cancellation in additively twisted sums on gl n in a previous paper with schmid we considered the regularity of automorphic distributions for gl 2 r and its connections to other topics in number theory and analysis in this paper we turn to the higher rank setting establishing the nontrivial bound n t a n e 2 i n o e t 3 4 e uniformly in r for a n the coefficients of the l function of a cusp form on gl 3 z gl 3 r we also derive an equivalence theorem 7 1 between analogous cancellation statements for cusp forms on gl n r and the sizes of certain period integrals these in turn imply estimates for the second moment of cusp form l functions

quantum unique ergodicity for sl2 z h we eliminate the possibility of escape of mass for hecke maass forms of large eigenvalue for the modular group combined with the work of lindenstrauss this establishes the quantum unique ergodicity conjecture of rudnick and sarnak for hecke maass forms on the modular surface sl 2

zeros of l functions attached to maass forms this paper considers psl 2 automorphic cusp forms the maass forms and their associated dirichlet series there are two major results the first uses real variable techniques to give an explicit reconstruction of the cusp form in terms of the dirichlet series this representation is valid in the entire upper half plane the second uses this representation to show that the dirichlet series has many zeros on its critical line this is the analogue of the hardy littlewood result for the riemann zeta function

equidistribution in shrinking sets and l 4 norm bounds for automorphic forms we study two closely related problems stemming from the random wave conjecture for maas forms the first problem is bounding the l 4 norm of a maas form in the large eigenvalue limit we complete the work of spinu to show that the l 4 norm of an eisenstein series e z 1 2 it g restricted to compact sets is bounded by sqrt log t g the second problem is quantum unique ergodicity in shrinking sets we show that by averaging over the centre of hyperbolic balls in gamma backslash mathbb h quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the planck scale this result is conditional on the generalised lindelof hypothesis for hecke maas eigenforms but is unconditional for eisenstein series we also show that equidistribution for hecke maas eigenforms need not hold at or below the planck scale finally we prove similar equidistribution results in shrinking sets for heegner points and closed geodesics associated to ideal classes of quadratic fields

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

l infty norms of eigenfunctions of arithmetic surfaces here and elsewhere a l1o0 as a xc is a basic problem of quantum chaos s in any case almost nothing beyond 0 2 is known about 110 joc when the curvature is negative one can push the standard techniques and replace a1 4 by a1 4 log a in this case in this paper we use arithmetic techniques in particular modular correspondences to obtain the first improvement in the exponent 0 2 we also obtain lower bounds on the l norms in these arithmetic cases in more detail let a b be a quaternion division algebra over q a is linearly generated by 1 w q wq over q and w2 a q2 b wq qw 0 here a b e z are square free and we will assume that a 0 as usual the norm and trace are defined by n a a d tr a a oy where if a xo x1w x2q x3wq

on the voronoi formula for gl n we prove a general vorono u i formula for cuspidal automorphic representations of rm gl n over number fields this generalizes recent work by miller schmid and goldfeld li on maass forms our method follows closely the adelic framework of integral representations of l functions the proof is flexible enough to allow ramification and we propose possible variants for example the assumption that the additive twist is trivial at places where the representation is ramified is sufficient to obtain an explicit final statement with hyper kloosterman sums

simple zeros of the ramanujan τ dirichlet series it is known that l s has all of its non real zeros in the strip l l 2 t 6 t l simple zeros in the region 0 t t the methods of these papers however will probably not give any results for l s the levinson method requires a mean value theorem for the square of the modulus of s multiplied by a dirichlet polynomial the length of this polynomial is critical and it seems unlikely that such a theorem with a long enough polynomial could be proven at this time in the case of l as far as the method of 1 is concerned we relied on a factorization of the dedekind zeta function as a product of the

planck scale mass equidistribution of toral laplace eigenfunctions we study the small scale distribution of the l 2 mass of eigenfunctions of the laplacian on the two dimensional flat torus given an orthonormal basis of eigenfunctions lester and rudnick commun math phys 350 1 279 300 2017 showed the existence of a density one subsequence whose l 2 mass equidistributes more or less down to the planck scale we give a more precise version of their result showing equidistribution holds down to a small power of log above planck scale and also showing that the l 2 mass fails to equidistribute at a slightly smaller power of log above the planck scale this article rests on a number of results about the proximity of lattice points on circles much of it based on foundational work of javier cilleruelo

the quantum unique ergodicity conjecture for thin sets abstract we consider some analogs of the quantum unique ergodicity conjecture for geodesics and shrinking families of sets in particular we prove the analog of the que conjecture for eisenstein series restricted to the infinite geodesic connecting 0 and inside the modular surface

secular determinants of random unitary matrices we consider the characteristic polynomials of random unitary matrices u drawn from various circular ensembles in particular the statistics of the coefficients of these polynomials are studied the variances of these secular coefficients are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent the latter secular coefficients are related to the traces of powers of u by newton s well known formulae while the traces tend to have gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large the secular coefficients exhibit strong mutual correlations due to newton s mixing of traces to coefficients these results might become relevant for current efforts at combining semiclassics and random matrix theory in quantum treatments of classically chaotic dynamics

on mean values of mollifiers and l functions associated to primitive cusp forms we study the second moment of the l function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers this family is a natural extension of the mollifers considered by conrey and by bui conrey and young as an application we improve the current lower bound on critical zeros of holomorphic primitive cusp forms

on random matrices from the compact classical groups if m is a matrix taken randomly with respect to normalized haar measure on u n o n or sp n then the real and imaginary parts of the random variables tr mk k 1 converge to independent normal random variables with mean zero and variance k 2 as the size n of the matrix goes to infinity for the unitary group this is a direct consequence of the strong szeg6 theorem for toeplitz determinants we will prove a conjecture of diaconis

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

linear functions on the classical matrix groups let m be a random matrix in the orthogonal group on distributed according to haar measure and let a be a fixed n x n matrix over r such that tr aa t n then the total variation distance of the random variable tr am to a standard normal random variable is bounded by 2 3 n 1 and this rate is sharp up to the constant analogous results are obtained for m a random unitary matrix and a a fixed n x n matrix over c the proofs are applications of a new abstract normal approximation theorem which extends stein s method of exchangeable pairs to situations in which continuous symmetries are present

the influence of noise on the logistic model the behavior of the logistic system which is generated by the functionf x ax 1 x changes in an interesting way if it is perturbed by external noise it turns out that the chaotic behavior which was predicted by li and yorke for orbits of period 3 becomes visible and that a sequence of mergence transitions occurs at the critical parameter the change of the invariant probability density and the lyapunov exponents are examined numerically the power spectrum for the period 3 orbit for different fluctuations is calculated and a recursion formula for the time evolution of the probability density is presented as a discrete time analog of a chapman kolmogorov equation

the circle method and bounds for l functions i let f be a hecke maass or holomorphic primitive cusp form of arbitrary level and nebentypus and let chi be a primitive character of conductor m for the twisted l function l s f otimes chi we establish the hybrid subconvex bound begin aligned l left frac 1 2 it f otimes chi right ll m 3 t frac 1 2 frac 1 18 varepsilon end aligned for t in mathbb r the implied constant depends only on the form f and varepsilon

equidistribution of eisenstein series in the level aspect we prove an equidistribution property of the eisenstein series for congruence subgroups as the level goes to infinity this is an analogy of the phenomenon called quantum ergodicity

asymptotic theory of multidimensional chaos a delay differential equation u t u t f u t 1 0 t and its generalization are investigated in the limit 0 when the attractor s dimension increases infinitely it is shown that a number of statistical characteristics are asymptotically independent of as for the attractor it can be regarded as a direct product ofo 1 equivalent subattractors their statistical characteristics being asymptotically independent of the results enable one to predict some characteristics of the attractor with fractal dimensiond 1 for the case 1 when they are inaccessible numerically the approach developed seems to be applicable for a wide class of spatiotemporal systems

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

an additive problem in the fourier coefficients of cusp forms we establish an estimate on sums of shifted products of fourier coefficients coming from holomorphic or maass cusp forms of arbitrary level and nebentypus these sums are analogous to the binary additive divisor sum which has been studied extensively as an application we derive extending work of duke friedlander and iwaniec a subconvex estimate on the critical line for l functions associated to character twists of these cusp forms

distributions and analytic continuation of dirichlet series this is the second in a series of three papers the other two are summation formulas from poisson and voronoi to the present progr math 220 2004 419 440 and automorphic distributions l functions and voronoi summation for gl 3 preprint the first paper is primarily expository while the third proves a voronoi style summation formula for the coefficients of a cusp form on gl 3 z gl 3 r the present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula and also for the proof of the gl 3 gl 1 converse theorem given in the third paper the primary concept studied is a notion of the order of vanishing of a distribution along a closed submanifold applications are given to the analytic continuation of riemann s zeta function degree 1 and degree 2 l functions the converse theorem for gl 2 and a characterization of the classical mellin transform inversion relations on functions with specified singularities

planck scale number of nodal domains for toral eigenfunctions abstract we study the number of nodal domains in balls shrinking slightly above the planck scale for generic toral eigenfunctions we prove that up to the natural scaling the nodal domains count obeys the same asymptotic law as the global number of nodal domains the proof on one hand uses new arithmetic information to refine bourgain s de randomisation technique at planck scale and on the other hand it requires a planck scale version of yau s conjecture which we believe to be of independent interest

the spectral decomposition of shifted convolution sums let pi 1 pi 2 be cuspidal automorphic representations of pgl 2 r qf conductor 1 and hecke eigenvalues lambda pi 1 2 n and let h 0 be an integer for any smooth compactly supported weight functions w 1 w 2 r x c and any y 0 aspectral decomposition of the shifted convolution sum sigma m n h lambda pi 1 vertical bar m vertical bar lambda pi 2 vertical bar n vertical bar root vertical bar mn vertical bar w 1 m y w 2 n y is obtained as an application aspectral decomposition of the dirichlet series graphics is proved for rs 1 2 with polynomial growth on vertical lines in the s aspect and uniformity in the h aspect

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

analytic continuation of representations and estimates of automorphic forms 0 1 analytic vectors and their analytic continuation let g be a lie group and g v a continuous representation of g in a topological vector space v a vector v v is called analytic if the functionv g 7 g v is a real analytic function on g with values in v this means that there exists a neighborhood u of g in its complexification gc such thatv extends to a holomorphic function on u in other words for each element g u we can unambiguously define the vector g v asv g i e we can extend the action of g to a somewhat larger set in this paper we will show that the possibility of such an extension sometimes allows one to prove some highly nontrivial estimates unless otherwise stated g sl 2 r so gc sl 2 c we consider a typical representation ofg i e a representation of the principal series namely fix c and consider the space dof smooth homogeneous functions of degree 1 on r 2 0 i e d c r 2 0 ax ay a 1 x y we

bounds for gl 3 gl 2 l functions and gl 3 l functions in this paper we will give the subconvexity bounds for self dual gl 3 l functions in the t aspect as well as subconvexity bounds for self dual gl 3 gl 2 l functions in the gl 2 spectral aspect

the circle method and bounds for l functions ii subconvexity for twists of gl 3 l functions let pi be a rm sl 3 bbb z hecke maass cusp form let chi chi 1 chi 2 be a dirichlet character with chi i primitive modulo m i suppose m 1 m 2 are primes such that sqrt m 2 m 4 delta m 1 m 2m 3 delta where m m 1m 2 and 0 delta 1 28 in this paper we will prove the following subconvex bound l left 1 over 2 pi otimes chi right ll pi varepsilon m 3 over 4 delta varepsilon

rate of convergence of linear functions on the unitary group we study the rate of convergence to a normal random variable of the real and imaginary parts of where u is an n n random unitary matrix and an is a deterministic complex matrix we show that the rate of convergence is o n 2 b with 0 b 1 depending only on the asymptotic behaviour of the singular values of an for example if the singular values are non degenerate different from zero and o 1 as n then b 0 the proof uses a berry esseen inequality for linear combinations of eigenvalues of random unitary matrices and so appropriate for strongly dependent random variables

generalizations of some integrals over the unitary group using the character expansion method we generalize several well known integrals over the unitary group to the case where general complex matrices appear in the integrand these integrals are of interest in the theory of random matrices and may also find applications in lattice gauge theory

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

the twisted fourth moment of the riemann zeta function we compute the asymptotics of the fourth moment of the riemann zeta func tion times an arbitrary dirichlet polynomial of length t 1 11 the study of the moments of the riemann zeta function has a long and distinguished history starting with the work of hardy and littlewood in 1918 and continuing to the present day one motivation for understanding moments is that they yield information about the maximum size of the zeta function the lindelof hypothesis another application is to zero density estimates which in turn have consequences for primes in short intervals however they have become an interesting topic in their own right very few rigorous results are known just the second and fourth power moments indeed it is only recently that a believable conjecture for higher powers has been made the twisted moments that is moments of the riemann zeta function times an arbitrary dirichlet polynomial are important too for example levinson s method of detecting zeros of the zeta function lying on the critical line requires knowing the asymptotics of the mollified second moment in a series of papers duke friedlander and iwaniec used estimates for amplified moments of a family of l functions in order to deduce a subconvexity bound for an individual member of the family of course there are far easier methods to give a subconvexity bound for zeta but there are close analogies between different families and it is desirable to understand the structure of these amplified moments in general in this paper we prove an asymptotic formula for the twisted fourth moment of the riemann zeta function where we may take a dirichlet polynomial of length up to t 1 11 e

distribution of mass of holomorphic cusp forms we prove an upper bound for the l4 norm and for the l2 norm restricted to the vertical geodesic of a holomorphic hecke cusp form f of large weight k the method is based on watson s formula and estimating a mean value of certain l functions of degree 6 further applications to restriction problems of siegel modular forms and subconvexity bounds of degree 8 l functions are given

on the moments of traces of matrices of classical groups we consider random matrices belonging to the groups u n o n so n and sp n and distributed according to the corresponding unit haar measure we prove that the moments of traces of powers of the matrices coincide with the moments of certain gaussian random variables if the order of moments is low enough corresponding formulas proved partly before by various methods are obtained here in the framework of a unique method reminiscent of the method of correlation equations of statistical mechanics the equations are derived by using a version of the integration by parts

moments of the riemann zeta function assuming the riemann hypothesis we obtain an upper bound for the moments of the riemann zeta function on the critical line our bound is nearly as sharp as the conjectured asymptotic formulae for these moments the method extends to moments in other families of l functions

trace formula for compact gamma setminus psl_ 2 mathbb r and the equidistribution theory of closed geodesics soit un sous groupe cocompact discret d elements hyperboliques de g psl 2 r et soit x t la surface hyperbolique associee g k le fibre unite co tangent s x t peut s identifier a g des classes de conjugaison d elements hyperboliques determinent des geodesiques fermees dans s x t pour chaque geodesique fermee soit la mesure positive sur c s x definie par p uf soit t ld t ou l est la longueur de on donne des developpements asymptotiques de t f quand f est une fonction propre de casimir de poids m

on the uniform equidistribution of long closed horocycles it is well known that on any given hyperbolic surface of finite area a closed horocycle of length becomes asymptotically equidistributed as in this paper we prove that any subsegment of length greater than 1 2 e of such a closed horocycle also becomes equidistributed as the exponent 1 2 e is the best possible and improves upon a recent result by hejhal he3 we give two proofs of the above result our second proof leads to explicit information on the rate of convergence we also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to

spatio temporal structure in coupled map lattices two point correlations versus mutual information a one dimensional lattice of logistic maps is investigated in the case of strong nonlinearity and strong coupling although the dynamics may be classified as fully developed turbulence spatio temporal structure can be detected by computing time delayed mutual information and two point correlations the correlation is found to be superior in detecting weak structure an improved algorithm for mutual information is described

mock gaussian behaviour for linear statistics of classical compact groups we consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups we compute their moments and find that the first few moments are gaussian whereas the limiting distribution is not the precise number of gaussian moments depends upon the particular statistic considered

turan inequalities and zeros of dirichlet series associated with certain cusp forms the turan inequalities are a countably infinite set of conditions about the power series coefficients of certain entire functions which are necessary in order for the function to have only real zeros we give a one parameter family of generalized dirichlet series each with functional equation for which the turan inequalities hold for the associated 4 function normalized so that the critical line is the real axis for a discrete set of values of the parameter the dirichlet series has an euler product and is the l series associated to a modular form for these we expect the analogue of the riemann hypothesis to hold for the rest of the values of the parameter we do not expect an analogue of the riemann hypothesis we show for one particular value of the parameter that the dirichlet series in fact has zeros within the region of absolute convergence 1 riemann s integral representation for the function one approach to the study of the zeros of the riemann zeta function begins with the integral representation for riemann s 4 function the function is given by s is s l f s21f s 2 c s where 4 is riemann s zeta function and s a it is a complex variable the mellin transform of 4 involves the theta function h z e n z 2 n oo where e z exp 27riz we have s 2f 4 a00 v x xs 2 2 jox where u e 2 0 iu i yiku e n 2 n 1 using the transformation formula for h h l z z1 26 z received by the editors april 10 1992 1991 mathematics subject classification primary 1 1f66 secondary 30c1 5 research of the second author supported in part by the alfred p sloan foundation research of both authors supported in part by a grant from the nsf 1994 american mathematical society 0002 9947 94 1 00 25 per page

borel theorems for random matrices from the classical compact symmetric spaces we study random vectors of the form tr a 1 v tr a r v where v is a uniformly distributed element of a matrix version of a classical compact symmetric space and the a v are deterministic parameter matrices we show that for increasing matrix sizes these random vectors converge to a joint gaussian limit and compute its covariances this generalizes previous work of diaconis et al for haar distributed matrices from the classical compact groups the proof uses integration formulas due to collins and sniady for polynomial functions on the classical compact groups

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

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

the mean square of the error term for the fourth power moment of the zeta function we prove that j e2 t 2 dt t 2 log c t where e2 t denotes the error term in the asymptotic formula for fo k of dt the proof depends on the second author s explicit formula for a modified version of the latter integral several related results are also proved including the analogue of the above result for the binary additive divisor problem

coupled map lattices as models of deterministic and stochastic differential delay equations we discuss the probabilistic properties of a class of differential delay equations dde s by first reducing the equations to coupled map lattices and then considering the spectral properties of the associated transfer operators the analysis is carried out for the deterministic case and a stochastic case perturbed by additive or multiplicative white noise this scheme provides an explicit description of the evolution of phase space densities in dde s and yields an evolution equation that approximates the analog for delay equations of the generalized liouville and fokker planck equations it is shown that in many cases of interest for both stochastic and deterministic delay equations the phase space densities reach a limit cycle in the asymptotic regime this statistical cycling is observed numerically in continuous time systems with delay and discussed in light of our analytical description of the transfer operators 1995 the american physical society

study of a high dimensional chaotic attractor the nature of a very high dimensional chaotic attractor in an infinite dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high dimensional attractor in the phase space we introduce two complementary bases from which the attractor is observed one the lyapunov basis composed of the lyapunov vectors and the another the fourier basis composed of the fourier modes we introduce the exterior subspaces on the basis of the lyapunov vectors and observe the chaotic motion projected onto these exteriors it is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace goes out of the attractor the origin of such a phenomenon is attributed to more fundamental features of our attractor which become manifest when the attractor is observed from the lyapunov basis a counterpart of the phenomenon can be observed also on the fourier basis because there is a statistical one to one correspondence between the lyapunov vectors and the fourier modes in particular a statistical property of the high pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor

hybrid bounds for twisted l functions the aim of this paper is to derive bounds on the critical line s 1 for l functions attached to twists f of a primitive cusp form f of level n and a primitive character modulo q that break convexity simultaneously in the s and q aspects if f has trivial nebentypus it is shown that

mass equidistribution for hecke eigenforms in this work we study the mass equidistribution for holomorphic hecke eigenforms and establish by employing incomplete poincare series and the petersson formula sharp equidistribution results when the average is performed over intervals much shorter than before a key feature is the analysis of the off diagonal terms that result from this shortening of intervals 2003 wiley periodicals inc

the fourth moment of dirichlet l functions we compute the fourth moment of dirichlet l functions at the central point for prime moduli with a power savings in the error term

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

mass equidistribution for hecke eigenforms we prove a conjecture of rudnick and sarnak on the mass equidistribution of hecke eigenforms this builds upon independent work of the authors