding to eigenfunctions these measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions we present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries as an application of these results we give a simple proof of the fact that in a manifold of constant positive sectional curvature every measure which is invariant by the geodesic flow is an invariant semiclassical measure

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

weighted norm inequalities for the continuous square function we prove new weighted norm inequalities for real variable analogues of the lusin area function we apply our results to obtain new i weighted norm inequalities for singular integral operators ii weighted sobolev inequalities iii eigenvalue estimates for degenerate schrodinger operators

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

joint quasimodes positive entropy and quantum unique ergodicity we study joint quasimodes of the laplacian and one hecke operator on compact congruence surfaces and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding semiclassical measures together with the measure classification result of lindenstrauss ann math 2 163 1 165 219 2006 this implies quantum unique ergodicity for such functions our result is optimal with respect to the dimension of the space from which the quasi mode is constructed we also study equidistribution for sequences of joint quasimodes of the two partial laplacians on compact irreducible quotients of mathbb h times mathbb h

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

quantum decay rates in chaotic scattering we study quantum scattering on manifolds equivalent to the euclidean space near infinity in the semiclassical regime we assume that the corresponding classical flow admits a non trivial trapped set and that the dynamics on this set is of axiom a type uniformly hyperbolic we are interested in the distribution of quantum resonances near the real axis in two dimensions we prove that if the trapped set is sufficiently thin then there exists a gap between the resonances and the real axis that is quantum decay rates are bounded from below in higher dimension the condition for this gap is given in terms of a certain topological pressure associated with the classical flow under the same assumption we also prove a resolvent estimate with a logarithmic loss compared to non trapping situations

dynamical entropy of c algebras and von neumann algebras the definition of the dynamical entropy is extended for automorphism groups ofc algebras as an example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

ergodic billiards that are not quantum unique ergodic partially rectangular domains are compact two dimensional riemannian manifolds x either closed or with boundary that contain a flat rectangle or cylinder in this paper we are interested in partially rectangular domains with ergodic billiard flow examples are the bunimovich stadium the sinai billiard or donnelly surfaces we consider a one parameter family x t of such domains parametrized by the aspect ratio t of their rectangular part there is convincing theoretical and numerical evidence that the laplacian on x t with dirichlet or neumann boundary conditions is not quantum unique ergodic que we prove that this is true for all t in 1 2 excluding possibly a set of lebesgue measure zero this yields the first examples of ergodic billiard systems proven to be non que

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

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

pseudo differential analysis on hyperbolic surfaces soit d le disque unite psu 1 1 un sous groupe discret et x d une surface hyperbolique on considere le comportement asymptotique des valeurs propres et des fonctions propres de sur x on presente un calcul d operateur pseudo differentiel pour obtenir ce resultat

near sharp strichartz estimates with loss in the presence of degenerate hyperbolic trapping we consider an n dimensional spherically symmetric asymptotically euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic by separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond ehrenfest time we show that solutions to the linear schrodinger equation with initial conditions localized on a spherical harmonic satisfy strichartz estimates with a loss depending only on the dimension n and independent of the degeneracy the strichartz estimates are sharp up to an arbitrary 0 loss this is in contrast to christianson and wunsch amer j math 2013 where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set independent of the dimension

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

entropy and the localization of eigenfunctions we study the large eigenvalue limit for the eigenfunctions of the laplacian on a compact manifold of negative curvature in fact we only assume that the geodesic flow has the anosov property in the semi classical limit we prove that the wigner measures associated to eigenfunctions have positive metric entropy in particular they cannot concentrate entirely on closed geodesies 1 introduction statement of results we consider a compact riemannian manifold m of dimension d 2 and assume that the geodesic flow ter acting on the unit tangent bundle of m has a chaotic behaviour this refers to the asymptotic properties of the flow when time t tends to infinity ergodicity mixing hyperbolicity we assume here that the geodesic flow has the anosov property the main example being the case of negatively curved manifolds the words quantum chaos express the intuitive idea that the chaotic features of the geodesic flow should imply certain special features for the corresponding quantum dynamical system that is according to schrodinger the unitary flow exp iht t r acting on the hilbert space l2 m where a stands for the laplacian on m and h is proportional to the planck constant recall that the quantum flow converges in a sense to the classical flow g in the so called semi classical limit h 0 one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow one expects for instance a very different behaviour of eigenfunctions of the laplacian or the distribution of its eigenvalues if the geodesic flow is anosov or in the other extreme completely integrable see sa95 the convergence of the quantum flow to the classical flow is stated in the egorov theorem consider one of the usual quantization procedures op which associates an operator oph a acting on l2 m to every smooth compactly supported function a e c t m on the cotangent bundle t m according to the egorov theorem we have for any fixed t

on the entropy of quantum limits for 2 dimensional cat maps we study semiclassical measures or quantum limits for quantized hyperbolic automorphisms of mathbb t 2 we show that any quantum limit has the following property if a weight is carried on ergodic components of low entropy say entropy less than h0 then a weight must be carried on ergodic components of high entropy hmax h0 where hmax is the maximal entropy this combines some existing partial results towards the classification of quantum limits

the discrete prolate spheroidal filter as a digital signal processing tool the discrete prolate spheroidal dps filter is one of the class of nonrecursive finite impulse response fir filters the dps filter first introduced by tufts and francis 1 is superior to other filters in this class in that it has maximum energy concentration in the frequency passband and minimum ringing in the time domain slepian 2 gives a complete discussion of dps function properties we give a mathematical development of the dps filter properties provide information required to easily and accurately construct even many coefficient filters and compare the properties of this filter with those of the more commonly used filters of the same class we note that use of the dps filter allows for particularly useful statements of data time frequency resolution cell values and that overall it forms an especially useful though little known tool for digital signal processing

equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for frenkel kontorova models there is a well known correspondence between the dynamics of symplectic twist maps which represent an important class of hamiltonian systems and the equilibrium states of a class of variational problems in solid state physics known as frenkel kontorova models in this paper it is shown that the key concepts of uniform hyperbolicity in the first context and phonon gap in the second context are equivalent this allows one to transfer many ideas between the two and hence to deduce new results for example we prove the uniform hyperbolicity of certain invariant sets for symplectic twist maps constructed from so called non degenerate anti integrable limits by aubry and abramovici using the concept of phonon gap and deduce that they have measure zero we prove existence of many multi defect equilibrium states for a frenkel kontorova system if it has a single non degenerate defect equilibrium state via standard results in the theory of uniform hyperbolicity

wigner measures and observability for the schrodinger equation on the disk we analyse the structure of semiclassical and microlocal wigner measures for solutions to the linear schrodinger equation on the disk with dirichlet boundary conditions our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk we prove a structure theorem expressing the restriction of the wigner measures on each invariant torus in terms of second microlocal measures they are obtained by performing a finer localization in phase space around each of these tori at the limit of the uncertainty principle and are shown to propagate according to heisenberg equations on the circle our construction yields as corollaries a that the disintegration of the wigner measures is absolutely continuous in the angular variable which is an expression of the dispersive properties of the equation b an observability inequality saying that the l 2 norm of a solution on any open subset intersecting the boundary resp the l 2 norm of the neumann trace on any nonempty open set of the boundary controls its full l 2 norm resp h 1 norm these results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary

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

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