on the inverse problem of statistical physics from irreversible semigroups to chaotic dynamics abstract we show that all measure preserving stationary markov processes arise as projections of kolmogorov dynamical systems which have positive entropy production and are prototypes of chaos this result not only contributes to the clarification of the relation of dynamics with stochastic processes but also shows the physical significance of the misra prigogine courbage theory of irreversibility in the more general context of the inverse problem of statistical physics because we want positivity preserving transformations our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of positive dilations

on the discrete time version of the brussels formalism we develop a discrete time version of the so called brussels formalism in nonequilibrium statistical mechanics for continuous endomorphisms of a banach space we show that if the evolution operator u and the projector p are such that pu is a compact operator and the spectral radius of i p u i p is strictly less than the spectral radius of u then the formalism holds and the evolution operator is quasicompact

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

jordan blocks in a one dimensional markov map abstract a simple one dimensional chaotic map whose spectral decomposition of the frobenius perron operator in the space of polynomials has jordan block structure is given and analyzed the jordan block structure implies that some correlation functions have exponential decay modified by coefficients which are polynomials in t such a correlation is given for this system and a numerical simulation which agrees with the theoretical prediction is presented

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

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

probabilistic approach to homoclinic chaos three dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues i and giving rise to homoclinic chaos when the shil nikov condition is satisfied are studied the 2d poincare map and its 1d contractions capturing the essential features of the flow are given at homoclinicity these 1d maps are found to be piecewise linear this property allows one to reduce the frobenius perron equation to a master equation whose solution is analytically known the probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived

spectral decomposition of chebyshev maps abstract we construct a spectral decomposition of the frobenius perron operator for the chebyshev polynomials of the first kind we defined a suitable dual pair or rigged hilbert space which provides mathematical meaning to the spectral decomposition the spectra of the even chebyshev maps do not contain the odd powers of 1 m and the odd eigenfunctions are in the null space of the frobenius perron operator moreover the odd chebyshev maps have degenerate spectra without jordan blocks the eigenvalues in the decompositions are the resonances of power spectrum and have magnitudes less than one as in the case of the family of tent maps

rigged hilbert spaces associated with misra prigogine courbage theory of irreversibility abstract it is proved that in the misra prigogine courbage theory of irreversibility using the internal time superoperator fixing its associated non unitary transformation amounts to rigging the corresponding hilbert liouville space more precisely it is demonstrated that any determines three canonical riggings of the liouville space l first one with a hilbert space with a norm greater than the relative one from l a second one with a hilbertian space which is a kothe space if is compact and is a nuclear space if has certain nuclear properties and finally a third one with a smaller hilbertian space with a still stronger topology which is nuclear if n is hilbert schmidt for some positive integer n in contrast any rigging of this type originated in a dynamical system having an internal time superoperator defines a in a canonical way

unitary quasi affine transforms of contractions abstract the unitary quasi affine transforms of contractions are the cornerstone of the exact theory of irreversibility introduced by misra prigogine and courbage this work shows that the class of contractions with unitary quasi affine transforms is just c 1 and that every unitary quasi affine transform of a contraction is unitarily equivalent to the residual part of its sz nagy foias dilation some connections with the grossmann theory of nested hilbert spaces and generalized eigenvalues are established these results are applied to the study of the baker map

from irreversible markov semigroups to chaotic dynamics abstract we answer qualitatively the inverse coarse graining problem of statistical physics namely which microscopic dynamics give rise to a given physically observed markov semigroup as a result of exact coarse graining we prove the fact that all markov chains arise as projections of dynamical systems in larger spaces and show that in particular the irreversible markov chains arise as projections of chaotic systems of kolmogorov type this result generalizes our previous results on the misra prigogine courbage semigroups because we want positivity preserving transformations our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of positive dilations

from probabilistic descriptions to deterministic dynamics abstract we answer the question of dilating a markov semigroup to a deterministic evolution in a larger space we show that certain markov processes may arise as projections of kolmogorov dynamical systems this result shows in particular the physical significance of the misra prigogine courbage theory of irreversibility because we want positivity preserving transformations in both case our procedure although analogous to the sz nagy foias dilation theory has a different viewpoint that of the natural extension of exact systems

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

ensemble dynamics of intermittency and power law decay a spectral decomposition of the frobenius perron operator is constructed for one dimensional maps with intermittent chaos using the method of coherent states a technique using the spectral density function is applied to the the well known cusp map which generates weak type ii intermittency higher order corrections are obtained to the leading 1 t long time behavior of the x x autocorrelation

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

resonances of dynamical systems and fredholm riesz operators on rigged hilbert spaces abstract resonances of dynamical systems are defined as the singularities of the analytically continued resolvent of the restriction of the frobenius perron operator to suitable test function spaces a sufficient condition for resonances to arise from a meromorphic continuation to the entire plane is that the frobenius perron operator is a fredholm riesz operator on a rigged hilbert space after a discussion of spectral theory in locally convex topological vector spaces we illustrate the approach for a simple chaotic system namely the renyi map

non unitary transformation of conservative to dissipative evolutions the authors briefly review in a general context the theory of non unitary transformations between conservative dynamics with internal time and dissipative evolutions developed previously in the context of statistical mechanics and apply the results to show the existence of a similarity transformation between the wave equation and the dissipative telegraphist equation

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

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

analysis and adaptive modeling of highly heterogeneous elastic structures in this paper we develop a theory and methodology for obtaining approximate solutions to boundary value problems describing the deformation of highly heterogeneous linearly elastic structures the method which represents a significant departure from traditional homogenization methods provides a systematic and rigorous approach towards resolving the effects of microstructure of different scales on the macroscopic response of complex heterogeneous structures an early variant of this method was first introduced in io the method referred to as hdpm homogenized dirichlet projection method proceeds by first solving an auxiliary homogenized boundary value problem describing the deformation of a structure with the same exterior geometry but with a selected set of uniform material properties the adequacy of this homogenized solution is then determined using an a posteriori error estimate that provides a measure of the error of the homogenized solution in subdomains of the structure compared to the solution of the fine scale heterogeneous problem without specific knowledge of this fine scale solution in those subdomains where the homogenized solution is deemed acceptable it is retained however in subdomains where the homogenized solution is inadequate as is determined when the estimated error exceeds a preset tolerance a local boundary value problem is constructed by projecting the homogenized displacements onto a partition designed to isolate these subdomains these local boundary value problems are then solved in those subdomains where the homogenized solution is inaccurate using the exact microstructure with the approximate local boundary conditions a posteriori error estimates are made to ascertain the quality of the resulting solution if the quality of the solution remains inadequate above a preset error tolerance a two stage adaptive procedure is implemented stage i material adaptivity corresponds to modifying the homogenized structure s material properties if after stage i the solution quality is still inadequate the subdomains of local solution are enlarged thereby modeling in greater detail the actual microstructure and the local solution process is repeated stage ii subdomain unrefinement the main feature of this method is that only in subdomains where the error in the usual homogenized solution is above a preset tolerance is the microstructure taken into account the cost of this method is shown to be orders of magnitude cheaper than direct huge scale computational simulations of micromechanical events the results of several numerical experiments are provided to demonstrate the method and verify theoretical estimates in this work we present a method for the solution of boundary value problems modeling the deformation of structures composed of highly heterogeneous linearly elastic materials we concentrate on materials formed by a homogeneous matrix embedded with particulate matter of different properties fig 1 however the results presented are not limited to materials of this type and can in theory be used for materials with virtually any microstructure a straightforward calculation reveals that an accurate numerical approximation of fine scale solutions of

hierarchical modeling of heterogeneous solids in this paper a methodology is introduced for the development of adaptive methods for hierarchical modeling of elastic heterogeneous bodies the approach is based on the idea of computing an estimate of the modeling error introduced by replacing the actual fine scale material tensor with that of a homogenized material and to adaptively refine the material description until a prespecified error tolerance is met this process generates a family of coarse scale solutions in which the solution corresponding to the fine scale model of the body which embodies the exact microstructure loading and boundary conditions represents the highest level of sophistication in a family of continuum models the adaptive strategy developed can lead to a new non uniform description of material properties which reflects the loading and boundary conditions a post processing technique is also introduced which endows the coarse scale solutions with fine scale information through a local solution process convergence of the adaptive algorithm is proven and modeling error estimates as a function of scale of material description are presented preliminary results of several numerical experiments are given to confirm estimates and to illustrate the promise of the approach in practical applications

hausdorff volume in non equiregular sub riemannian manifolds in this paper we study the hausdorff volume in a non equiregular sub riemannian manifold and we compare it with a smooth volume we first give the lebesgue decomposition of the hausdorff volume then we study the regular part show that it is not commensurable with the smooth volume and give conditions under which it is a radon measure we finally give a complete characterization of the singular part we illustrate our results and techniques on numerous examples and cases e g to generic sub riemannian structures

resonances and the extension of dynamics to rigged hilbert space abstract we propose a unified operator theoretic formulation of resonances and resonance states in rigged hilbert space our approach shows how resonances are the basis for the extension of dynamics for unstable systems and in a synthesis and a generalization of the work of the brussels austin groups directed by prigogine with the work of bohm and gadella on gamow vectors and with the extension theory developed by the st petersburg group directed by pavlov we illustrate the approach for prototypes of resonances namely resonances of the power spectrum scattering resonances and resonances associated with the unstable bloch states

quantum systems with diagonal singularity their logic and intrinsic irreversibility abstract the work of the brussels austin group on irreversibility over the last years has shown that quantum large poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional hilbert space framework and logic we characterize the algebra of observables and the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

theory of fast nondeterministic physical random bit generation with chaotic lasers we theoretically show that completely stochastic fast physical random bit generation at a rate of more than one gigabit per second can be realized by using lasers with optical delayed feedback which creates high dimensional chaos of laser light outputs the theory is based on the mixing property of chaos which transduces microscopic quantum noise of spontaneous emission in lasers into random transitions between discrete macroscopic states

the extension of classical dynamics for unstable hamiltonian systems abstract classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the liouville equation it is shown that for the class of large nonintegrable poincare systems lps the two descriptions are not equivalent practically all dynamical systems studied in statistical mechanics belong to this class the basic step is the extension of the liouville operator l h outside the hilbert space to functions singular in their fourier transformation this function space plays an important role in statistical mechanics as functions of the hamiltonian and therefore equilibrium distribution functions belong to this class physically these functions correspond to situations characterized by persistent interactions as they are realized in macroscopic physics persistent interactions are introduced in contrast to transient interactions studied in quantum mechanics by the s matrix approach asymptotically free in and out states the eigenvalue problem for the liouville operator l h is solved in this generalized function space for lps we obtain a complex irreducible spectral representation complex means that the eigenvalues are complex numbers whose imaginary parts refer to the various irreversible processes such as relaxation times diffusion irreducible means that these representations cannot be implemented by trajectory theory as the result the dynamical group of evolution splits into two semigroups moreover the laws of classical dynamics take a new form as they have to be formulated on the statistical level they express possibilities and no more certitudes two examples of typical classical systems i e interacting particles and anharmonic lattices are studied

concurrent multi scale analysis of elastic composites by a multi level computational model abstract in this paper an adaptive multi level computational model is developed to comprehensively address issues of discretization and physically based modeling error in structures consisting of heterogeneous micro structures the model consists of three essential levels i a purely macroscopic domain level 0 with homogenized material parameters ii a macro micro domain level 1 with the micro domain represented by the periodic repetition of a representative volume element rve and iii purely microscopic domain level 2 where the rve ceases to exist and extended micro structural regions need to be modeled all micro structural computations of arbitrary heterogeneous domains are conducted using the adaptive voronoi cell finite element model an intermediate layer is used between the macroscopic and microscopic regions for smooth transition the level 1 domain is utilized to estimate criteria for switching from macroscopic to microscopic calculations physically based error indicators are developed for transition from macroscopic to microscopic domain statistical functions are used to assess the extent of the rve domains for non uniform micro structures for the macroscopic domains h and hp adaptations are executed using conventional error indicators based on the energy norm the numerical simulations are compared with results from pure micro mechanics and other multi level models in the literature the model is also tested for a realistic problem with macroscopic stress risers

sub riemannian vs euclidean dimension comparison and fractal geometry on carnot groups we solve gromov s dimension comparison problem for hausdorff and box counting dimension on carnot groups equipped with a carnot caratheodory metric and an adapted euclidean metric the proofs use sharp covering theorems relating optimal mutual coverings of euclidean and carnot caratheodory balls and elements of sub riemannian fractal geometry associated to horizontal self similar iterated function systems on carnot groups inspired by falconer s work on almost sure dimensions of euclidean self affine fractals we show that carnot caratheodory self similar fractals are almost surely horizontal as a consequence we obtain explicit dimension formulae for invariant sets of euclidean iterated function systems of polynomial type jet space carnot groups provide a rich source of examples

numerical model reduction with error control in computational homogenization of transient heat flow numerical model reduction nmr is exploited for solving the finite element problem on a representative volume element rve that arises from the computational homogenization of a model problem of transient heat flow since the problem is linear an orthogonal basis is obtained via the classical method of spectral decomposition a symmetrized version of the space time variational format is adopted for estimating the error from the model reduction in i energy norm and in ii given quantities of interest this technique which was recently developed in the context of the non selfadjoint stationary diffusion convection problem is novel in the present context of nmr by considering the discrete unreduced model as exact we are able to obtain guaranteed bounds on the error while using only the reduced basis and with minor computational effort the performance of the error estimates is demonstrated via numerical results where the subscale is modeled in both one and three spatial dimensions

spectrum and eigenfunctions of the frobenius perron operator of the tent map the spectrum and eigenfunctions of the frobenius perron operator induced by the tent map are discussed in detail special attention is paid to the case where the critical point of the map lies on an aperiodic trajectory and the differences from maps with a periodic critical trajectory are stressed it is shown that the relevant eigenvalues of the spectrum are not sensitive to the aperiodicity of the critical trajectory all other parts of the spectrum and all eigenfunctions in particular are changed drastically if the critical trajectory becomes aperiodic the intimate connection between the point spectrum and the kneading invariant is established and the critical slowing down as well as the band splitting are a consequence of its properties the structure of the infinite sequence of null spaces and its implications on the spectrum of the operator are discussed it is shown that any initial distributionp 0 x of bounded variation can be projected uniquely onto the eigenfunctions of the relevant eigenvalues and that the time dependence ofp n x is determined by this expansion up to an erroro n from this the stationary and the asymptotic behavior of the correlation function x n x can be derived exactly

polynomial shift states of a chaotic map we present a piecewise linear map of the unit interval in which the resolvent of the frobenius perron operator considered in a polynomial basis has an essen tial singularity at the origin associated with the essential singularity are polynomial shift states which are obtained from creation and annihilation operators in non self dual function spaces correlation functions of general polynomial observables have decay components that vanish in a finite time

adaptive finite element heterogeneous multiscale method for homogenization problems in this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method unlike standard finite element methods our discretization scheme relies on macro and microfinite elements the desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data as the macroscopic data such as the macroscopic diffusion tensor are not available beforehand appropriate error indicators have to be defined for designing adaptive methods we show that such indicators based only on the available macro and microsolutions used to compute the actual macrosolution can be defined allowing for a macroscopic mesh refinement strategy which is both reliable and efficient the corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm in the case of a uniformly oscillating tensor we recover the standard residual based a posteriori error estimate for the finite element method applied to the homogenized problem numerical experiments confirm the efficiency and reliability of the adaptive multiscale method

rigorous computation of diffusion coefficients for expanding maps for real analytic expanding interval maps a novel method is given for rigorously approximating the diffusion coefficient of real analytic observables as a theoretical algorithm our approximation scheme is shown to give quadratic exponential convergence to the diffusion coefficient the method for converting this rapid convergence into explicit high precision rigorous bounds is illustrated in the setting of lanford s map x mapsto 2x frac 1 2 x 1 x pmod 1

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

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

a ruelle operator for a real julia set letr be an expanding rational function with a real bounded julia set and let left lg right left x right sum limits ry x frac g left y right left r left y right right 2 be a ruelle operator acting in a space of functions analytic in a neighbourhood of the julia set we obtain explicit expressions for the resolvent function e left x z lambda right left i lambda l right 1 frac 1 z x and in particular for the fredholm determinantd det i l it gives us an equation for calculating the escape rate we relate our results to orthogonal polynomials with respect to the balanced measure ofr two examples are considered

on the use of the pulsed convection approach for modelling advection diffusion in chaotic flows a prototypical example and direct numerical simulations abstract this article addresses the application of pulsed system models in which the advection operator is decoupled from the diffusion operator for investigating the physics of dispersion homogenization in deterministic chaotic flows the analysis is organized along to main directions i the development of a simplified time continuous model which can be viewed as a generalization in a time continuous frame of the baker s transformation and which is amenable to analytical investigation and ii the comparison of the results deriving from several typical pulsed system models with the direct numerical simulation of the advection diffusion equation both these approaches reveal the intrinsic ambiguity of the pulsed system approach in describing advection diffusion problems

verification of stochastic models in uncertain environments using the constitutive relation error method abstract this paper deals with local stress errors which occur in linear elastic fe analysis in an uncertain environment material loads the estimation technique described here uses the concept of error in constitutive relation but it can be adapted to other estimators first we review the principle of the method then extend it to the stochastic case this approach yields bounds on the stresses associated with a given confidence level we illustrate this presentation through a test example

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

gamow vectors for degenerate scattering resonances in this paper we construct gamow vectors for resonances given by poles of the analytic continuation of the s matrix of any finite order we study their modes of decay or growth we obtain jordan block structures for the extended hamiltonians and evolution operators on the subspaces spanned by these gamow vectors we perform this study within the context of the rigged hilbert space extension of quantum theory we construct an explicit friedrichs model with a double pole resonance to illustrate the general formulation

guaranteed error bounds in homogenisation an optimum stochastic approach to preserve the numerical separation of scales this paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of pdes with random fast evolving diffusion coefficients more precisely in the context of linear elliptic diffusion problems in randomly packed particulate composites we develop an approach to strictly bound the error in the expectation and second moment of quantities of interest without ever solving the fine scale intractable stochastic problem the most attractive feature of our approach is that the error bounds are computed without any integration of the fine scale features our computations are purely macroscopic deterministic and remain tractable even for small scale ratios the second contribution of the paper is an alternative derivation of modelling error bounds through the prager synge hypercircle theorem we show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie we interpret our optimum result as an extension of reuss voigt approaches which are classically used to estimate the homogenised diffusion coefficients of composites to the estimation of macroscopic engineering quantities of interest finally we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation

smoothness of subriemannian isometries we show that the group of isometries i e distance preserving homeomorphisms of an equiregular subriemannian manifold is a finite dimensional lie group of smooth transformations the proof is based on a new pde argument in the spirit of harmonic coordinates establishing that in an arbitrary subriemannian manifold there exists an open dense subset where all isometries are smooth

on the relaxation time of gauss s continued fraction map i the hilbert space approach koopmanism it is shown that u the adjoint of koopman s isometric operator uf x f tx corresponding to the map tx x sup 1 mod 1 of the unit interval is isomorphic to a symmetric integral operator when restricted to a hilbert space of holomorphic functions f this result also obtained by babenko in a different setting allows them to derive new trace formulas using generalized temple s inequalities they determine the relaxation time of the above system with great accuracy in contrast to a widespread belief it appears to be unrelated to the entropy of the map t

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

on the discrete frobenius perron operator of the bernoulli map we study the spectra of a finite dimensional frobenius perron operator matrix of the bernoulli map derived from phase space discretization the eigenvalues and right and left eigenvectors are analytically calculated which are closely related to periodic orbits on the partition points in the degenerate case jordan decomposition of the matrix is explicitly constructed except for the isolated eigenvalue 1 there is no definite limit with respect to eigenvalues when n the behaviour of the eigenvectors is discussed in the limit of large n

pauls rectifiable and purely pauls unrectifiable smooth hypersurfaces abstract this paper is related to the problem of finding a good notion of rectifiability in sub riemannian geometry in particular we study which kind of results can be expected for smooth hypersurfaces in carnot groups our main contribution will be a consequence of the following result there exists a c hypersurface s without characteristic points that has uncountably many pairwise non isomorphic tangent groups on every positive measure subset the example is found in a carnot group of topological dimension 8 it has hausdorff dimension 12 and so we use on it the hausdorff measure h 12 as a consequence we show that any lipschitz map defined on a subset of a carnot group of hausdorff dimension 12 with values in s has negligible image with respect to the hausdorff measure h 12 in particular we deduce that s cannot be lipschitz parametrizable by countably many maps each defined on some subset of some carnot group of hausdorff dimension 12 as main consequence we have that a notion of rectifiability proposed by s pauls is not equivalent to one proposed by b franchi r serapioni and f serra cassano at least for arbitrary carnot groups in addition we show that given a subset u of a homogeneous subgroup of hausdorff dimension 12 of a carnot group every bi lipschitz map f u s satisfies h 12 f u 0 finally we prove that such an example does not exist in heisenberg groups we prove that all c hypersurfaces in h n with n 2 are countably h n 1 r rectifiable according to pauls definition even with bi lipschitz maps

estimation of local modeling error and goal oriented adaptive modeling of heterogeneous materials i error estimates and adaptive algorithms abstract a theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented these quantities may for example represent averaged stresses on the surface of inclusions or mollifications of pointwise stresses or displacements or in general local features of the fine scale solution characterized by continuous linear functionals these estimators are used to construct goal oriented adaptive procedures in which models of the microstructure are adapted to deliver local features to a preset level of accuracy algorithms for implementing these procedures are discussed and preliminary numerical results are given the analysis is restricted to linear static heterogeneous elastic materials

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

spectral decomposition of tent maps using symmetry considerations the spectral decomposition of the frobenius perron operator of maps composed of many tents is determined from symmetry considerations the eigenstates involve euler as well as bernoulli polynomials

subdynamics and nonintegrable systems work on the application of poincare s theorem to large classical or quantum systems with a continuous spectrum is continued in situations where it is applicable poincare s theorem prevents the construction of a complete set of eigenprojectors which would be hermitian as well as analytic in the coupling constant in contrast the theory of subdynamics as developed by the brussels group permits the construction of a unique set of projectors for t 0 giving up the requirement of hermiticity which is replaced by star hermiticity the theory of subdynamics is presented in a new self contained way starting from the commutation relation lh lh where lh is the liouvillian this presentation is far more direct and avoids some of the lengthy discussions associated with previous presentations based mainly on the resolvent of the liouvillian subdynamics appears to be of interest from many points of view it generalizes the concept of spectral representation while permitting to retain all the degrees of freedom present in the unperturbed hamiltonian in contrast degrees of freedom are lost when going to the spectral representation e g in the friedrichs model subdynamics permits us to solve the initial value problem associated with the liouville equation retaining the non markovian contributions which appear in the standard presentation finally it introduces a classification of large dynamical systems classical or quantum into integrable and nonintegrable ones it is therefore of direct interest for a number of basic problems which belong to the class of nonintegrable dynamical systems such as the interaction of matter with light the applications of this technique to these problems will be worked out in subsequent papers

investigation of the interactions between the numerical and the modeling errors in the homogenized dirichlet projection method abstract the homogenized dirichlet projection method hdpm was developed in 1 2 as a systematic technique for analyzing highly heterogeneous elastic structures the method can provide an analysis of structures composed of composite materials with very complex microstructure at a fraction of the cost of solving the full fine scale model in the present investigation the hdpm is revisited to take into account the unavoidable numerical errors produced in finite element approximations of the associated boundary value problems the total error of the hdpm which now takes into account both for the modeling and numerical errors is split into several terms each accounting for a parameter in the method the parameters are the choice of the homogenized material property the partition into subdomains the coarse finite element mesh used to solve the homogenized problem and the fine meshes used to solve the subdomain problems numerical experiments are carried out on 1 d problems for which the exact solutions are easily calculated the experiments reveal that the influence of the coarse and fine meshes are very different when free of numerical error the hdpm is based on four main results we rewrite these in the framework of the error in the constitutive law this leads to a clear mechanical interpretation of the results moreover it allows us to extend the results to nonlinear constitutive laws and to find new properties of the hdpm some of the theoretical results are validated for specific cases involving known numerical error finally explicit a posteriori upper bounds are derived for the total and numerical error a simple adaptive strategy is presented for choosing the fine mesh and the subdomain partition the strategy is tested on a 1 d model problem

on the relaxation time of gauss continued fraction map ii the banach space approach transfer operator method the spectrum of the transfer operator for the maptx 1 x 1 x when restricted to a certain banach space of holomorphic functions is shown to coincide with the spectrum of the adjointu of koopman s isometric operatoruf x f t x when the former is restricted to the hilbert space introduced in part i of this work ifn denotes the operator p1 withp1 the projector onto the eigenfunction to the dominant eigenvalue 1 1 of then n is au0 positive operator with respect to some cone and therefore has a dominant positive simple eigenvalue 2 a minimax principle holds giving rigorous upper and lower bounds both for 2 and the relaxation time of the mapt

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

cad model simplification error estimation for electrostatics problems simplifying the geometry of a cad model using defeaturing techniques enables more efficient discretization and subsequent simulation for engineering analysis problems understanding the effect this simplification has on the solution helps one to decide whether the simplification is suitable for a specific simulation problem it can also help one to understand the functional effect of a geometry feature the effect of the simplification is quantified by a user defined quantity of interest which is assumed to be approximately linear in the solution a bound on the difference between the quantity of interest of the original and simplified solutions based on the energy norm is derived the approach is presented in the context of electrostatics problems but can be applied in general to a range of elliptic partial differential equations numerical results on the efficiency of the bound are provided for electrostatics problems with simplifications involving changes inside the problem domain as well as changes

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

intrinsic irreversibility and the validity of the kinetic description of chaotic systems irreversibility for a class of chaotic systems is seen to be an exact consequence of the dynamics through the use of a generalized spectral representation of the time evolution operator of probability densities the generalized representation is valid for one dimensional systems when the initial probability density satisfies certain physical conditions of smoothness the formalism is first applied to the one dimensional multi bernoulli map which is a simple map displaying deterministic diffusion the two dimensional invertible baker and multibaker transformations are then studied and the physical conditions determining which discrete spectral values are realized are seen to depend on the smoothness of both the density as well as the observable considered the generalized representation is constructed using a resolvent formalism the eigenstates of the diffusive systems are seen to be of a fractal nature

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

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

rigged hilbert spaces for chaotic dynamical systems we consider the problem of rigging for the koopman operators of the renyi and the baker maps we show that the rigged hilbert space for the renyi maps has some of the properties of a strict inductive limit and give a detailed description of the rigged hilbert space for the baker maps

estimation of local modeling error and goal oriented adaptive modeling of heterogeneous materials part ii a computational environment for adaptive modeling of heterogeneous elastic solids this paper addresses a classical and largely unsolved problem given a structural component constructed of a heterogeneous elastic material that is in equilibrium under the action of applied loads determine local micromechanical features of its response e g local stresses and displacements in or around phase boundaries or in inclusions to an arbitrary preset level of accuracy it being understood that the microstructure is a priori unknown may be randomly distributed may exist at multiple spatial scales and may contain millions even billions of microscale components the approach described in this work begins with a mathematical abstraction of this problem in which the material body is modeled as an elastic solid with highly variable possibly randomly distributed elastic properties information on the actual character of the microstructure of given material bodies is determined by computerized tomography ct imaging a procedure is given for determining the effective material properties from imaging data using either deterministic or stochastic methods an algorithm is then described for determining local quantities of interest such as average stresses on inclusion boundaries to arbitrary accuracy relative to the fine scale model a new computational environment for implementing such analyses is presented which employs parallel adaptive hp finite element methods ct interfaces automatic meshing procedures and effectively adaptive modeling schemes within the basic premises on which the approach is based results of any specified accuracy can be obtained independently of the number of microscale components and constituents the results of several numerical experiments are presented

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

nonlocality of the misra prigogine courbage semigroup we show that the markov semigroups constructed by misra prigogine and courbage through nonunitary similarity transformations of kolmogorov systems are not implementable by local point transformations i e they are not the frobenius perron semigroups associated with noninvertible point transformations in contrast with the semigroups obtained by coarse graining projections our result is a straightforward generalization of the proof of the nonlocality of the similarity transformation given by goldstein misra and courbage and also of the previous illustration by misra and prigogine for the baker transformation and completes the characterization of the misra prigogine courbage semigroups

moment equations and closure schemes in chaotic dynamics closure schemes for the infinite hierarchy of moment equations describing the time evolution of averages of observables over a non equilibrium ensemble of initial conditions for the logistic map model are derived with the aid the frobenius perron equation in the long time regime for the growth parameter r 4 the approximations compare well with exact results derived for specific initial probability densities

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

probabilistic control of chaos chaotic maps under control recently we have proposed a new probabilistic method for the control of chaotic systems 1 in this paper we apply our method to characteristic cases of chaotic maps one and two dimensional examples as these chaotic maps are structurally stable they cannot be controlled using conventional control methods without significant change of the dynamics our method consists in the probabilistic coupling of the original system with a controlling system this coupling can be understood as a feedback control of probabilistic nature the chosen periodic orbit of the original system is a global attractor for the probability densities the generalized spectral decomposition of the associated frobenius perron operator provides a spectral condition of controllability for chaotic dynamical systems

alternative formulation of classical and quantum dynamics for non integrable systems we continue our work on large poincare systems lps characterized by a continuous spectrum and a continuous set of resonances poincare s theorem prevents the existence of solutions of the eigenvalue problem associated with the hamiltonian as well as the liouville operator which would be analytic in the coupling constant in simple cases such as the friedrichs model studied in a recent paper poincare s divergences can be avoided using a natural time ordering of the dynamical states in the hilbert space however in general the time ordering has to be performed in the liouville space of density matrices

computational uncertainty analysis in multiresolution materials via stochastic constitutive theory abstract a stochastic constitutive theory is proposed in this work to propagate microstructure uncertainties in computational multiscale continuum models to bulk multiresolution material behavior ubiquitous fine resolution uncertainty sources influencing prediction of material properties based on their structures are categorized in detail and this research transmits these uncertainties to coarser material resolutions by introducing a stochastic constitutive theory deduced from volume element simulations to implement the stochastic upscaling process two advanced uncertainty quantification methods are examined statistical copula functions and random process polynomial chaos expansion both methods confront the mathematical difficulty in randomizing constitutive laws by capturing the marked correlation among constitutive parameters seen in complex materials thus the results proffer a more accurate probabilistic estimation of constitutive material behavior the contribution of this work is twofold uncertainty is propagated from heterogeneous material structure to material property via the stochastic constitutive theory and rigorous data driven mathematics are formalized to represent complicated dependence structures in multivariate statistical distributions to the authors knowledge this is the first work in multiresolution mechanics that presents an approach to computationally derive correlation functions from numerical experiments as opposed for instance to assuming one a priori the method put forth in this research though quite general is applied to a mathematical example and plastic high strength steel alloy for demonstration results include stochastic constitutive curve confidence intervals for the material stress strain response and qualitative comparisons of the two stochastic methods detailed herein

on the long time behaviour of ensembles in a model of deterministic diffusion we describe a large model system based on the baker transformation where deterministic diffusion occurs the model is similar to one recently considered by gaspard and by hasegawa and driebe we point out the close relationship between this system and a simple random walk and analyse the evolution with time of ensembles in the system using the resolvent based version of the subdynamics formalism developed by prigogine and his collaborators we obtain an exact and rigorous description of the long time behaviour of ensembles including the irreversible approach to equilibrium for the case where the system has finite size we also consider the thermodynamic limit where the size of the system becomes infinite and derive a description of the long time behaviour in this case where correlations decay non exponentially with time

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

dynamics of coarse grained variables in spatially extended systems abstract some new ways to probe the dynamics of spatially extended systems giving rise to spatio temporal chaos are explored coarse grained observables suitable for describing large scale behavior are introduced by locally averaging the original variables over a spatial region and their stability spatial correlation function and statistical properties are compared to those of the original variables in general these observables do not obey closed form equations truncation schemes in which high order fourier modes or variables other than the coarse grained ones are discarded are developed and their sensitivity to the parameters is assessed

quantum theory of non integrable systems in 1889 h poincare introduced a basic distinction between integrable and non integrable dynamical systems this distinction refers to the role of resonances which may lead to divergences a specially important class of non integrable systems are large poincare systems lps which have a continuous spectrum and present continuous sets of resonances as has been shown earlier lps play an essential role both in classical and quantum mechanics essentially all nontrivial problems of field theory as well as kinetic theory belong this class we consider here the well known friedrichs model in which an unstable discrete level is coupled to a continuum poincare s theorem prevents the existence of solutions of the eigenvalue problem associated to the hamiltonian which would be analytic in the coupling constant