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

singular hyperbolic attractors are chaotic we prove that a singular hyperbolic attractor of a 3 dimensional flow is chaotic in two different strong senses first the flow is expansive if two points remain close at all times possibly with time reparametrization then their orbits coincide second there exists a physical or sinai ruelle bowen measure supported on the attractor whose ergodic basin covers a full lebesgue volume measure subset of the topological basin of attraction moreover this measure has absolutely continuous conditional measures along the center unstable direction is a u gibbs state and is an equilibrium state for the logarithm of the jacobian of the time one map of the flow along the strong unstable direction this extends to the class of singular hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic or axiom a attractors for flows in particular these results can be applied i to the flow defined by the lorenz equations ii to the geometric lorenz flows iii to the attractors appearing in the unfolding of certain resonant double homoclinic loops iv in the unfolding of certain singular cycles and v in some geometrical models which are singular hyperbolic but of a different topological type from the geometric lorenz models in all these cases the results show that these attractors are expansive and have physical measures which are u gibbs states

ergodic theory for markov fibred systems and parabolic rational maps a paraboiic rational map of the riemann sphere admits a non atomic h conformal measure on its julia set where h the hausdorff dimension of the julia set and satisfies 1 2 h 2 with respect to this measure the rational map is conservative exact and there is an equivalent a finite invariant measure finiteness of the measure is characterised central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one a theory of markov fibred systems is developed and parabolic rational maps are considered within this framework

maximum likelihood adaptive filter for partially observed boolean dynamical systems we present a framework for the simultaneous estimation of state and parameters of partially observed boolean dynamical systems pobds simultaneous state and parameter estimation is achieved through the combined use of the boolean kalman filter and boolean kalman smoother which provide the minimum mean square error state estimators for the pobds model and maximum likelihood ml parameter estimation in the presence of continuous parameters ml estimation is performed using the expectation maximization algorithm the performance of the proposed ml adaptive filter is demonstrated by numerical experiments with a pobds model of gene regulatory networks observed through noisy next generation sequencing rna seq time series data using the well known p53 mdm2 negative feedback loop gene regulatory model

on the central limit theorem for dynamical systems etant donne un systeme dynamique aperiodique x t il y a un f l 2 avec fd 0 satisfaisant le theoreme de la limite centrale

limiting spectral distribution of gram matrices associated with functionals of β mixing processes we give asymptotic spectral results for gram matrices of the form n 1 x n x t n where the entries of x n are dependent across both rows and columns and that are functionals of absolutely regular sequences and have only finite second moments we derive under mild dependence conditions in addition to an arithmetical decay condition on the mixing coefficients an integral equation of the stieltjes transform of the limiting spectral distribution of n 1 x n x t n in terms of the spectral density of the underlying process applications to examples of positive recurrent markov chains and dynamical systems are also given

limit theorems for functionals of mixing processes with applications to u statistics and dimension estimation in this paper we develop a general approach for investigating the asymptotic distribution of functional xn f zn k k z of absolutely regular stochastic processes zn n z such functional occur naturally as orbits of chaotic dynamical systems and thus our results can be used to study probabilistic aspects of dynamical systems we first prove some moment inequalities that are analogous to those for mixing sequences with their help several limit theorems can be proved in a rather straightforward manner we illustrate this by re proving a central limit theorem of ibragimov and linnik then we apply our techniques to u statistics matrix equation with symmetric kernel h r r r we prove a law of large numbers extending results of aaronson burton dehling gilat hill and weiss for absolutely regular processes we also prove a central limit theorem under a different set of conditions than the known results of denker and keller as our main application we establish an invariance principle for u processes un h h indexed by some class of functions we finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system

a vector valued almost sure invariance principle for hyperbolic dynamical systems we prove an almost sure invariance principle approximation by d dimensional brownian motion for vector valued holder observables of large classes of nonuniformly hyperbolic dynamical systems these systems include axiom a dieomorphisms and flows as well as systems modelled by young towers with moderate tail decay rates in particular the position variable of the planar periodic lorentz gas with finite horizon approximates a 2 dimensional brownian motion

asymptotic periodicity and banded chaos abstract treating one dimensional maps as dynamical systems we examine their evolution in terms of density flow in particular a type of density evolution known as asymptotic periodicity is studied unlike statistically stable or exact systems asymptotically periodic systems do not in general evolve to an invariant density even though they possess one consequently a nonequilibrium formalism using several metastable states rather than one invariant density is examined as an alternative to study the physical properties of asymptotically periodic systems asymptotic periodicity is demonstrated for the hat map and the quadratic map at the parameters where these maps generate banded chaos or quasiperiodicity using the ergodic properties of asymptotic periodicity a compact expression for the time correlation function is obtained that does not make any assumptions about decoupling the periodic and stochastic components of c t finally a generalization to the boltzmann gibbs entropy known as the conditional entropy is studied as an index characterizing the asymptotic density sequence that emerges in asymptotically periodic systems

even the first iterate of a markov operator is contracting in anl2 norm a weightedl2 norm is introduced in which markov operators e g associated with noisy maps are contracting provided the kernel i e the transitional distribution is smooth enough this results in strong relaxational properties of noisy maps similar to this norm integral functionals appear useful when studying spatiotemporal chaos and random fields

statistical properties of a nonuniformly hyperbolic map of the interval we prove a power law upper bound for the decay of the correlations for holder observables in the case of a nonuniformly hyperbolic map of the interval introduced by gaspard and wang as a piecewise linear approximation of the intermittent map of manneville pomeau the result is then applied to compute the central limit theorem for the same class of observables

rigorous approximation of diffusion coefficients for expanding maps we use ulam s method to provide rigorous approximation of diffusion coefficients for uniformly expanding maps an algorithm is provided and its implementation is illustrated using lanford s map

regularity of srb entropy for geometric lorenz attractors we consider the classical geometric lorenz attractors showing that the srb entropy admits gamma holder continuity for any 0 gamma 1

equilibrium states of weakly hyperbolic one dimensional maps for holder potentials there is a wealth of results in the literature on the thermodynamic formalism for potentials that are in some sense hyperbolic we show that for a sufficiently regular one dimensional map satisfying a weak hyperbolicity assumption every holder continuous potential is hyperbolic a sample consequence is the absence of phase transitions the pressure function is real analytic on the space of holder continuous functions another consequence is that every holder continuous potential has a unique equilibrium state and that this measure has exponential decay of correlations

a robust method for shift detection in time series we present a robust test for change points in time series which is based on the two sample hodges lehmann estimator we develop new limit theory for a class of statistics based on the two sample u quantile processes in the case of short range dependent observations using this theory we can derive the asymptotic distribution of our test statistic under the null hypothesis we study the finite sample properties of our test via a simulation study and compare the test with the classical cusum test and a test based on the wilcoxon mann whitney statistic

approach to gaussian stochastic behavior for systems driven by deterministic chaotic forces we consider skew product dynamical systems that describe the stroboscopic dynamics of a damped particle subjected to a chaotic kick force in a suitable scaling limit the dynamics converges to the ornstein uhlenbeck process we investigate the deterministic chaotic corrections in the vicinity of this gaussian limit case for various examples of chaotic forces we present numerical evidence that for certain classes of chaotic forces the deterministic chaotic corrections of the invariant density are universal we provide analytical results for forces generated by tchebyscheff maps and sketch a renormalization group theory in the space of probability densities

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

exponential inequalities for dynamical measures of expanding maps of the interval we prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval as an immediate corollary we obtain a concentration inequality we apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

slow rates of mixing for dynamical systems with hyperbolic structures we consider invertible discrete time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable return time we show that the decay of correlations of the srb measure associated to that hyperbolic structure is related to the tail of the recurrence times we also give sufficient conditions for the validity of the central limit theorem this extends previous results by young in ann math 147 585 650 1998 israel j math 110 153 188 1999

an empirical central limit theorem for intermittent maps we prove an empirical central limit theorem for the distribution function of a stationary sequence under a dependence condition involving only indicators of half line we show that the result applies to the empirical distribution function of iterates of expanding maps with a neutral fixed point at zero as soon as the correlations are summable

gibbs measures for fibred systems abstract we consider a topological dynamical system t y y on a metric space y which forms a fibre bundle over another dynamical system if t is fibrewise expanding and exact along fibres and if is a holder continuous function we prove the existence of a system of conditional measures called a family of gibbs measures where the jacobian is determined by this theorem reduces to ruelle s perron frobenius theorem when the base of the fibred system consists of a single point the method of proof does not use any form of symbolic representation we also study continuity properties of a family of gibbs measures over the base and give applications to the equilibrium theory of higher dimensional complex dynamics

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

piecewise invertible dynamical systems the aim of the paper is the investigation of piecewise monotonic maps t of an interval x the main tool is an isomorphism of x t with a topological markov chain with countable state space which is described by a 0 1 transition matrix m the behavior of the orbits of points in x under t is very similar to the behavior of the paths of the markov chain every irreducible submatrix of m gives rise to a t invariant subset l of x such that l is the set x of all limit points of the orbit of an x x the topological entropy of l is the logarithm of the spectral radius of the irreducible submatrix which is a l 1 operator besides these sets l there are two t invariant sets y and p such that for all x x the set x is either contained in one of the sets l or in y or in p the set p is a union of periodic orbits and y is contained in a finite union of sets y with y x and has topological entropy zero this isomorphism of x t with a topological markov chain is also an important tool for the investigation of t invariant measures on x results in this direction which are published elsewhere are described at the end of the paper furthermore a part of the proofs in the paper is purely topological without using the order relation of the interval x so that some results hold for more general dynamical systems x t

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

global solutions for a hyperbolic parabolic system of chemotaxis abstract we study a hyperbolic parabolic model of chemotaxis related to tumor angiogenesis in dimensions one and two we consider diffusions given by the fractional laplacian and in particular we prove the global existence of classical solutions in certain dissipation regimes

an invariance principle for maps with polynomial decay of correlations we give a general method of deriving statistical limit theorems such as the central limit theorem and its functional version in the setting of ergodic measure preserving transformations this method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations

asymptotic limit law for the close approach of two trajectories in expanding maps of the circle given two pointsx y s 1 randomly chosen independently by a mixing absolutely continuous invariant measure of a piecewise expanding and smooth mapf of the circle we consider for each e 0 the point process obtained by recording the timesn 0 such that f n x f n y e with the further assumption that the density of is bounded away from zero we show that when e tends to zero the above point process scaled by e 1 converges in law to a marked poisson point process with constant parameter measure this parameter measure is given explicity by an average on the rate of expansion off

regularity and other properties of absolutely continuous invariant measures for the quadratic family in the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of jakobson s theorem in particular we show that the density of the invariant measure is continuous at misiurewicz points from this we deduce that the lyapunov exponent is also continuous at these points our considerations apply just to the parameters constructed in the proof of jakobson s theorem other properties like the positivity of the lyapunov exponent uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system are also proved

almost sure invariance principle for nonuniformly hyperbolic systems we prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems discrete time systems and flows are covered by this result in particular the result applies to the planar periodic lorentz flow with finite horizon statistical limit laws such as the central limit theorem the law of the iterated logarithm and their functional versions are immediate consequences

iterates of expanding maps the iterates of expanding maps of the unit interval into itself have many of the properties of a more conventional stochastic process when the expanding map satisfies some regularity conditions and when the starting point is suitably chosen at random in this paper we show that the sequence of iterates can be closely tied to an m dependent process this enables us to prove good bounds on the accuracy of gaussian approximations our main tools are coupling and stein s method

statistical analysis and spectral estimation techniques for one dimensional chaotic signals signals arising out of nonlinear dynamics are compelling models for a wide range of both natural and man made phenomena in contrast to signals arising out of linear dynamics extremely rich behavior is obtained even when we restrict our attention to one dimensional 1 d chaotic systems with certain smoothness constraints an important class of such systems are the so called markov maps we develop several properties of signals obtained from markov maps and present analytical techniques for computing a broad class of their statistics in closed form these statistics include for example correlations of arbitrary order and all moments of such signals among several results we demonstrate that all markov maps produce signals with rational spectra and we can therefore view the associated signals as chaotic arma processes with chaotic white noise as a special case finally we also demonstrate how markov maps can be used to approximate to arbitrary accuracy the statistics any of a broad class of non markov chaotic maps

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

system identification of nonlinear state space models this paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state space form more specifically a maximum likelihood ml framework is employed and an expectation maximisation em algorithm is derived to compute these ml estimates the expectation e step involves solving a nonlinear state estimation problem where the smoothed estimates of the states are required this problem lends itself perfectly to the particle smoother which provides arbitrarily good estimates the maximisation m step is solved using standard techniques from numerical optimisation theory simulation examples demonstrate the efficacy of our proposed solution

specification on the interval we study the consequences of discontinuities on the specification property for interval maps after giving a necessary and sufficient condition for a piecewise monotonic piecewise continuous map to have this property we show that for a large and natural class of families of such maps including the 3 transformations the set of parameters for which the specification property holds though dense has zero lebesgue measure thus regarding the specification property the general case is at the opposite of the continuous case solved by a m blokh russian math surveys 38 1983 133 134 for which we give a proof

almost sure invariance principle for dynamical systems by spectral methods we prove the almost sure invariance principle for stationary d valued random processes with very precise dimension independent error terms solely under a strong assumption concerning the characteristic functions of these processes this assumption is easy to check for large classes of dynamical systems or markov chains using strong or weak spectral perturbation arguments

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

asymptotic periodicity of the iterates of positivity preserving operators soit x un espace de banach reel soit x un sous ensemble ferme de x tel que 1 si x x y x 0 alors x y x et x x 2 il existe m 0 0 tel que pour x x il existe x x et x x qui satisfont x x x x m 0 x m 0 x x et si x y y pour y x y x alors y x x 3 si x x y x alors x x y soit b un operateur lineaire borne sur x soit bx cx f0 est un sous ensemble compact non vide de x et lim n dist b n x f 0 0 quand x x et x 1 alors b n x est asymptotiquement periodique pour x x

law of iterated logarithm and invariance principle for one parameter families of interval maps we show that for almost every map in a transversal one parameter family of piecewise expanding unimodal maps the birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm this result will follow from an almost sure invariance principle for the birkhoff sum as a function on the parameter space furthermore we obtain a similar result for general one parameter families of piecewise expanding maps on the interval

rigorous statistical procedures for data from dynamical systems various questions about the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations this includes symmetry tests and the determination of dimension coefficients the paper contains the theoretical results and several simulations explain the methods

chains of infinite order chains with memory of variable length and maps of the interval we show how to construct a topological markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order in particular we characterize the maps corresponding to stochastic chains with memory of variable length the problem treated here is the converse of the classical construction of the gibbs formalism for markov expanding maps of the interval

limit theorems and markov approximations for chaotic dynamical systems we prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients we also develop techniques of markov approximations for dynamical systems we apply our results to expanding interval maps axiom a diffeomorphisms chaotic billiards and hyperbolic attractors

new dependence coefficients examples and applications to statistics to measure the dependence between a real valued random variable x and a algebra we consider four distances between the conditional distribution function of x given and the distribution function of x the coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations in particular we show that they are well adapted to functions of mixing sequences iterated random functions and dynamical systems starting from a new covariance inequality we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence we obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis under a minimal condition on the coefficients using recent results we show that our coefficients may be also used to obtain various exponential inequalities a concentration inequality for lipschitz functions and a berry esseen type inequality

brownian motion from deterministic dynamics abstract certain deterministic dynamical systems exhibit a transition from non gaussian chaotic to brownian motion behavior when a suitable scaling limit is applied for a simple hyperbolic model system we analyze the structure of the corresponding strange attractor determine the invariant measure and elucidate the transition scenario leading to a gaussian stochastic process

a strong pair correlation bound implies the clt for sinai billiards we investigate the possibility of proving the central limit theorem clt for dynamical systems using only information on pair correlations a strong bound on multiple correlations is known to imply the clt chernov and markarian in chaotic billiards 2006 in chernov s paper j stat phys 122 6 2006 such a bound is derived for dynamically holder continuous observables of dispersing billiards here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations thus a strong bound on pair correlations alone implies the clt for a wider class of observables the result is extended to anosov diffeomorphisms in any dimension some non invertible maps are also considered

asymptotic periodicity of the iterates of markov operators on dit que p l 1 l 1 est un operateur de markov si 1 pf 0 pour f 0 et 2 pf f si f 0 on montre que tout operateur de markov p a une certaine decomposition spectrale si pout tout f l 1 avec f 0 et f 1 p n f 7b f quand n ou 7b f est un sous ensemble fortement compact de l 1

statistics of close visits to the indifferent fixed point of an interval map we study a dynamical system defined by a map of the interval 0 1 which has 0 as an indifferent fixed point but is otherwise expanding we prove that the sequence of successive entrance times in a small neighborhood 0 a converges in law when suitably normalized to a homogeneous poisson point process

rate of convergence in the weak invariance principle for deterministic systems we obtain the first results on convergence rates in the prokhorov metric for the weak invariance principle functional central limit theorem for deterministic dynamical systems our results hold for uniformly expanding hyperbolic axiom a systems as well as nonuniformly expanding hyperbolic systems such as dispersing billiards henon like attractors viana maps and intermittent maps as an application we obtain convergence rates for deterministic homogenization in multiscale systems

on mean field fluctuations in globally coupled maps abstract we consider the mean field fluctuations in the globally coupled map x n 1 i f x n i 1 n j 1 n x n j with an expanding piecewise linear local map f in the thermodynamic limit n using the reduced master equation self consistent frobenius perron operator it is shown that correlations between the elements are absent in simultaneous ensembles and only arise in time series for small the equilibrium state with constant mean field is proved to be unstable so the mean field fluctuates corresponding to a complex attractor of the reduced master equation it is chaotic and its lyapunov exponent is estimated to be 2 the order of magnitude of stationary fluctuations is estimated as exp o 2 the theoretical approach used enables us to develop high precision numerical methods able to detect such tiny fluctuations

decay of correlations dedicated to micheline ishay i would like to thank p boyland l chierchia v donnay g de martino c gole j l lebowitz m lyubich m rychlik i g schwarz s vaienti and especially g gallavotti for helpful and enlightening discussions particularly warm thanks go to n chernov for carefully reading and finding a mistake in an early version the present paper benefits from several improvements due to his sharp criticism in addition i am indebted to p collet and most of all m wojtkowski for introducing me to the magical world of cones finally i thank j milnor director of the institute for mathematical sciences at stony brook university where i was visiting during part of this work and the italian c n r gnfm for providing travel funds

particle filters for partially observed boolean dynamical systems abstract partially observed boolean dynamical systems pobds are a general class of nonlinear models with application in estimation and control of boolean processes based on noisy and incomplete measurements the optimal minimum mean square error mmse algorithms for pobds state estimation namely the boolean kalman filter bkf and boolean kalman smoother bks are intractable in the case of large systems due to computational and memory requirements to address this we introduce approximate mmse filtering and smoothing algorithms based on the auxiliary particle filter apf method which are called apf bkf and apf bks respectively for joint state and parameter estimation the apf bkf is used jointly with maximum likelihood ml methods for simultaneous state and parameter estimation in pobds models in the case the unknown parameters are discrete the proposed ml adaptive filter consists of multiple apf bkfs running in parallel in a manner reminiscent of the multiple model adaptive estimation mmae method in classical linear filtering theory in the presence of continuous parameters the proposed ml adaptive filter is based on an efficient particle based expectation maximization em algorithm for the pobds model which is based on a modified forward filter backward simulation ffbsi in combination with the apf bks the performance of the proposed particle based adaptive filters is assessed through numerical experiments using a pobds model of the well known cell cycle gene regulatory network observed through noisy rna seq time series data

regularity of the metric entropy for expanding maps the main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the perron frobenius operator for expanding mappings as a consequence we are able to show that the metric entropy of an expanding map has modulus of continuity x log 1 x on the space of c2 expandings we also give an explicit estimate of the rate of mixing for cl functions in terms of natural constants it seems that the method we present can be generalized to other classes of dynamical systems which have a distinguished invariant measure like axiom a diffeomorphisms it also can be adopted to show that the entropy of the quadratic family f x 1 ux2 computed with respect to the absolutely continuous invariant measure found in jakobson s theorem varies continuously the last result is going to appear somewhere else

dynamical systems of langevin type abstract dynamical systems of langevin type are deterministic mappings that arise if the gaussian white noise of a langevin equation is replaced by a deterministic chaotic dynamics we describe the various types of mappings that can be obtained in this way and investigate typical transition scenarios from complicated non gaussian to langevin like behavior that occur if a time scale parameter is changed we define a new characteristic quantify fir ergodic mappings called effective langevin radius which is related to the critical time scale where the marginal invariant density of the dynamical system of langevin type loses its differentiability the invariant density is shown to contain useful information on the entire characteristic functional of the chaotic driving force some general symmetry properties of the invariant densities are discussed

interval maps quasi symmetrically conjugate to a piecewise affine map abstract consider a multimodal interval map f of c 3 with non flat critical points we establish several characterizations of the map f that is quasi symmetrically conjugate to a piecewise affine map in the case f is topologically exact and all of its periodic points are hyperbolic repelling in particular we give a negative answer to a question posed by henk bruin in 1

a bernstein type inequality for some mixing processes and dynamical systems with an application to learning we establish a bernstein type inequality for a class of stochastic processes that include the classical geometrically mixing processes rio s generalization of these processes as well as many time discrete dynamical systems modulo a logarithmic factor and some constants our bernsteintype inequality coincides with the classical bernstein inequality for i i d data we further use this new bernstein type inequality to derive an oracle inequality for generic regularized empirical risk minimization algorithms and data generated by such processes applying this oracle inequality to support vector machines using the gaussian kernels for both least squares and quantile regression it turns out that the resulting learning rates match up to some arbitrarily small extra term in the exponent the optimal rates for i i d processes

approach to equilibrium for locally expanding maps in ℝ k by using a well known technique from classical statistical mechanics of one dimensional lattice spin systems we prove existence of an absolutely continuous invariant asymptotic measure for certain locally expanding mapst of the unit cube in k we generalize herewith in a certain sense the results of lasota and yorke on piecewise expanding maps of the unit interval to higher dimensions we show a kuzmin type theorem for these systems from which exponential approach to equilibrium and strong mixing properties follow