quasi multiplicativity of typical cocycles we show that typical in the sense of bonatti and viana ergod theory dyn syst 24 5 1295 1330 2004 and avila and viana port math 64 311 376 2007 holder and fiber bunched text gl d mathbb r valued cocycles over subshifts of finite type are uniformly quasi multiplicative with respect to all singular value potentials we prove the continuity of the singular value pressure and its corresponding necessarily unique equilibrium state for such cocycles and apply this result to repellers moreover we show that the pointwise lyapunov spectrum is closed and convex and establish partial multifractal analysis on the level sets of pointwise lyapunov exponents for such cocycles

analyticity of smooth eigenfunctions and spectral analysis of the gauss map we provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form uf x a x y f b x y dy acting on functions f u user1 v to mathbb c evolution operators of one dimensional dynamical systems and markov processes have this form we estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the frobenius perron operator of the continuous fraction gauss map we prove that any infinitely differentiable eigenfunction f of this frobenius perron operator corresponding to a non zero eigenvalue admits a unique analytic extension to the set mathbb c backslash infty 1 analyzing the spectrum of the frobenius perron operator in spaces of smooth functions we extend significantly the domain of validity of the mayer and ropstorff asymptotic formula for the decay of correlations of the gauss map

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

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

simple deterministic dynamical systems with fractal diffusion coefficients we analyze a simple model of deterministic diffusion the model consists of a one dimensional array of scatterers with moving point particles the particles move from one scatterer to the next according to a piecewise linear expanding deterministic map on unit intervals the microscopic chaotic scattering process of the map can be changed by a control parameter the macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter we calculate the diffusion coefficent and the largest eigenmodes of the system by using markov partitions and by solving the eigenvalue problems of respective topological transition matrices for different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter we provide qualitative and quantitative arguments to explain features of this fractal structure

expanding direction of the period doubling operator we prove that the period doubling operator has an expanding direction at the fixed point we use the induced operator a perron frobenius type operator to study the linearization of the period doubling operator at its fixed point we then use a sequence of linear operators with finite ranks to study this induced operator the proof is constructive one can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point

on thermodynamics of rational mapsi negative spectrum we study the pressure spectrum p t of the maximal measure for arbitrary rational maps we also consider its modified version which is defined by means of the variational principle with respect to non atomic invariant measures it is shown that for negative values of t the modified spectrum has all major features of the hyperbolic case analyticity the existence of a spectral gap for the corresponding transfer operator rigidity properties etc the spectrum p t can be computed in terms of their legendre transforms are the hausdorff and the box counting dimension spectra of the maximal measure respectively this work is closely related to a paper 32 by d ruelle

the algebra of harmonic functions for a matrix valued transfer operator abstract we analyze matrix valued transfer operators we prove that the fixed points of transfer operators form a finite dimensional c algebra for matrix weights satisfying a low pass condition we identify the minimal projections in this algebra as correlations of scaling functions i e limits of cascade algorithms

new families of fourier eigenfunctions for steerable filtering a new diadic family of eigenfunctions of the 2 d fourier transform has been discovered specifically new wavelets are derived by steering the elongated hermite gauss filters with respect to rotations thus obtaining a natural generalization of the laguerre gauss harmonics interestingly these functions are also proportional to their 2 d fourier transform their analytical expression is provided in a compact and treatable form by means of a new ad hoc matrix notation in which the cases of even and odd orders of the hermite polynomials are unified moreover these functions can be efficiently implemented by means of a recursive formula that is derived in this paper the proposed filters are applied to the problem of gradient estimation to improve the theoretical canny tradeoff of position accuracy versus noise rejection that occurs in edge detection experimental results show considerable improvements in using the new wavelets over both isotropic gaussian derivatives and other elongated steerable filters more recently introduced finally being the proposed wavelets a set of fourier eigenfunctions they can be of interest in other fields of science such as optics and quantum mechanics

measures of controllability and observability and residues the relationship between two measures of controllability and observability is derived given a linear time invariant system both of the measures are based on angles between various subspaces of the state space it is shown that these measures can be used to bound the norm of the residues of the transfer function

on the distribution of zeros of a ruelle zeta function we study the limit distribution of zeros of a ruelle function for the dynamical systemz z2 c whenc is real andc 2 0 and apply the results to the correlation functions of this dynamical system

orthonormal expansions of invariant densities for expanding maps abstract we give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding c markov maps this is a classical problem with one of the standard approaches being ulam s method problems in modern mathematics interscience new york 1960 of phase space discretisation our method hinges instead on the expansion of the density function with respect to an l 2 orthonormal basis and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map the efficiency of the method and its extension to c k expanding maps are also discussed

unwrapping eigenfunctions to discover the geometry of almost invariant sets in hyperbolic maps the numerical approximation of perron frobenius operators allows efficient determination of the physical invariant measure of chaotic dynamical systems as a fixed point of the operator eigenfunctions of the perron frobenius operator corresponding to large subunit eigenvalues have been shown to describe almost invariant dynamics in one dimensional expanding maps we extend these ideas to hyperbolic maps in higher dimensions while the eigendistributions of the operator are relatively uninformative applying a new procedure called unwrapping to regularised versions of the eigendistributions clearly reveals the geometric structures associated with almost invariant dynamics this unwrapping procedure is applied to a uniformly hyperbolic map of the unit square to discover this map s dominant underlying dynamical structure and to the standard map to pinpoint clusters of period 6 orbits

the differential zeta function for axiom a attractors closed orbit of least period x t an important problem is to construct the meromorphic extensions for these zeta functions a detailed knowledge of the domain of s gives information on the long term behaviour of the flow for example this is a standard approach to deriving asymptotic formulae for closed orbits cf 9 for the case of geodesic flows on surfaces of constant negative curvature the zeta function is known to have a meromorphic extension to the entire complex plane unfortunately an obstruction to a general theory along these lines is the existence of examples of axiom a flows for which d s does not have a meromorphic extension to c in a recent article william parry proposed a modified definition of a zeta function for axiom a attractors he defined the differential zeta function by ns s h i e s t l where the least period x x of a closed orbit is replaced by the expansion xu t along the unstable manifold around the orbit 8 parry also defined the useful notion of a synchronized flow i e one for which the measure of maximal entropy and the srb measure coincide the following remark is taken from page 273 of his article when the flow is synchronized the zeta functions coincide this is the case for example with a geodesic flow on a surface of constant negative curvature one wonders whether synchronization has something to do with the very desirable features of the zeta function of such a geodesic flow in this article we shall give a solution to parry s question which shows that in many cases du s has a meromorphic extension to the entire complex plane our main result can be stated as follows

zeta functions and transfer operators for piecewise monotone transformations given a piecewise monotone transformationt of the interval and a piecewise continuous complex weight functiong of bounded variation we prove that the ruelle zeta function z of t g extends meromorphically to z 1 where lim g tn 1 g tg 1 n and thatz is a pole of if and only ifz 1 is an eigenvalue of the corresponding transfer operator l we do not assume that l leaves a reference measure invariant

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

linear response and consequences for differentiability of statistical quantities and multifractal analysis in this article we initially prove the differentiability of the topological pressure equilibrium states and their densities with respect to smooth expanding dynamical systems and any smooth potential this is done by proving the regularity of the dominant eigenvalue of the transfer operator with respect to dynamics and potential from that we obtain strong consequences on the regularity of the dynamical system statistical properties that apply in more general contexts indeed we prove that the average and variance obtained from the central limit theorem vary c r 1 with respect to the c r expanding dynamics and c r potential and also there is a large deviations principle exhibiting a c r 1 rate with respect to the dynamics and the potential an application for multifractal analysis is given we also obtained asymptotic formulas for the derivatives of the topological pressure and other thermodynamical quantities

transfer operators acting on zygmund functions we obtain a formula for the essential spectral radius pess of transfer type operators associated with families of c1 6 diffeomorphisms of the line and zygmund or h6lder weights acting on banach spaces of zygmund respectively h6lder functions in the uniformly contracting case the essential spectral radius is strictly smaller than the spectral radius when the weights are positive

hilbert spaces built on a similarity and on dynamical renormalization we develop a hilbert space framework for a number of general multiscale problems from dynamics the aim is to identify a spectral theory for a class of systems based on iterations of a non invertible endomorphism we are motivated by the more familiar approach to wavelet theory which starts with the two to one endomorphism r z z2 in the one torus t a wavelet filter and an associated transfer operator this leads to a scaling function and a corresponding closed subspace v0 in the hilbert space l2 r using the dyadic scaling on the line r one has a nested family of closed subspaces vn n z with trivial intersection and with dense union in l2 r more generally we achieve the same outcome but in different hilbert spaces for a class of non linear problems in fact we see that the geometry of scales of subspaces in hilbert space is ubiquitous in the analysis of multiscale problems e g martingales complex iteration dynamical systems graph iterated function systems of affine type and subshifts in

explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions abstract we consider transfer operators acting on spaces of holomorphic functions and provide explicit bounds for their eigenvalues more precisely if is any open set in c d and l is a suitable transfer operator acting on bergman space a 2 its eigenvalue sequence n l is bounded by n l a exp a n 1 d where a a 0 are explicitly given

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

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

from ruelle s transfer operator to the schro dinger operator abstract we show that ruelle s transfer operator defined for an appropriate class of strongly chaotic mappings reduces to the schrodinger operator with imaginary time in an appropriate scaling limit we also derive the equations for the next order corrections to the schrodinger behavior if the scaling limit is not completely performed in the simplest case the first order corrections turns out to obey a schrodinger equation with a source term this means a possible chaotic dynamics underlying the feynman kac formula on a very small time scale e g the planc scale leads generalized quantum mechanics with an additional source term

one dimensional gibbs states and axiom a diffeomorphisms on etudie la mecanique statistique en equilibre de systemes reticules classiques a une dimension a interactions a decroissance exponentielle pour de tels systemes les transformees de fourier des fonctions de correlation de paires sont meromorphes dans une bande et les residus des poles peuvent s exprimer en termes de distributions de gibbs on applique ces resultats aux diffeomorphismes de l axiome a de smale

crossed products by finite index endomorphisms and kms states abstract given a unital c algebra a an injective endomorphism a a preserving the unit and a conditional expectation e from a to the range of we consider the crossed product of a by relative to the transfer operator l 1e when e is of index finite type we show that there exists a conditional expectation g from the crossed product to a which is unique under certain hypothesis we define a gauge action on the crossed product algebra in terms of a central positive element h and study its kms states the main result is if h 1 and e ab e ba for all a b a e g when a is commutative then the kms states are precisely those of the form g where is a trace on a satisfying the identity a l h ind e a where ind e is the jones kosaki watatani index of e

entropy volume growth and srb measures for banach space mappings we consider c 2 frechet differentiable mappings of banach spaces leaving invariant compactly supported borel probability measures and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces srb measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds equivalently the sum of positive lyapunov exponents of the map in addition to numerous difficulties incurred by our infinite dimensional setting a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions the results here generalize previously known results for diffeomorphisms of finite dimensional riemannian manifolds and are applicable to dynamical systems defined by large classes of dissipative parabolic pdes

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

correlation analysis of dynamical chaos abstract we study correlation and spectral properties of chaotic self sustained oscillations of different types it is shown that some classical models of stochastic processes can be used to describe behavior of autocorrelation functions of chaos the influence of noise on chaotic systems is also considered

sharp estimates in ruelle theorems for matrix transfer operators a matrix coefficient transfer operator on the space of sections of an m dimensional vector bundle over n dimensional compact manifold is considered the spectral radius of is estimated by para and the essential spectral radius by here is the set of ergodic f invariant measures and for is the measure theoretic entropy of f is the largest lyapunov exponent of the cocycle over f generated by and is the smallest lyapunov exponent of the differential of f

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

spectrum of the ruelle operator and exponential decay of correlations for open billiard flows the paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional visibility condition using a modification of the technique of dolgopyat we get spectral estimates for the ruelle operator related to a markov family for the nonwandering trapping set of the flow similar to those of dolgopyat in the case of transitive anosov flows on compact manifolds with smooth jointly non integrable horocycle foliations as a consequence we get exponential decay of correlation for holder continuous potentials on the nonwandering set combining the spectral estimate for the ruelle operator with an argument of pollicott and sharp we also derive the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a half plane re s h t e where h t is the topological entropy of the billiard flow and an asymptotic formula with an error term for the number of closed orbits of least period 0

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

the resonance spectrum of the cusp map in the space of analytic functions we prove that the frobenius perron operator u of the cusp map f 1 1 1 1 f x 1 2 x which is an approximation of the poincare section of the lorenz attractor has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1 we also prove that for any q 0 1 the spectrum of u in the hardy space in the disk z c z q 1 q is the union of the segment 0 1 and some finite or countably infinite set of isolated eigenvalues of finite multiplicity