stability of statistical properties in two dimensional piecewise hyperbolic maps we investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions we accomplish such a program in the case of two dimensional maps with uniformly bounded second derivative but we are confident that the present approach can be successful in much greater generality we hope including higher dimensional billiards for the class of systems at hand we obtain a complete description of the srb measures their statistical properties and their stability with respect to many types of perturbations including deterministic and random perturbations and holes

on the essential spectrum of the transfer operator for expanding markov maps the essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail to this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inc k is a disk whose radius is related to the free energy of the liapunov exponent

spectral decomposition of the renyi map the authors construct a generalized spectral decomposition of the frobenius perron operator of the general beta adic renyi map using a general iterative operator method applicable in principle to any mixing dynamical system they also explicitly define appropriate rigged hilbert spaces which provide mathematical meaning to the formally obtained spectral decomposition the explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the frobenius perron operator decreases as the smoothness of the domain increases the reason for the change of the spectrum from the unit disk to isolated eigenvalues is the existence of coherent states of the frobenius perron operator which are not infinitely differentiable

a hybrid method for calculation of ruelle pollicott resonances we present a numerical method for calculation of ruelle pollicott resonances of dynamical systems it constructs an effective coarse grained propagator by considering the correlations of multiple observables over multiple time steps the method is compared to the usual approaches to the example of the perturbed cat map and is shown to be numerically efficient and robust

spectral analysis and an area preserving extension of a piecewise linear intermittent map we investigate the spectral properties of a one dimensional piecewise linear intermittent map which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point we explicitly derive generalized eigenvalues and eigenfunctions of the frobenius perron operator of the map for classes of observables and piecewise constant initial densities and it is found that the frobenius perron operator has two simple real eigenvalues 1 and ensuremath lambda d ensuremath 1 0 and a continuous spectrum on the real line 0 1 from these spectral properties we also found that this system exhibits a power law decay of correlations this analytical result is found to be in a good agreement with numerical simulations moreover the system can be extended to an area preserving invertible map defined on the unit square this extended system is similar to the baker transformation but does not satisfy hyperbolicity a relation between this area preserving map and a billiard system is also discussed

an analogue of the prime number theorem for closed orbits of axiom a flows for an axiom a flow restricted to a basic set we extend the zeta function to an open set containing w s h where h is the topological entropy this enables us to give an asymptotic formula for the number of closed orbits by adapting the wiener ikehara proof of the prime number theorem

exact correlation functions for dual unitary lattice models in 1 1 dimensions we consider a class of quantum lattice models in 1 1 dimensions represented as local quantum circuits that enjoy a particular dual unitarity property in essence this property ensures that both the evolution in time and that in space are given in terms of unitary transfer matrices we show that for this class of circuits generically nonintegrable one can compute explicitly all dynamical correlations of local observables our result is exact nonpertubative and holds for any dimension d of the local hilbert space in the minimal case of qubits d 2 we also present a classification of all dual unitary circuits which allows us to single out a number of distinct classes for the behavior of the dynamical correlations we find noninteracting classes where all correlations are preserved the ergodic and mixing one where all correlations decay and interestingly also classes that are both interacting and nonergodic

decay of correlations in one and two dimensions an exposition of some methods of proving exponential stretched exponential decay of correlations is given one dimensional strictly hyperbolic and quadratic maps and two dimensional piecewise smooth uniformly hyperbolic maps are considered the emphasis is on the fundamental constructions of the markov sieve method due to bunimovich chernov sinai and those of liverani s hilbert metric method

periodic orbit expansions for classical smooth flows the authors derive a generalized selberg type zeta function for a smooth deterministic flow which relates the spectrum of an evolution operator to the periodic orbits of the flow this relation is a classical analogue of the quantum trace formulae and selberg type zeta functions

gibbs measures and power spectra for type i intermittent maps abstract gibbs measures of dynamical systems are investigated using the corresponding correlation function and power spectrum the method is shown to be equivalent to the concept of order q power spectra which has been developed previously a technique for the computation of these spectra is presented as an application we specially focus on the treatment of type i intermittent model maps and investigate their phase transition in the vicinity of the phase transition point the order q power spectra are shown to obey some typical scaling relation which is captured neither by the topological pressure nor by the ordinary power spectrum

relative entropies for kinetic equations in bounded domains irreversibility stationary solutions uniqueness the relative entropy method describes the irreversibility of the vlasov poisson and vlasov boltzmann poisson systems in bounded domains with incoming boundary conditions uniform in time estimates are deduced from the entropy in some cases these estimates are sufficient to prove the convergence of the solution to a unique stationary solution as time goes to infinity the method is also used to analyse other types of boundary conditions such as mass and energy preserving diffuse reflection boundary conditions and to prove the uniqueness of stationary solutions for some special collision terms

sharp vanishing order of solutions to stationary schrodinger equations on carnot groups of arbitrary step abstract based on a variant of the frequency function approach of almgren 1 under appropriate assumptions we establish an optimal upper bound on the vanishing order of solutions to stationary schrodinger equations associated to sub laplacian on carnot groups of arbitrary step such a bound provides a quantitative form of strong unique continuation and can be thought of as a subelliptic analogue of the recent results obtained by bakri 3 and zhu 27 for the standard laplacian

computing the topological pressure for intermittent maps the topological pressure is obtained as the leading zero of a dynamical zeta function we consider the problem of computing this zero when it is close to a singularity in particular we study a family of intermittent maps which we argue exhibit a branch point singularity in its zeta functions the convergence of the cycle expansion close to this point is extremely slow to deal with this problem we consider a resummation of the cycle expansion the idea is quite similar to that of pade approximants but the ansatz is a generalized series expansion around the branch point rather than a rational function the improvement on convergence of the leading zero is considerable we also briefly discuss the relation between correlation decay and the nature of the branch point

ruelle resonances in kicked quantum spin chain abstract we study exponential decay of high temperature time correlation functions in a non integrable quantum spin chain problem namely ising spin 1 2 chain kicked with tilted homogeneous magnetic field for this purpose we define a master propagator over a suitable banach space of quantum observables quantum many body analogue of perron frobenius operator whose leading eigenvalue determines the asymptotic decay of correlations this is demonstrated with explicit calculation for which a fast algorithm for the construction of the master propagator is developed

optimal three spheres inequality at the boundary for the kirchhoff love plate s equation with dirichlet conditions we prove a three spheres inequality with optimal exponent at the boundary for solutions to the kirchhoff love plate s equation satisfying homogeneous dirichlet conditions this result implies the strong unique continuation property at the boundary sucpb our approach is based on the method of carleman estimates and involves the construction of an ad hoc conformal mapping preserving the structure of the operator and the employment of a suitable reflection of the solution with respect to the flattened boundary which ensures the needed regularity of the extended solution to the authors knowledge this is the first nontrivial sucpb result for fourth order equations with a bi laplacian principal part

on the rate of convergence to equilibrium in one dimensional systems we determine the essential spectral radius of the perron frobenius operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such one dimensional systems to the greatest eigenvalues of generalized perron frobenius operators of the transformations operators which yield singular invariant measures

lyapunov exponents and anomalous diffusion of a lorentz gas with infinite horizon using approximate zeta functions we compute the lyapunov exponent the generalized lyapunov exponents and the diffusion constant for a lorentz gas on a square lattice thus having infinite horizon approximate zeta functions written in terms of probabilities rather than periodic orbits are used in order to avoid the convergence problems of cycle expansions the emphasis is on the relation between the analytic structure of the zeta function where a branch cut plays an important role and the asymptotic dynamics of the system the lyapunov exponent for the corresponding map agrees with the conjectured limit map 2 log r c o r and we derive an approximate value for the constantc in good agreement with numerical simulations we also find a diverging diffusion constantd t logt and a phase transition for the generalized lyapunov exponents

locating resonances for axiom a dynamical systems for a class of differentiable dynamical systems called axiom a systems it has been shown by pollicott and the author that correlation functions have fourier transforms which are meromorphic in a strip the poles or resonances are however not easy to locate this note reviews the results which are known and discusses a simple model where the position of resonances can be estimated the effect of noise is also discussed

relaxation in finite and isolated classical systems an extension of onsager s regression hypothesis in order to derive the reciprocity relations onsager formulated a relation between thermal equilibrium fluctuations and relaxation widely known as regression hypothesis it is shown in the present work how such a relation can be extended to finite and isolated classical systems this extension is derived from the fluctuation dissipation theorem for the microcanonical ensemble the results are exemplified with a nonintegrable system in order to motivate possible applications to dynamical systems and statistical mechanics of finite systems

approach to equilibrium for a class of random quantum models of infinite range we consider random generalizations of a quantum model of infinite range introduced by emch and radin the generalizations allow a neat extension from the class l 1 of absolutely summable lattice potentials to the optimal class l 2 of square summable potentials first considered by khanin and sinai and generalised by van enter and van hemmen the approach to equilibrium in the case of a gaussian distribution is proved to be faster than for a bernoulli distribution for both short range and long range lattice potentials while exponential decay to equilibrium is excluded in the nonrandom l 1 case it is proved to occur for both short and long range potentials for gaussian distributions and for potentials of class l 2 in the bernoulli case open problems are discussed

hierarchical resolution of power spectra abstract we identify the basic ingredients determining the structure of the power spectra of non linear dynamical systems in a hierarchical order of importance the analysis performed with the help of symbolic methods shows that dynamical invariants such as topological and metric properties of the symbolic orbits explain the main qualitative features of the spectra whereas the coordinate dependent values of the observable itself represent a less relevant contribution consideration of simple dynamical models with increasing number of topological transition rules evidences the formation of coherent structures peaks and explains their position and size by constructing the parse tree of the allowed symbolic itineraries it is possible to estimate conditional probabilities by considering orbits belonging to adjacent tree levels accordingly a markov transition matrix is obtained for each level l and is used to generate signals with statistical properties which approximate those of the actual one increasingly better for l a considerable improvement is achieved by recoding the original signal in terms of variable length words and by re applying the above procedure to the transformed signal which is equivalent to a renormalization operation of the associated dynamical map the accuracy of the estimates is directly related to the convergence of the scaling function for the conditional probabilities analytic results are presented for the simplest five markov models arising from piecewise linear continuous one dimensional maps numerical studies have been performed for the logistic and henon maps and for the lorenz system

quantitative uniqueness for elliptic equations at the boundary of c1 dini domains abstract based on a variant of the frequency function approach of almgren we establish an optimal upper bound on the vanishing order of solutions to variable coefficient schrodinger equations at a portion of the boundary of a c 1 dini domain such bound provides a quantitative form of strong unique continuation at the boundary it can be thought of as a boundary analogue of an interior result recently obtained by bakri and zhu for the standard laplacian

correlation decay and return time statistics abstract we address the general problem of possible modes of correlation decay for low dimensional dynamical systems in particular we focus on systems lacking full hyperbolicity where typically power laws occur

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

locating pollicott ruelle resonances in chaotic dynamical systems a class of numerical schemes a class of numerical methods to determine pollicott ruelle resonances in chaotic dynamical systems is proposed this is achieved by relating some existing procedures that make use of the pade approximants and interpolating exponentials to both the memory function techniques used in the theory of relaxation and the filter diagonalization method used in the harmonic inversion of time correlation functions this relationship leads to a theoretical framework in which all these methods become equivalent and which allows for new and improved numerical schemes

ruelle resonances in quantum many body dynamics we define a quantum perron frobenius master operator over a suitable normed space of translationally invariant states adjoint to the quasi local c algebra of quantum lattice gasses e g spin chains whose spectrum determines the exponents of decay of time correlation functions the theoretical ideas are applied to a generic example of kicked ising spin 1 2 chains we show that the chaotic eigenmodes corresponding to leading eigenvalue resonances have fractal structure in the basis of local operators

long range correlation properties of area preserving chaotic systems abstract this article shows numerically that the variance of the stretching exponents for two dimensional chaotic area preserving systems grows asymptotically as a linear function of time although an intermediate anomalous power law scaling may occur this implies that the autocorrelation function of the stretching exponents is integrable this result is a generic property of 2 d mixing systems generated by diffeomorphisms the physical significance of the non persistent anomalous behavior in the decay of fluctuations is briefly addressed

c1 domains and unique continuation at the boundary it is shown that the square of a nonconstant harmonic function u that either vanishes continuously on an open subset v contained in the boundary of a dini domain or whose normal derivative vanishes on an open subset v in the boundary of a c1 1 domain in d satisfies the doubling property with respect to balls centered at points q v under any of the above conditions the module of the gradient of u is a b2 d weight when restricted to v and the hausdorff dimension of the set of points q v u q 0 is less than or equal to d 2 these results are generalized to solutions to elliptic operators with lipschitz second order coefficients and bounded coefficients in the lower order terms 1997 john wiley sons inc

resonances in chaotic dynamics we present a discussion and some numerical results on the actual possibility of making accessible by numerical techniques the complex singularities of the power spectrum resonances for a chaotic signal henon s transformation is investigated in detail showing that the position of the leading resonance in the complex frequency plane determines the kind of mixing rate in the time evolution

linear stability and instability of relativistic vlasov maxwell systems we consider the linear stability problem for a symmetric equilibrium of the relativistic vlasov maxwell rvm system for an equilibrium whose distribution function depends monotonically on the particle energy we obtain a sharp linear stability criterion the growing mode is proved to be purely growing and we get a sharp estimate of the maximal growth rate in this paper we specifically treat the periodic 1 d case and the 3d whole space case with cylindrical symmetry we explicitly illustrate using the linear stability criterion in the 1 d case several stable and unstable examples 2006 wiley periodicals inc

spectral characterization of anomalous diffusion of a periodic piecewise linear intermittent map abstract for a piecewise linear version of the periodic map with anomalous diffusion the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the frobenius perron fp operator are explicitly derived the evolution of the averages is controlled by real eigenvalues as well as continuous spectra terminating at the unit circle appropriate scaling limits are shown to give a normal diffusion if the reduced map is in the stationary regime with normal fluctuations a levy flight if the reduced map is in the stationary regime with levy type fluctuations and a transport of ballistic type if the reduced map is in the non stationary regime

perturbations of geodesic flows on surface of constant negative curvature and their mixing properties we consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature we find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non perturbed ones one condition says that the hamilton jacobi equation introduced in this work for the conjugation problem should admit a solution as a formal power series not necessarily convergent in the perturbation parameter the alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem the relation with the similar problems arising in the kam theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed as a byproduct of our analysis we obtain some results on the livscic guillemin kazhdan equation and on the fourier series for the sl 2 group we also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property with respect to the geodesic flow and to the invariant volume measure which is exponential with a universal exponent independent on the particular function equal to the curvature of the surface divided by 2 this result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow in this case we find that the decay rate is the inverse of the time up to logarithms

resonances for axiom bf a flows etant donne un flot d axiome a sur m et des fonctions lisses b c m r on montre que la fonction de correlation en temps bc pour un etat de gibbs a une transformee de fourier bc mesomorphe dans une bande on etudie les residus des poles de bc

leading pollicott ruelle resonances for chaotic area preserving maps recent investigations in nonlinear sciences show that not only hyperbolic but also mixed dynamical systems may exhibit exponential relaxation in the chaotic regime the relaxation rates which lead the decay of probability distributions and correlation functions are related to the classical evolution resolvent perron frobenius operator pole logarithm the so called pollicott ruelle resonances in this brief report the leading pollicott ruelle resonances are calculated analytically for a general class of area preserving maps besides the leading resonances related to the diffusive modes of momentum dynamics slow rate we also calculate the leading faster rate related to the angular correlations the analytical results are compared to the existing results in the literature

markov shift in non commutative probability abstract we consider a class of quantum dissipative semigroup on a von neumann algebra which admits a normal invariant state we investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical markov shift in case the normal invariant state is also faithful we also extend the notion of quantum detailed balance introduced by frigerio gorini and prove that forward weak markov process and backward weak markov process are equivalent by an anti unitary operator

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

classical nonlinear response of a chaotic system ii langevin dynamics and spectral decomposition the spectrum of a strongly chaotic system consists of discrete complex ruelle pollicott rp resonances we interpret the rp resonances as eigenstates and eigenvalues of the fokker planck operator obtained by adding an infinitesimal diffusion term to the first order liouville operator we demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise for the second order response function we establish an equivalence of the spectral decomposition in the limit of vanishing noise and the long time asymptotic expansion in the deterministic case

a spectral characterization of kms states let be a state on ac dynamical system mathfrak a mathbb r alpha for each of the following properties of 1 is k ms with respect to for some given 0 2 is either a kms state or a ground state necessary and sufficient conditions are given involving only the spectral subspaces of mathfrak a associated with the results provide a new insight in the concept of passivity introduced by w pusz and s l woronowicz

a sharp stability criterion for the vlasov maxwell system we consider the linear stability problem for a 3d cylindrically symmetric equilibrium of the relativistic vlasov maxwell system that describes a collisionless plasma for an equilibrium whose distribution function decreases monotonically with the particle energy we obtained a linear stability criterion in our previous paper 24 here we prove that this criterion is sharp that is there would otherwise be an exponentially growing solution to the linearized system we also treat the considerably simpler periodic 1 frac 1 2 d case the new formulation introduced here is applicable as well to the non relativistic case to other symmetries and to general equilibria

periodic orbits as the skeleton classical and quantum chaos a description of a low dimensional deterministic chaotic system in terms of unstable periodic orbits cycles is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos comparable to the familiar perturbation expansions for nearly integrable systems the infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a selberg product form dominated by the short cycles and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits methods for implementing this computation for finite subshift dynamics are introduced

r adic one dimensional maps and the euler summation formula for the r adic one dimensional maps the author explicitly constructs the decaying eigenstates and adjoint eigenstates associated with the ruelle resonances it is shown that the eigenfunctions of the corresponding frobenius perron operator are the well known bernoulli polynomials the adjoint eigendistributions are obtained as derivatives of the dirac distributions at the end points of the unit interval the resulting expansion of the initial density in terms of the decaying eigenstates is given by the euler summation formula

gentle perturbations of the free bose gas i it is demonstrated that the thermal structure of the noncritical free bose gas is completely described by certain periodic generalized gaussian stochastic process or equivalently by a certain periodic generalized gaussian random field elementary properties of this gaussian stochastic thermal structure are established gentle perturbations of several types of the free thermal stochastic structure are studied in particular new models of non gaussian thermal structures are constructed and a new functional integral representation of the corresponding euclidean time green functions is obtained rigorously

statistical approach to nonhyperbolic chaotic systems spectral properties of the evolution operator for probability densities are obtained for unimodal maps for which all periodic orbits are unstable and the lyapunov exponent calculated from the first iterate of the critical point converges to a positive constant the method is applied to the logistic map both for parameter values at which finite markov partitions can be found as well as for more typical parameter values a universal behavior is found for the spectral gap in the period doubling inverse cascade of chaotic band merging bifurcations full agreement with numerical simulation is obtained textcopyright 1996 the american physical society

on the spectra of randomly perturbed expanding maps we consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing we do this by proving the robustness of the spectra of their perron frobenius operators

generalized spectral decomposition of the β adic baker s transformation and intrinsic irreversibility abstract we construct a generalized spectral decomposition of the frobenius perron operator of the adic baker s transformation using a general iterative operator method applicable in principle for any mixing dynamical system the eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one we explicitly define appropriate generalized function spaces which provide mathematical meaning to the formally obtained spectral decomposition the unitary frobenius perron evolution of densities when extended to the generalized function spaces splits into two semigroups one decaying in the future and the other in the past this split which reflects the asymptotic evolution of the forward and backward k partitions shows the instrinsic irreversibility of the baker s transformation

action principle in nonequilibrium statistical dynamics we introduce a variational method for approximating distribution functions of dynamics with a liouville operator l in terms of a nonequilibriumaction functional for two independent left and right trial states the method is valid for deterministic or stochastic markov dynamics and for stationary or time dependent distributions a practical rayleigh ritz procedure is advanced whose inputs are a finitely parametrized ansatz for the trial states leading to a parametric action for their evolution the euler lagrange equations of the action principle are hamiltonian in form generally noncanonical this permits a simple identification of fixed points as critical points of the parametric hamiltonian we also establish a variational principle for low order statistics such as mean values and correlation functions by means of the least effective action the latter is a functional of the given variable which is positive and convex as a consequence of h older realizability inequalities its value measures the cost for a fluctuation from the average to occur and in a weak noise limit it reduces to the onsager machlup action in general the effective action is shown to arise from the nonequilibrium action functional by a constrained variation this result provides a rayleigh ritz scheme for calculating just the desired low order statistics with internal consistency checks less demanding than for the full distribution textcopyright 1996 the american physical society

generalized markov coarse graining and spectral decompositions of chaotic piecewise linear maps spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear markov maps and a large class of repellers the eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal markov partition equal the reciprocals of the zeros of the ruelle zeta functions the logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing the space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit s so that local behavior of observables can be considered the new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit s finally arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed

strange objects in the complex plane julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems technically they are the closure of the set of unstable cycles of analytic maps physically they are sets of points which lead to chaotic behavior the mapf z z2 p is analyzed for smallp where the julia set is a closed curve and for largep where the julia set is completely disconnected in both cases the hausdorff dimension is calculated in perturbation theory and numerically an expression for the rate at which points escape from the neighborhood of the julia set is derived and tested in a numerical simulation of the escape

stable magnetic equilibria in collisionless plasmas there are many steady states in collisionless plasmas their dynamical stability has been one of the main interests in the study of plasmas in one dimensional space we discover some new types of stable equilibria by constructing new invariants to the vlasov maxwell system these stable equilibria include new flat tail solutions that connect different densities pressures and strengths of constant magnetic fields at x 1997 john wiley sons inc

leading ruelle resonances of chaotic maps the leading ruelle resonances of typical chaotic maps the perturbed cat map and the standard map are calculated by variation it is found that excluding the resonance associated with the invariant density the next subleading resonances are approximately the roots of the equation z 4 ensuremath gamma where ensuremath gamma is a positive number that characterizes the amount of stochasticity of the map the results are verified by numerical computations and the implications to the form factor of the corresponding quantum maps are discussed

some geometrical models of chaotic dynamics the free motion of a particle on a surface of constant negative curvature a pseudosphere was one of the first models of chaotic motion it became the prototype for the theory of hyperbolic systems developed by bowen and sinai in these models geometry suggests a symbolic coding which already exhibits fully chaotic behaviour one can return to these models to seek possible manifestations of quantum chaos here the mathematical technique is harmonic analysis on hyperbolic space chaotic behaviour seems to appear both in the behaviour of individual eigenfunctions and in the sequence of spectral values

on the effects of an uncertainty on the evolution law in dynamical systems abstract we show that a small perturbation of the evolution law of chaotic dynamical systems is practically equivalent to the finite precision on the knowledge of the initial condition in spite of the short predictability time t p on a single trajectory we found that the statistical properties are not sensitive to small changes in the evolution law or initial probability distribution this feature holds also for correlation functions at a delay larger than t p we discuss the roundoff effects on integrable systems

meromorphic extensions of generalised zeta functions in this paper we give a full description of the spectrum of the ruelle perron frobenius operator acting on the banach space of holder continuous functions on a subshift of finite type theorem 1 these results are then used to extend the meromorphic domain of generalised zeta functions theorem 2 the most important application of these results is to the domain of the smale zeta function for axiom a flows theorem 3 in the course of this paper we settle questions raised by ruelle and sunada

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

mixing rates and exterior forms in chaotic systems the authors propose a numerical method for extracting the spectrum of decay rates of time correlations in chaotic dynamical systems the sum of the first p decay rates is related to the asymptotic behaviour of the time correlation of suitable exterior forms of order p the method is applied to maps of the interval and to the henon map

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m