projective modules over noncommutative tori are multi window gabor frames for modulation spaces in the present investigation we link noncommutative geometry over noncommutative tori with gabor analysis where the first has its roots in operator algebras and the second in time frequency analysis we are therefore in the position to invoke modern methods of operator algebras e g topological stable rank of banach algebras to display the deeper properties of gabor frames furthermore we are able to extend results due to connes and rieffel on projective modules over noncommutative tori to banach algebras which arise in a natural manner in gabor analysis the main goal of this investigation is twofold i an interpretation of projective modules over noncommutative tori in terms of gabor analysis and ii to show that the morita rieffel equivalence between noncommutative tori is the natural framework for the duality theory of gabor frames more concretely we interpret generators of projective modules over noncommutative tori as the gabor atoms of multi window gabor frames for modulation spaces moreover we show that this implies the existence of good multi window gabor frames for modulation spaces with gabor atoms in e g feichtinger s algebra or in schwartz space

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

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

some counterexamples in the theory of weyl heisenberg frames we present an example of a positive function g with a positive fourier transform g spl circ and reasonable smoothness and decay properties such that 1 sup nm exp spl pi itm g t n n m spl isin z does not constitute a frame for l sup 2 r we also give counterexamples for the statement that one can tell in definiteness of a weyl heisenberg frame operator from in definiteness of its weyl symbol

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

chaos and crises in more than two dimensions instituto de astronomi ayfi sica del espacio conicet casilla de correo 67 socursal 28 1428 buenos aires argentina received 23 december 1996 noisy chaotic trajectories with nite time lyapunov exponents that uctuate about zero are basicallyunshadowable s dawson c grebogi t sauer and j a yorke phys rev lett 73 1927 1994 this canoccur when periodic orbits with different numbers of unstable directions coexist inside the attractor thepresence of a he non type chaotic saddle i e a nonattracting chaotic set with a structure similar to that of thehe non attractor guarantees such coexistence in a persistent manner s p dawson phys rev lett 76 4348 1996 in this paper we describe how these sets appear naturally in maps of more than two dimensions howthey can be found and what crises they produce s1063 651x 97 13005 7 pacs number s 05 45 1b 05 40 1ji introduction

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

toward a theory of rank one attractors introduction 1 statement of results part i preparation 2 relevant results from one dimension 3 tools for analyzing rank one maps part ii phase space dynamics 4 critical structure and orbits 5 properties of orbits controlled by critical set 6 identification of hyperbolic behavior formal inductive procedure 7 global geometry via monotone branches 8 completion of induction 9 construction of srb measures part iii parameter issues 10 dependence of dynamical structures on parameter 11 dynamics of curves of critical points 12 derivative growth via statistics 13 positive measure sets of good parameters appendices

varying the time frequency lattice of gabor frames a gabor or weyl heisenberg frame for l 2 r d is generated by time frequency shifts of a square integrable function the gabor atom along a time frequency lattice the dual frame is again a gabor frame generated by the dual atom in general gabor frames are not stable under a perturbation of the lattice constants that is even for arbitrarily small changes of the parameters the frame property can be lost in contrast as a main result we show that this kind of stability does hold for gabor frames generated by a gabor atom from the modulation space m 1 r d which is a dense subspace of l 2 r d moreover in this case the dual atom depends continuously on the lattice constants in fact we prove these results for more general weighted modulation spaces as a consequence we obtain for gabor atoms from the schwartz class that the continuous dependence of the dual atom holds even in the schwartz topology also we complement these main results by corresponding statements for gabor riesz sequences and their biorthogonal system

on the structure and interpolation properties of quasi shift invariant spaces abstract the structure of certain types of quasi shift invariant spaces which take the form v x span l 2 x j j z for an infinite discrete set x x j r is investigated additionally the relation is explored between pairs x and y such that interpolation of functions in v x via interpolants in v y solely from the samples of the original function is possible and stable some conditions are given for which the sampling problem is stable and for which recovery of functions from their interpolants from a family of spaces v y is possible

gabor super frames with hermite functions we investigate vector valued gabor frames sometimes called gabor superframes based on hermite functions hn let h h0 h1 hn be the vector of the first n 1 hermite functions we give a complete characterization of all lattices lambda subseteq mathbb r 2 such that the gabor system rm e 2 pi i lambda 2 t bf h t lambda 1 lambda lambda 1 lambda 2 in lambda is a frame for l 2 mathbb r mathbb c n 1 as a corollary we obtain sufficient conditions for a single hermite function to generate a gabor frame and a new estimate for the lower frame bound the main tools are growth estimates for the weierstrass function a new type of interpolation problem for entire functions on the bargmann fock space and structural results about vector valued gabor frames

sampling theorems for shift invariant spaces gabor frames and totally positive functions we study nonuniform sampling in shift invariant spaces and the construction of gabor frames with respect to the class of totally positive functions whose fourier transform factors as hat g xi prod j 1 n 1 2 pi i delta j xi 1 e c xi 2 for delta 1 ldots delta n in mathbb r c 0 in which case g is called totally positive of gaussian type in analogy to beurling s sampling theorem for the paley wiener space of entire functions we prove that every separated set with lower beurling density 1 is a sampling set for the shift invariant space generated by such a g in view of the known necessary density conditions this result is optimal and validates the heuristic reasonings in the engineering literature using a subtle connection between sampling in shift invariant spaces and the theory of gabor frames we show that the set of phase space shifts of g with respect to a rectangular lattice alpha mathbb z times beta mathbb z forms a frame if and only if alpha beta 1 this solves an open problem going back to daubechies in 1990 for the class of totally positive functions of gaussian type the proof strategy involves the connection between sampling in shift invariant spaces and gabor frames a new characterization of sampling sets without inequalities in the style of beurling new properties of totally positive functions and the interplay between zero sets of functions in a shift invariant space and functions in the bargmann fock space

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

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

gabor frames and time frequency analysis of distributions abstract this paper lays the foundation for a quantitative theory of gabor expansions f x k n c k n e 2 in x g x k in analogy to wavelet expansions of besov triebel lizorkin spaces we show that the correct class of spaces which can be characterized by the magnitude of the coefficients c k n is the class of modulation spaces to analyze the behavior of the coefficients it is necessary to invert the gabor frame operator on these spaces we show that the frame operator is invertible on modulation spaces if and only if it is invertible on l 2 and the atom g is in a suitable space of test functions a similar statement for wavelet theory is false the second part is devoted to gabor analysis on general time frequency lattices

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

discretized gabor frames of totally positive functions in this paper a large class of universal windows for gabor frames weyl heisenberg frames is constructed these windows have the fundamental property that every overcritical rectangular lattice generates a gabor frame likewise every undercritical rectangular lattice generates a riesz sequence

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

persistent homoclinic tangencies in the he non family abstract homoclinic tangencies in the henon family f x y x 2 by x for the parameter values b 0 3 and 1 270 1 420 are investigated our main observation is that there exist three intervals comprising 93 percent of the values of the parameter such that for a dense set of parameter values in these intervals the henon family possesses a homoclinic tangency therefore one should expect long parameter intervals where the henon family is not structurally stable strong numerical support for this observation is provided

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

srb measures for partially hyperbolic systems whose central direction is mostly expanding we construct sinai ruelle bowen srb measures supported on partially hyperbolic sets of diffeomorphisms the tangent bundle splits into two invariant subbundles one of which is uniformly contracting under the assumption that the complementary subbundle is non uniformly expanding if the rate of expansion lyapunov exponents is bounded away from zero then there are only finitely many srb measures our techniques extend to other situations including certain maps with singularities or critical points as well as diffeomorphisms having only a dominated splitting and no uniformly hyperbolic subbundle

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

from invariant curves to strange attractors we prove that simple mechanical systems when subjected to external periodic forcing can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied in particular the existence of global strange attractors with fully stochastic properties is proved for a class of second order odes

implementation of discretized gabor frames and their duals the usefulness of gabor frames depends on the easy computability of a suitable dual window this question is addressed under several aspects several versions of schulz s iterative algorithm for the approximation of the canonical dual window are analyzed for their numerical stability for gabor frames with totally positive windows or with exponential b splines a direct algorithm yields a family of exact dual windows with compact support it is shown that these dual windows converge exponentially fast to the canonical dual window

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

almost sure invariance principle for nonuniformly hyperbolic systems we prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems discrete time systems and flows are covered by this result in particular the result applies to the planar periodic lorentz flow with finite horizon statistical limit laws such as the central limit theorem the law of the iterated logarithm and their functional versions are immediate consequences

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

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

large deviations for non uniformly expanding maps we obtain large deviation bounds for non uniformly expanding maps with non flat singularities or criticalities and for partially hyperbolic non uniformly expanding attracting sets that is given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map showing that the lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved as easy by products we deduce escape rates from subsets of the basins of physical measures for these types of maps the rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states

robust ergodic properties in partially hyperbolic dynamics we study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting earlier work of bonatti and viana 2000 about existence and finitude of physical measures is extended to the case of local diffeomorphisms moreover we prove that such systems constitute a c 2 open set in which statistical stability is a dense property in contrast all mostly contracting systems are shown to be stable under small random perturbations

gabor orthonormal bases generated by the unit cubes abstract we consider the problem in determining the countable sets in the time frequency plane such that the gabor system generated by the time frequency shifts of the window 0 1 d associated with forms a gabor orthonormal basis for l 2 r d we show that if this is the case the translates by elements of the unit cube in r 2 d must tile the time frequency space r 2 d by studying the possible structure of such tiling sets we completely classify all such admissible sets of time frequency shifts when d 1 2 moreover an inductive procedure for constructing such sets in dimension d 3 is also given an interesting and surprising consequence of our results is the existence for d 2 of discrete sets with g 0 1 d forming a gabor orthonormal basis but with the associated time translates of the window 0 1 d having significant overlaps

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

an optimal result for sampling density in shift invariant spaces generated by meyer scaling function abstract for a class of continuously differentiable function satisfying certain decay conditions it is shown that if the maximum gap sup i x i 1 x i between the consecutive sample points is smaller than a certain number b 0 then any f v can be reconstructed uniquely and stably as a consequence of this result it is shown that if 1 then x i i z is a stable set of sampling for v with respect to the weight w i i z where w i x i 1 x i 1 2 and is the scaling function associated with meyer wavelet further the maximum gap condition 1 is sharp

rapid decay of correlations for nonuniformly hyperbolic flows we show that superpolynomial decay of correlations rapid mixing is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which young proved exponential decay of correlations the proof combines techniques of dolgopyat and operator renewal theory it follows from our results that planar periodic lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing

permutation tests for centre effect on survival endpoints with application in an acute myeloid leukaemia multicentre study when analysing multicentre data it may be of interest to test whether the distribution of the endpoint varies among centres in a mixed effect model testing for such a centre effect consists in testing to zero a random centre effect variance component it has been shown that the usual asymptotic 2 distribution of the likelihood ratio and score statistics under the null does not necessarily hold in the case of censored data mixed effects cox models have been used to account for random effects but few works have concentrated on testing to zero the variance component of the random effects we propose a permutation test using random permutation of the cluster indices to test for a centre effect in multilevel censored data results from a simulation study indicate that the permutation tests have correct type i error rates contrary to standard likelihood ratio tests and are more powerful the proposed tests are illustrated using data of a multicentre clinical trial of induction therapy in acute myeloid leukaemia patients copyright 2014 john wiley sons ltd

frames in the bargmann space of entire functions we look at the decomposition of arbitrary f in l2 r in terms of the family of functions cp x w exp iimnab iman i x nb 2 with a b 0 we derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab 2w n n an integer 2 2 transported to the hilbert space f of entire functions introduced by v bargmann these results are expressed as inequalities of the form

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

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

regular families of kernels for nonlinear approximation abstract this article studies sufficient conditions on families of approximating kernels which provide n term approximation errors from an associated nonlinear approximation space which match the best known orders of n term wavelet expansion these conditions provide a framework which encompasses some notable approximation kernels including splines cardinal functions and many radial basis functions such as the gaussians and general multiquadrics examples of such kernels are given to justify the criteria additionally the techniques involved allow for some new results on n term greedy interpolation of sobolev functions via radial basis functions

an endpoint balian low theorem for schauder bases abstract we prove two new results on the validity of the balian low theorem in the setting of schauder bases our first main result proves an endpoint balian low theorem for gabor systems that form schauder bases for l 2 r if g l 2 r is compactly supported and 2 g 2 d then the gabor system g g 1 1 cannot be a schauder basis of so called type our second main result proves that the classical balian low theorem for orthonormal bases and riesz bases fails in the setting of schauder bases in particular given 0 there exists g l 2 r such that g g 1 1 is a schauder basis for l 2 r and such that t 3 g t 2 d t and 3 g 2 d

wiener s lemma for twisted convolution and gabor frames as a consequence ca is invertible and bounded on all p zd for 1 p oo simultaneously in this article we study several non commutative generalizations of wiener s lemma and their application to gabor theory the paper is divided into two parts the first part sections 2 and 3 is devoted to abstract harmonic analysis and extends wiener s lemma to twisted convolution the second part section 4 is devoted to the theory of gabor frames specifically to the design of dual windows with good time frequency localization in particular we solve a conjecture of janssen feichtinger and one of us 17 18 9 these two topics appear to be completely disjoint but they are not the solution of the conjectures about gabor frames is an unexpected application of methods from non commutative harmonic analysis to application oriented mathematics it turns out that the connection between twisted convolution and the heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis to be more concrete we formulate some of our main results first and will deal with the details and the technical background later

statistical stability and continuity of srb entropy for systems with gibbs markov structures we present conditions on families of diffeomorphisms that guarantee statistical stability and srb entropy continuity they rely on the existence of horseshoe like sets with infinitely many branches and variable return times as an application we consider the family of henon maps within the set of benedicks carleson parameters

continuity properties of the gabor frame operator abstract for a gabor frame with small lattice it is shown that the frame operator is continuous and invertible on many banach spaces defined by smoothness and decay properties as a consequence it is shown that there exist gabor type frame dual frame pairs and tight frames with good decay and smoothness properties also it is shown that the iteration scheme by which a function can be recovered from its frame coefficients converges robustly e g in lp and sobolev norms also explicit estimates on lattice sizes for which such results hold can be obtained

generic family with robustly infinitely many sinks we show for every r d ge 0 or r d ge 2 the existence of a baire generic set of c d families of c r maps f a a in mathbb r k of a manifold m of dimension ge 2 so that for every a small the map f a has infinitely many sinks when the dimension of the manifold is ge 3 the generic set is formed by families of diffeomorphisms when m is the annulus this generic set is formed by local diffeomorphisms this is a counter example to a conjecture of pugh and shub

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

generalized physical and srb measures for hyperbolic diffeomorphisms in this paper we introduce the notion of generalized physical and srb measures these measures naturally generalize classical physical and srb measures to measures which are supported on invariant sets that are not necessarily attractors we then perform a detailed case study of these measures for hyperbolic henon maps for this class of systems we are able to develop a complete theory about the existence uniqueness finiteness and properties of these natural measures moreover we derive a classification for the existence of a measure of full dimension we also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of as well as the differences to the results for henon maps finally we study the regular dependence of the dimension of the generalized physical srb measure on the diffeomorphism for the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism

on the smooth dependence of srb measures for partially hyperbolic systems in this paper we study the differentiability of srb measures for partially hyperbolic systems we show that for any s geq 1 for any integer ell geq 2 any sufficiently large r any varphi in c r mathbb t mathbb r such that the map f mathbb t 2 to mathbb t 2 f x y ell x y varphi x is c r stably ergodic there exists an open neighbourhood of f in c r mathbb t 2 mathbb t 2 such that any map in this neighbourhood has a unique srb measure with c s 1 density which depends on the dynamics in a c s fashion we also construct a c infty mostly contracting partially hyperbolic diffeomorphism f mathbb t 3 to mathbb t 3 such that all f in a c2 open neighbourhood of f possess a unique srb measure mu f and the map f mapsto mu f is strictly holder at f in particular non differentiable this gives a partial answer to dolgopyat s question 13 3 in dolgopyat commun math phys 213 181 201 2000

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

the geometric approach for constructing sinai ruelle bowen measures an important class of physically relevant measures for dynamical systems with hyperbolic behavior is given by sinai ruelle bowen srb measures we survey various techniques for constructing srb measures and studying their properties paying special attention to the geometric push forward approach after describing this approach in the uniformly hyperbolic setting we review recent work that extends it to non uniformly hyperbolic systems