on the existence of a solution to a spectral estimation problem a la byrnes georgiou lindquist a parametric spectral estimation problem in the style of byrnes georgiou and lindquist was posed in 1 but the existence of a solution was only proved in a special case based on their results we show that a solution indeed exists given an arbitrary matrix valued prior density the main tool in our proof is the topological degree theory

on the partial stochastic realization problem we describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix riccati equation our analysis of this covariance extension equation cee is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem answering a conjecture due to georgiou 1987 in the affirmative we also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the cee yielding some insights into the structure of solutions to the minimal partial stochastic realization problem by combining this parameterization with some of the classical approaches in partial realization theory we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations as a corollary to these results we note that in sharp contrast with the deterministic case there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length

a complete parameterization of all positive rational extensions of a covariance sequence in this paper we formalize the observation that filtering and interpolation induce complementary or dual decompositions of the space of positive real rational functions of degree less than or equal to n from this basic result about the geometry of the space of positive real functions we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence on the one hand by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space we are able to develop estimates on the asymptotic behavior of the schur parameters 1918 of positive rational extensions on the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence indeed motivated by its application to signal processing speech processing and stochastic realization theory this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by georgiou 1983 1987 our basic result however also enables us to analyze the robustness of this parameterization with respect to variations in the problem data the methodology employed is a combination of complex analysis geometry linear systems and nonlinear dynamics

on the estimation of structured covariance matrices this paper discusses a method for estimating the covariance matrix of a multivariate stationary process w generated as the output of a given linear filter fed by a stationary process y the estimated covariance matrix must satisfy two constraints it must be positive semi definite and it must be consistent with the fact that w is the output of the given linear filter it turns out that these constraints force the estimated covariance to lie in the intersection of a cone with a linear space while imposing only the first of the two constraints is rather straightforward guaranteeing that both are satisfied is a non trivial issue to which quite a bit of attention has already been devoted in the literature our approach extends the method for estimating the toeplitz covariance matrix of order m of a process y based on the biased spectral estimator stoica moses 1997 this extension is based on the characterization of the output covariance matrix in terms of the filter parameters and the sequence of covariance lags of the input process after introducing our estimation method we propose a comparison performance between this one and other methods proposed in the literature simulation results show that our approach constitutes a valid estimation procedure

uncertainty bounds for spectral estimation the purpose of this paper is to study metrics suitable for assessing uncertainty of power spectra when these are based on finite second order statistics the family of power spectra which is consistent with a given range of values for the estimated statistics represents the uncertainty set about the true power spectrum our aim is to quantify the size of this uncertainty set using suitable notions of distance and in particular to compute the diameter of the set since this represents an upper bound on the distance between any choice of a nominal element in the set and the true power spectrum since the uncertainty set may contain power spectra with lines and discontinuities it is natural to quantify distances in the weak topology the topology defined by continuity of moments we provide examples of such weakly continuous metrics and focus on particular metrics for which we can explicitly quantify spectral uncertainty we then consider certain high resolution techniques which utilize filter banks for preprocessing and compute worst case a priori uncertainty bounds solely on the basis of the filter dynamics this allows the a priori tuning of the filter banks for improved resolution over selected frequency bands

regularity of the maxwell equations in heterogeneous media and lipschitz domains abstract this note establishes regularity estimates for the solution of the maxwell equations in lipschitz domains with non smooth coefficients and minimal regularity assumptions the argumentation relies on elliptic regularity estimates for the poisson problem with non smooth coefficients

on the georgiou lindquist approach to constrained kullback leibler approximation of spectral densities we consider the georgiou lindquist constrained approximation of spectra in the kullback leibler sense we propose an alternative iterative algorithm to solve the corresponding convex optimization problem the lagrange multiplier is computed as a fixed point of a nonlinear matricial map simulation indicates that the algorithm is extremely effective

on the uniqueness result of theorem 6 in relative entropy and the multivariable multidimensional moment problem matrix valued covariance extension and multivariate spectral estimation are formulated as generalized moment problems in the three approach and its extensions under this context we discuss theorem 6 in the title paper concerning the bijectivity of a moment map defined over a parametric family of spectral densities in particular we provide a counterexample in which the moment map under consideration is shown to have a critical point namely a point at which the jacobian loses rank then with standard techniques in bifurcation theory we conclude further that the computed critical point is a bifurcation point which means that the moment map is not injective

non generic spectral statistics in the quantized stadium billiard we consider the effect of a continuous family of neutral bouncing ball orbits on the energy spectrum of the quantized stadium billiard using a semiclassical approximation we derive analytic expressions for standard two point spectral measures the corrections due to the bouncing ball orbits account for some of the non generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured once the bouncing ball contributions are subtracted the spectrum is shown to be well reproduced by the semi classical trace formula based on unstable periodic orbits we also study special patterns in the spectrum which are due to other non generic features such as edge effects and whispering gallery trajectories

on the geometry of maximum entropy problems we show that a simple geometric result suffices to derive the form of the optimal solution in a large class of finite and infinite dimensional maximum entropy problems concerning probability distributions spectral densities and covariance matrices these include burg s spectral estimation method and dempster s covariance completion as well as various recent generalizations of the above we then apply this orthogonality principle to the new problem of completing a block circulant covariance matrix when an a priori estimate is available

unstable periodic orbits and semiclassical quantisation the bohr sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic unstable orbits are isolated it provides a semiclassical euler factorisation for the functional determinant of the quantal hamiltonian in contrast to the hadamard infinite product over the eigenvalues by which the exact determinant is defined

minimum relative entropy for quantum estimation feasibility and general solution we propose a general framework for solving quantum state estimation problems using the minimum relative entropy criterion a convex optimization approach allows us to decide the feasibility of the problem given the data find the maximal common kernel of all admissible states and whenever necessary to relax the constraints in order to allow for a physically admissible solution building on these results the variational analysis can be completed ensuring existence and uniqueness of the optimum the latter can then be computed by efficient standard algorithms for convex optimization without resorting to approximate methods or restrictive assumptions on its rank

remarks on control design with degree constraint the purpose of this note is to highlight similarities and differences between two alternative methodologies for feedback control design under constraints on the mcmillan degree of the feedback system both sets of techniques focus on uniformly optimal designs the first is based on the work of gahinet and apkarian and that of skelton while the other is based on earlier joint work of the authors with c i byrnes

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

generalized interpolation in h infty with a complexity constraint in a seminal paper sarason generalized some classical interpolation problems for h infinity functions on the unit disc to problems concerning lifting onto h 2 of an operator t that is defined on k

a convex optimization approach to the rational covariance extension problem in this paper we present a convex optimization problem for solving the rational covariance extension problem given a partial covariance sequence and the desired zeros of the modeling filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s seminal existence result and his conjecture a stronger version of which we have resolved in byrnes et al ieee trans automat control ac 40 1995 pp 1841 1857

a maximum entropy solution of the covariance extension problem for reciprocal processes stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of markov random fields restricted to one dimension nonstationary reciprocal processes have been extensively studied in the past especially by jamison et al the specialization of the nonstationary theory to the stationary case however does not seem to have been pursued in sufficient depth in the literature stationary reciprocal processes and reciprocal stochastic models are potentially useful for describing signals which naturally live in a finite region of the time or space line estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing in this paper we discuss a class of reciprocal processes which is the acausal analog of auto regressive ar processes familiar in control and signal processing we show that maximum likelihood identification of these processes leads to a covariance extension problem for block circulant covariance matrices this generalizes the famous covariance band extension problem for stationary processes on the integer line as in the usual stationary setting on the integer line the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem we show that the maximum entropy principle leads to a complete solution of the problem

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

on the well posedness of a parametric spectral estimation problem and its numerical solution this paper concerns a spectral estimation problem in which we want to find a spectral density function that is consistent with estimated second order statistics it is an inverse problem admitting multiple solutions and selection of a solution can be based on prior functions we show that the problem is well posed when formulated in a parametric fashion and that the solution parameter depends continuously on the prior function in this way we are able to obtain a smooth parametrization of admissible spectral densities based on this result the problem is reparametrized via a bijective change of variables out of a numerical consideration and then a continuation method is used to compute the unique solution parameter numerical aspects such as convergence of the proposed algorithm and certain computational procedures are addressed a simple example is provided to show the effectiveness of the algorithm

a globally convergent matricial algorithm for multivariate spectral estimation in this paper we first describe a matricial newton type algorithm designed to solve the multivariable spectrum approximation problem we then prove its global convergence finally we apply this approximation procedure to multivariate spectral estimation and test its effectiveness through simulation simulation shows that in the case of short observation records this method may provide a valid alternative to standard multivariable identification techniques such as matlab s pem and matlab s n4sid

matrix valued nevanlinna pick interpolation with complexity constraint an optimization approach over the last several years a new theory of nevanlinna pick interpolation with complexity constraint has been developed for scalar interpolants in this paper we generalize this theory to the matrix valued case also allowing for multiple interpolation points we parameterize a class of interpolants consisting of most interpolants of no higher degree than the central solution in terms of spectral zeros this is a complete parameterization and for each choice of interpolant we provide a convex optimization problem for determining it this is derived in the context of duality theory of mathematical programming to solve the convex optimization problem we employ a homotopy continuation technique previously developed for the scalar case these results can be applied to many classes of engineering problems and to illustrate this we provide some examples in particular we apply our method to a benchmark problem in multivariate robust control by constructing a controller satisfying all design specifications but having only half the mcmillan degree of conventional h sup spl infin controllers we demonstrate the advantage of the proposed method

multivariate spectral estimation based on the concept of optimal prediction in this technical note we deal with a spectrum approximation problem arising in three like multivariate spectral estimation approaches the solution to the problem minimizes a suitable divergence index with respect to an a priori spectral density we derive a new divergence family between multivariate spectral densities which takes root in the prediction theory under mild assumptions on the a priori spectral density the approximation problem based on this new divergence family admits a family of solutions moreover an upper bound on the complexity degree of these solutions is provided

a new family of high resolution multivariate spectral estimators in this paper we extend the beta divergence family to multivariate power spectral densities similarly to the scalar case we show that it smoothly connects the multivariate kullback leibler divergence with the multivariate itakura saito distance we successively study a spectrum approximation problem based on the beta divergence family which is related to a multivariate extension of the three spectral estimation technique it is then possible to characterize a family of solutions to the problem an upper bound on the complexity of these solutions will also be provided finally we will show that the most suitable solution of this family depends on the specific features required from the estimation problem

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

a convex optimization approach to arma modeling we formulate a convex optimization problem for approximating any given spectral density with a rational one having a prescribed number of poles and zeros n poles and m zeros inside the unit disc and their conjugates the approximation utilizes the kullback leibler divergence as a distance measure the stationarity condition for optimality requires that the approximant matches n 1 covariance moments of the given power spectrum and m cepstral moments of the corresponding logarithm although the latter with possible slack the solution coincides with one derived by byrnes enqvist and lindquist who addressed directly the question of covariance and cepstral matching thus the present paper provides an approximation theoretic justification of such a problem since the approximation requires only moments of spectral densities and of their logarithms it can also be used for system identification

spectral estimation via selective harmonic amplification the state covariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter and also imposes a generalized interpolation constraint on the power spectrum of the input process this algebraic property and the relationship between state covariance and the power spectrum of the input allow the use of matrix pencils and analytic interpolation theory for spectral analysis several algorithms for spectral estimation are developed with resolution higher than state of the art

an interpretation of the dual problem of the three like approaches spectral estimation can be performed using the so called three like approach such method leads to a convex optimization problem whose solution is characterized through its dual problem in this paper we show that the dual problem can be seen as a new parametric spectral estimation problem this interpretation implies that the three like solution is optimal in terms of closeness to the correlogram over a certain parametric class of spectral densities describing arma models enriching in this way its meaningfulness

resonant periodic orbits and the semiclassical energy spectrum the semiclassical density of states depends according to the periodic orbit sum formula on the linear stability of the orbits this means however that contributions from the marginally stable or resonant orbits which necessarily accompany stable ones diverge unphysically the remedy for a system of two degrees of freedom is found to lie in the classical non linear normal forms for periodic orbits which describe how satellite periodic orbits coalesce with the central one as resonance is approached in to 0 through these forms the resonant contributions are expressed as diffraction integrals the first few being diffraction catastrophes uniformly valid in in and h cross and finite even for in to 0 provided h cross 0 an extension is proposed to incorporate jointly multiple resonances found in repetitions of orbits

identifiability and well posedness of shaping filter parameterizations a global analysis approach in this paper we study the well posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients covariance lags or markov parameters for example we determine whether there exists a shaping filter with a prescribed window of markov parameters and a prescribed window of covariance lags we show that several such problems are well posed in the sense of hadamard that is one can prove existence uniqueness identifiability and continuous dependence of the model on the measurements our starting point is the global analysis of linear systems where one studies an entire class of systems or models as a whole and where one views measurements such as covariance lags and cepstral coefficients or markov parameters from data as functions on the entire class this enables one to pose such problems in a way that tools from calculus optimization geometry and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm independent fashion in this language we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona fide coordinate system on the space of shaping filters thereby establishing existence uniqueness and smooth dependence of the model parameters on the measurements from data

the structure of state covariances and its relation to the power spectrum of the input we study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances and identify the structure of positive semidefinite matrices that qualify as state covariances of the filter this structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices we then characterize all input power spectra consistent with any specific state covariance the parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous but not equivalent to a matricial nehari problem

on the well posedness of multivariate spectrum approximation and convergence of high resolution spectral estimators abstract in this paper we establish the well posedness of the generalized moment problems recently studied by byrnes georgiou lindquist and coworkers and by ferrante pavon ramponi we then apply these continuity results to prove the almost sure convergence of a sequence of high resolution spectral estimators indexed by the sample size

the interpolation problem with a degree constraint the author previously 1983 1987 showed that there is a correspondence between nonnegative hermitian trigonometric polynomials of degree spl les n and solutions to the standard nevanlinna pick caratheodory interpolation problem with n 1 constraints which are rational and also of degree spl les n it was conjectured that the correspondence under suitable normalization is bijective and thereby that it results in a complete parametrization of rational solutions of degree spl les n the conjecture was proven by byrnes et al 1995 along with a detailed study of this parametrization however byrnes et al used a slightly restrictive assumption that the trigonometric polynomials are positive and accordingly the corresponding solutions have positive real part the purpose of the present note is to extend the result to the case of nonnegative trigonometric polynomials as well we present the arguments in the context of the general nevanlinna pick caratheodory fejer interpolation

a rule for quantizing chaos the authors find a real function delta e whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories delta e is a finite sum over combinations of classical periodic orbits it is obtained from gutzwiller s infinite and divergent sum 1982 representing the spectral density in terms of periodic orbits by means of a resummation conjectured by analogy with a derivation of the riemann siegel formula for the riemann zeros they assess the practicality of the quantization condition

a maximum entropy enhancement for a family of high resolution spectral estimators structured covariances occurring in spectral analysis filtering and identification need to be estimated from a finite observation record the corresponding sample covariance usually fails to possess the required structure this is the case for instance in the byrnes georgiou lindquist three like tunable high resolution spectral estimators there the output covariance of a linear filter is needed to initialize the spectral estimation technique the sample covariance estimate however is usually not compatible with the filter in this paper we present a new systematic way to overcome this difficulty the new estimate is obtained by solving an ancillary problem with an entropic type criterion extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

realization of power spectra from partial covariance sequences in this paper we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process the standing assumption is the availability of an incomplete partial sequence of covariance samples of the process we study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples the classical theory of orthogonal polynomials with respect to the unit circle and the theory of moments have been utilized extensively in a variety of engineering problems including the one we are dealing with these have been known to provide a unifying framework for a variety of current spectral estimation techniques maximum entropy method pisarenko s harmonic decomposition etc in this work we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence and equal to the dimension of the maximum entropy realization the me solution is a point in this set other points correspond to pole zero models a general formula is obtained for recursively updated pole zero models of dimension increasing with the data record information about the zeros is obtained from the asymptotic behavior of the partial autocorrelation coefficients our approach combines techniques from the theory of orthogonal polynomials on the unit circle the theory of moments and also techniques from degree theory topology our objective is to develop a theory that will provide a frame for constructing recursively pole zero realizations of increasing dimension

solution of the general moment problem via a one parameter imbedding this paper presents a computational theory for the general scalar moment problem the formalism is sufficiently general to encompass problems in sensor arrays with arbitrary geometry and dynamics and in nonuniform multidimensional sampling given a finite set of moments the theory provides a test for the existence of a positive measure which is consistent with such data at the same time the theory also provides a characterization of all such consistent positive measures it should be noted that classical results e g in the theory of the trigonometric moment problem hamburger stieljes nevanlinna pick interpolation etc are not applicable to the general setting sought herein where there is no natural shift operator in the space spanned by the integration kernels the centerpiece of the theory is a differential equation which depends on the given finite set of moments and on an arbitrary positive function spl psi which plays the role of a free parameter the differential equation has an exponentially attractive point of equilibrium if and only if there exists a consistent positive measure for each spl psi the fixed point determines a corresponding measure suitable choice of spl psi allows recovering any measure which is consistent with the data the fixed point of the differential equation corresponds to an extremum of an entropy like functional and the differential equation is constructed via an appropriate homotopy that follows changes in the lagrange multipliers from a convenient starting value to a value for the multipliers that corresponds to the given moments

further results on the convergence of the pavon ferrante algorithm for spectral estimation in this paper we provide a detailed analysis of the global convergence properties of an extensively studied and extremely effective fixed point algorithm for the kullback leibler approximation of spectral densities proposed by pavon and ferrante in their paper on the georgiou lindquist approach to constrained kullback leibler approximation of spectral densities our main result states that the algorithm globally converges to one of its fixed points

how chaotic is the stadium billiard a semiclassical analysis the impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described semiclassically by the gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits i show that this belief is erroneous the gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the green s function semiclassical contributions to the trace show an dependent transition from hard chaos to integrable behaviour for trajectories approaching the bouncing ball orbits a whole region in phase space surrounding the marginal stable family acts semiclassically like a stable island with boundaries being explicitly dependent the localized bouncing ball states found in the billiard derive from this semiclassically stable island the bouncing ball orbits themselves however do not contribute to individual eigenvalues in the spectrum an ebk like quantization of the regular bouncing ball eigenstates in the stadium can be derived the stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics the behaviour is generically found at the border of classically stable islands in systems with a mixed phase space structure

time and spectral domain relative entropy a new approach to multivariate spectral estimation the concept of spectral relative entropy rate is introduced for jointly stationary gaussian processes using classical information theoretic results we establish a remarkable connection between time and spectral domain relative entropy rates this naturally leads to a new spectral estimation technique where a multivariate version of the itakura saito distance is employed it may be viewed as an extension of the approach called three introduced by byrnes georgiou and lindquist in 2000 which in turn followed in the footsteps of the burg jaynes maximum entropy method spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate the corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework indeed it is equal to the one featured by three in the scalar case the solution is computed via a globally convergent matricial newton type algorithm simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks especially in the case of short data records

important moments in systems and control the moment problem matured from its various special forms in the late 19th and early 20th centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields in particular the theory of systems and control is no exception where the applications have historically been to circuit theory optimal control robust control signal processing spectral estimation stochastic realization theory and the use of the moments of a probability density many of these applications are also still works in progress in this paper we consider the generalized moment problem expressed in terms of a basis of a finite dimensional subspace p of the banach space c a b and a positive sequence c but with a new wrinkle inspired by the applications to systems and control we seek to parameterize solutions which are positive rational measures in a suitably generalized sense our parameterization is given in terms of smooth objects in particular the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a euclidean space and which is the domain of some canonically defined functions the analysis of these functions and related maps yields interesting corollaries for the moment problem and its applications which we compare to those in the recent literature and which play a crucial role in part of our proof

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

determination of resonance spectra for bound chaotic systems we consider the computation of the eigenvalues of the evolution operator the resonance spectrum by means of the zeros of a zeta function in particular we address the problems of applying this formalism to bound chaotic systems caused by e g intermittency and non completeness of the symbolic dynamics for bound intermittent systems we derive an approximation of the zeta function with the aid of this zeta function it is argued that bound systems with long time tails have branch cuts in the zeta function and traces of the evolution operator approaching unity as a power law we also show that the dominant time scale can be much longer than the period of the shortest periodic orbit as is for example the case for the hyperbola billiard isolated zeros of the zeta function for the hyperbola billiard are evaluated by means of a cycle expansion crucial for the success of this approach is the identification of a sequence of periodic orbit responsible for a logarithmic branch cut in the zeta function semiclassical implications are briefly discussed