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

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

error control and mesh optimization for high order finite element approximation of incompressible viscous flow abstract we discuss the use of a posteriori error estimates for high order finite element methods during simulation of the flow of incompressible viscous fluids the correlation between the error estimator and actual error is used as a criterion for the error analysis efficiency we show how to use the error estimator for mesh optimization which improves computational efficiency for both steady state and unsteady flows the method is applied to two dimensional problems with known analytical solutions jeffrey hamel flow and more complex flows around a body both in a channel and in an open domain

on stochastic stability of non uniformly expanding interval maps we study the expanding properties of random perturbations of regular interval maps satisfying the summability condition of exponent one under mild conditions we prove strong stochastic stability non uniformly expanding interval maps play an important role in the theory of dynamical systems the statistical properties of these systems as well as the persistency of these properties have attracted much interest in particular the celebrated work of jakobson j showed that non uniformly expanding maps are abundant among interval maps extensions and generalizations of this work have produced many of the main examples of non uniformly hyperbolic dynamical sys tems in dimension one or higher see bc1 bc2 v wy r among others in this paper we study random perturbations of non uniformly expanding inter val maps f modeled by iterates of random maps so we shall study composition of maps of the form gn 1 g1 g0 where g0 g1 are independently chosen random maps from a small neighborhood of f in a suitable space of interval maps under very general conditions on the dynamics f and on perturbation types we shall prove stochastic stability a typical random orbit gn 1 g1 g0 x has roughly the same asymptotic distribution in the phase space as a typical orbit of f in a strong sense to be more precise let f 0 1 0 1 be a multimodal interval map of class c 3 with non flat critical points we shall assume that f has no attracting or neutral cycles in the deterministic case existence of an ergodic invariant probability mea sure which is absolutely continuous with respect to the lebesgue measure acip has been extensively studied such a measure is a physical measure in the sense that there exists a subset e of 0 1 with positive lebesgue measure such that for a e x e 1 n n 1 x

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

on the enhancement of diffusion by chaos escape rates and stochastic instability we consider stochastic perturbations of expanding maps of the interval where the noise can project the trajectory outside the interval we estimate the escape rate as a function of the amplitude of the noise and compare it with the purely diffusive case this is done under a technical hypothesis which corresponds to stability of the absolutely continuous invariant measure against small perturbations of the map we also discuss in detail a case of instability and show how stability can be recovered by considering another invariant measure

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

an a posteriori error estimate and adaptive timestep control for a backward euler discretization of a parabolic problem a simple algorithm for adaptive timestep control is presented for a backward euler discretization of a linear parabolic problem the algorithm is based on an a posteriori error estimate involving the computed approximate solution it is proved that with only very rough a priori information on the exact solution the algorithm will choose a sequence of timesteps for which the error will be controlled up to a constant uniformly in time on a given tolerance level

whitney holder continuity of the srb measure for transversal families of smooth unimodal maps we consider c 2 families t mapsto f t of c 4 nondegenerate unimodal maps we study the absolutely continuous invariant probability srb measure mu t of f t as a function of t on the set of collet eckmann ce parameters upper bounds assuming existence of a transversal ce parameter we find a positive measure set of ce parameters delta and for each t 0 in delta a set delta 0 subset delta of polynomially recurrent parameters containing t 0 as a lebesgue density point and constants c ge 1 gamma 4 so that for every 1 2 holder function a begin aligned big int a d mu t int a d mu t 0 big le c vert a vert c 1 2 t t 0 1 2 log t t 0 gamma forall t in delta 0 end aligned in addition for all t in delta 0 the renormalisation period p t of f t satisfies p t le p t 0 and there are uniform bounds on the rates of mixing of f t p t for all t with p t p t 0 if f t x tx 1 x the set delta contains almost all ce parameters lower bounds assuming existence of a transversal mixing misiurewicz thurston parameter t 0 we find a set of ce parameters delta mt accumulating at t 0 a constant c ge 1 and a c infty function a 0 so that begin aligned c t t 0 1 2 ge big int a 0 d mu t int a 0 d mu t 0 big ge c 1 t t 0 1 2 forall t in delta mt end aligned

solution of parabolic equations by backward euler mixed finite element methods on a dynamically changing mesh we develop and analyze methods based on combining the lowest order mixed finite element method with backward euler time discretization for the solution of diffusion problems on dynamically changing meshes the methods developed are shown to preserve the optimal rate error estimates that are well known for static meshes the novel aspect of the scheme is the construction of a linear approximation to the solution which is used in projecting the solution from one mesh to another extensions to advection diffusion equations are discussed where the advection is handled by upwinding numerical results validating the theory are also presented

annealed and quenched limit theorems for random expanding dynamical systems in this paper we investigate annealed and quenched limit theorems for random expanding dynamical systems making use of functional analytic techniques and more probabilistic arguments with martingales we prove annealed versions of a central limit theorem a large deviation principle a local limit theorem and an almost sure invariance principle we also discuss the quenched central limit theorem dynamical borel cantelli lemmas erdos renyi laws and concentration inequalities

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

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

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

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

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

a posteriori finite element error estimation for diffusion problems adjerid babuska and flaherty math models methods appl sci 9 1999 pp 261 286 and yu math numer sinica 13 1991 pp 89 101 and math numer sinica 13 1991 pp 307 314 show that a posteriori estimates of spatial discretization errors of piecewise bi p polynomial finite element solutions of elliptic and parabolic problems on meshes of square elements may be obtained from jumps in solution gradients at element vertices when p is odd and from local elliptic or parabolic problems when p is even we show that these simple error estimates are asymptotically correct for other finite element spaces the key requirement is that the trial space contain all monomial terms of degree p 1 except for x 1 p 1 and x 2 p 1 in a cartesian x1 x2 frame computational results show that the error estimates are accurate robust and efficient for a wide range of problems including some that are not supported by the present theory these involve quadrilateral element meshes problems with singularities and nonlinear problems

limit theorems in the stadium billiard we prove that the birkhoff sums for almost every relevant observable in the stadium billiard obey a non standard limit law more precisely the usual central limit theorem holds for an observable if and only if its integral along a one codimensional invariant set vanishes otherwise a open image in new window normalization is needed as one of the two key steps in the argument we obtain a limit theorem that holds in young towers with exponential return time statistics in general an abstract result that seems to be applicable to many other situations

differentiability of thermodynamical quantities in non uniformly expanding dynamics abstract in this paper we study the ergodic theory of a robust non uniformly expanding maps where no markov assumption is required we prove that the topological pressure is differentiable as a function of the dynamics and analytic with respect to the potential moreover we not only prove the continuity of the equilibrium states and their metric entropy as well as the differentiability of the maximal entropy measure and extremal lyapunov exponents with respect to the dynamics we also prove a local large deviations principle and central limit theorem and show that the rate function mean and variance vary continuously with respect to observables potentials and dynamics finally we show that the correlation function associated to the maximal entropy measure is differentiable with respect to the dynamics and it is c 1 convergent to zero in addition precise formulas for the derivatives of thermodynamical quantities are given

an asymptotic property of huisken s functional on minimal submanifolds of euclidean space abstract in this short note we study the asymptotic property of huisken s functional for mean curvature flow on the minimal submanifolds of euclidean space we prove that the limit of huisken s functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of euclidean space

floquet spectrum of weakly coupled map lattices we consider weakly coupled analytic expanding circle maps on the lattice d for d 1 with small coupling strength e and summable decay of the two sites coupling we study the spectrum of the associated perron frobenius transfer operators e on a suitable banach space perturbation theory applied to the difference of a high iterate n 0 of the transfer operators 0 n 0 and e n 0 yields localisation of the full spectrum of e n 0 the time transfer operator e n 0 commutes with the spatial translations and we provide a description of part of their joint eigenvalues more precisely of the nonresonant spectrum of e n 0 restricted to the eigenspaces of the spatial translations in particular we exhibit smooth curves of eigenvalues and eigenspaces of e n 0 as functions of the eigenvalues e i crystal momenta of the spatial translations

resonances for random highly oscillatory potentials we study discrete spectral quantities associated with schrodinger operators of the form rd vn d odd the potential vn models a highly disordered crystal it varies randomly at scale n 1 1 we use perturbation analysis to obtain almost sure convergence of the eigenvalues and scattering resonances of rd vn as n we identify a stochastic and a deterministic regime for the speed of convergence the type of regime depends whether the low frequency effects due to large deviations overcome the deterministic constructive interference between highly oscillatory terms

on almost sure versions of classical limit theorems for dynamical systems the purpose of this article is to support the idea that whenever we can prove a limit theorem in the classical sense for a dynamical system we can prove a suitable almost sure version based on an empirical measure with log average we follow three different approaches martingale methods spectral methods and induction arguments our results apply among others to axiom a maps or flows to systems inducing a gibbs markov map and to the stadium billiard

exponential decay of correlations for random lasota yorke maps we consider random piecewise smooth piecewise invertible maps mainly on the interval but also in higher dimensions we assume that on the average and possibly without any stochastic uniformity i the maps expand distances ii do not have too many pieces iii do not have too large a distortion and iv are strongly mixing we assume it no markov property we prove that as in the classical case of the iteration of a fixed piecewise expanding map of the interval we have it exponential decay of random correlations our proof builds on the one given by c liverani for deterministic mixing and piecewise expanding interval maps

decay of correlations for certain quadratic maps we prove exponential decay of correlations for f wheref belongs in a positive measure set of quadratic maps of the interval and is its absolutely continuous invariant measure these results generalize to other interval maps

entropy formula of pesin type for noninvertible random dynamical systems in this paper we consider random dynamical systems abbreviated henceforth as rds s generated by compositions of random endomorphisms maybe noninvertible and with singularities of class c 2 of a compact manifold entropy formula of pesin type is proved for such rds s under some absolute continuity conditions on the associated invariant measures

the finite element method for parabolic equations ii a posteriori error estimation and adaptive approach abstract we extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper 3 for time independent space meshes in the context of the model problem studied there results are given relating the effectivity of the error estimator to properties of the solution space meshes and manner in which the meshes change a procedure based upon this theory is presented for the adaptive construction of time dependent meshes the results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems author

pseudo orbit shadowing in the family of tent maps on montre que les applications en tente ont la propriete d ombrage pour presque tous les parametres s mais elles n ont pas cette propriete pour un ensemble dense non denombrable des parametres on montre que pour une application tente chaque pseudo orbite peut etre approchee par une orbite actuelle d une application en tente avec une boucle peut etre legerement plus grande

dispersion relation preserving stereo modeling method beyond nyquist frequency for acoustic wave equation abstractaccording to nyquist shannon sampling theory two sampling points of the displacements per minimal wavelength are required to reconstruct a wavefield however high frequency components above the nyquist frequency can be theoretically recovered by introducing spatial derivatives of the displacements we have developed a new finite difference scheme for solving the acoustic equation named the dispersion relation preserving stereo modeling method which is practical for seismic forward modeling beyond the nyquist frequency the key idea in this model is to approximate the high order spatial derivatives with wavenumber domain optimization in wavefields represented by the wave displacements and their gradients we investigated the theoretical properties of this method including the absolute error of spatial operators numerical error stability criterion and computational efficiency our results indicated that the method can effectively suppress the numerical dispersion on extremely coarse grids wi

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

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

volume growth and bernstein theorems for translating solitons abstract in this paper we study properties of the potential function of a translating soliton m in r n 1 and the volume growth of the intersection of euclidean balls with m we show that if m is a convex translating soliton then the infimum of the mean curvature function on m is zero i e inf m h 0 we give a condition to obtain the bernstein theorem for the translating solitons we also give an outline of a simple proof of the bernstein theorem due to bombieri de giorgi miranda

grid modification for the wave equation with attenuation the wave equation with attenuation due to a linear friction is approximated by a new mixed finite element method which allows one to use different grids and basis functions at different times when necessary this method enables one to track sharp moving wave fronts more efficiently and accurately error estimates with optimal convergent rates are established unconditional stability is also proved for this method

dimension theory for invariant measures of endomorphisms we establish the exact dimensional property of an ergodic hyperbolic measure for a c 2 non invertible but non degenerate endomorphism on a compact riemannian manifold without boundary based on this we give a new formula of lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms our results extend several well known theorems of barreira et al ann math 149 755 783 1999 and ledrappier and young commun math phys 117 4 529 548 1988 for diffeomorphisms to the case of endomorphisms

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems