non uniform specification and large deviations for weak gibbs measures we establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak gibbs measures under some mild assumptions we obtain upper and lower bounds for the measure of deviation sets of some non uniformly expanding maps including quadratic maps and robust multidimensional non uniformly expanding local diffeomorphisms for that purpose a measure theoretical weak form of specification is introduced and proved to hold for the robust classes of multidimensional non uniformly expanding local diffeomorphisms and viana maps

equilibrium states for the geodesic flow we give in this paper new results of large deviation type for the geodesic flow on a closed riemannian manifold which describe the proportion of geodesic arcs supporting measures close to the equilibrium states we introduce zeta functions in terms of geodesic arcs and show that they define holomorphic functions on a half plane given by the topological pressure

large deviation principle for benedicks carleson quadratic maps since the pioneering works of jakobson and benedicks carleson and others it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to lebesgue these measures allow one to statistically predict the asymptotic fate of lebesgue almost every initial condition estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space we use the sub exponential slow recurrence condition of benedicks carleson to build induced markov maps of arbitrarily small scale and associated towers to which the absolutely continuous measures can be lifted these various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle for the absolutely continuous measures this result encompasses dynamics far from equilibrium and thus significantly extends presently known local large deviations results for quadratic maps

nonconventional large deviations theorems we obtain large deviations theorems for both discrete time expressions of the form sum n 1 nf big x q 1 n ldots x q ell n big and similar expressions of the form int 0 tf big x q 1 t ldots x q ell t big dt in continuous time here x n n ge 0 or x t t ge 0 is a markov process satisfying doeblin s condition f is a bounded continuous function and q i n in for i le k while for i k they are positive functions taking on integer values on integers with some growth conditions which are satisfied for instance when q i s are polynomials of increasing degrees applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations will be obtained as well

multidimensional random subshifts of finite type and their large deviations i introduce random multidimensional subshifts of finite type which generalize models of spin glasses and establish the almost sure large deviations bounds for gibbs measures there the paper is sequel to ekw where the corresponding results were obtained for deterministic multidimensional subshifts of finite type

large deviations for probabilistic cellular automata we consider a generalized model of a probabilistic cellular automata described by a markov chain on an infinite dimensional space and derive certain large deviations bounds for corresponding occupational measures

homogeneous open quantum random walks on a lattice we study open quantum random walks oqrws for which the underlying graph is a lattice and the generators of the walk are homogeneous in space using the results recently obtained in carbone and pautrat ann henri poincare 2015 we study the quantum trajectory associated with the oqrw which is described by a position process and a state process we obtain a central limit theorem and a large deviation principle for the position process we study in detail the case of homogeneous oqrws on the lattice mathbb z d with internal space mathfrak h mathbb c 2

large deviations fluctuations and shrinking intervals this paper concerns the statistical properties of hyperbolic diffeomorphisms we obtain a large deviation result with respect to slowly shrinking intervals for a large class of holder continuous functions in case of time reversal symmetry we obtain a corresponding version of the fluctuation theorem

spatio temporal large deviations principle for coupled circle maps we consider the d 1 dimensional an dynamical system constituted by weakly coupled expanding circle maps on zd together with the spatial shifts this viewpoint allows us to use thermodynamic formalism and to describe the asymptotic behavior of the system in this setup we obtain a volume lemma which describes the exponential behavior of the size under lebesgue measure of dynamical balls around any orbit and then a large deviations principle for the empirical measure associated to this dynamical system the proofs are direct we do not use the coding constructed by jiang for such systems

large deviations for countable to one markov systems in this paper we study large deviation properties for countable to one markov systems associated to weak gibbs measures for non holder potentials furthermore we establish multifractal large deviation laws for countable to one piecewise conformal markov systems which are derived systems constructed over hyperbolic regions for certain nonhyperbolic systems exhibiting intermittency we apply our results to higher dimensional number theoretical transformations

weak expansion properties and large deviation principles for expanding thurston maps abstract in this paper we prove that an expanding thurston map f s 2 s 2 is asymptotically h expansive if and only if it has no periodic critical points and that no expanding thurston map is h expansive as a consequence for each expanding thurston map without periodic critical points and each real valued continuous potential on s 2 there exists at least one equilibrium state for such maps we also establish large deviation principles for iterated preimages and periodic points it follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding thurston map without periodic critical points and a potential that is holder continuous with respect to a visual metric on s 2

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

computation of extreme values of time averaged observables in climate models with large deviation techniques one of the goals of climate science is to characterize the statistics of extreme and potentially dangerous events in the present and future climate extreme events like heat waves droughts or floods due to persisting rains are characterized by large anomalies of the time average of an observable over a long time the framework of donsker varadhan large deviation theory could therefore be useful for their analysis in this paper we discuss how concepts and numerical algorithms developed in the context of with large deviation theory can be applied to study extreme rare fluctuations of time averages of surface temperatures at regional scale with comprehensive numerical climate models when performing this type of analysis unless a rigorous study of the convergence to the large deviation limit is performed it can be easy to be misled in thinking to have reached the asymptotic regime in this paper we provide a systematic protocol to study the convergence of large deviation functions tailored for applications to climate problems referring to the existing literature on the subject we provide explicit formulas to compute large deviation functions directly from time series of a deterministic dynamical system that can be applied to climate records and we describe how to study the convergence we show how using a rare event algorithm applied to a numerical model can improve the efficiency of the computation of the large deviation functions as a case study we consider the time averaged european surface temperature obtained with the numerical climate model plasim we show how a precise analysis of the convergence leads to the conclusion that the large deviation limit is nor properly reached for time scales shorter than a few years and is therefore of no practical interest to study midlatitude heat waves finally we show how even in a case like this rare event algorithms developed to study large deviation functions can be used to improve the statistics of events on time scales shorter than the one needed to reach the large deviation asymptotic regime

limiting measure of lee yang zeros for the cayley tree this paper is devoted to an in depth study of the limiting measure of lee yang zeroes for the ising model on the cayley tree we build on previous works of muller hartmann and zittartz z phys b 22 59 1975 muller hartmann z phys b 27 161 168 1977 barata and marchetti j stat phys 88 231 268 1997 and barata and goldbaum j stat phys 103 857 891 2001 to determine the support of the limiting measure prove that the limiting measure is not absolutely continuous with respect to lebesgue measure and determine the pointwise dimension of the measure at lebesgue a e point on the unit circle and every temperature the latter is related to the critical exponents for the phase transitions in the model as one crosses the unit circle at lebesgue a e point providing a global version of the phase transition of continuous order discovered by muller hartmann zittartz the key techniques are from dynamical systems because there is an explicit formula for the lee yang zeros of the finite cayley tree of level n in terms of the n th iterate of an expanding blaschke product a subtlety arises because the conjugacies between blaschke products at different parameter values are not absolutely continuous

specification and towers in shift spaces we show that a shift space on a finite alphabet with a non uniform specification property can be modeled by a strongly positive recurrent countable state markov shift to which every equilibrium state lifts in addition to uniqueness of the equilibrium state this gives strong statistical properties including the bernoulli property exponential decay of correlations central limit theorem and analyticity of pressure which are new even for uniform specification we give applications to shifts of quasi finite type synchronised and coded shifts and factors of beta shifts and s gap shifts

detection of influential observation in linear regression a new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models it is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values two examples are presented

large deviations averaging and periodic orbits of dynamical systems the paper deals with large deviation bounds for the proportion of periodic orbits with irregular behavior for expansive dynamical systems with specification in particular we obtain estimates for large deviations from the equidistribution for closed geodesics on negatively curved manifolds we derive also large deviation bounds in the averaging principle when the fast motion is the shift along periodic orbits

a diagnostic for cox regression and general conditional likelihoods abstract diagnostics for changes in the maximum likelihood estimate of due to deletion of single observations in linear and logistic regression models are equivalent to diagnostics obtained from fitting augmented regression models for the cox proportional hazards model and in conditional models for arbitrarily matched or stratified data diagnostics derived from augmented regression models are again useful and easy to compute these diagnostics are applicable to different formulations for the multiplicative covariable effects over time or over strata an exact or approximate treatment of tied failure times or multiple case matching and time dependent covariates

a spectral approach for quenched limit theorems for random hyperbolic dynamical systems we extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving to quenched random piecewise hyperbolic dynamics for general ergodic sequences of maps in a neighborhood of a hyperbolic map we prove a quenched large deviations principle ldp central limit theorem clt and local central limit theorem lclt

the large deviation principle and steady state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades there exist very few results for steady state fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes due to the technical difficulties here we give a proof for the steady state fluctuation theorem of a diffusion process in magnetic fields with explicit expressions of the free energy function and rate function the proof is based on the karhunen loeve expansion of complex valued ornstein uhlenbeck process

large deviations for expanding transformations with additive white noise large deviations estimates for the autocorrelations of order kof the random process zn xn n n 0 are obtained the processes xn n 0and n n 0are independent n n 0 are i i d bounded random variables xn tn x0 n mathbb n t m mis expanding leaving invariant a gibbs measure on a compact set m and m mathbb r is a continuous function a possible application of this result is the case where mis the unit circle and the gibbs measure is the one absolutely continuous with respect to the lebesgue measure on the circle the case when tis a uniquely ergodic map was studied in carmona et al 1998 in the present paper tis an expanding map however it is possible to derive large deviations properties for the autocorrelations samples 1 n n 1j 0zjzj k but the deviation function is quite different from the uniquely ergodic case because it is necessary to take into account the entropy of invariant measures for tas an important information the method employed here is a combination of the variational principle of the thermodynamic formalism with donsker and varadhan s large deviations approach

computing the pressure for axiom a attractors by time series and large deviations for the lyapunov exponent for the axiom a attractors a relation is given between the topological pressure and the spectrum of the generalized lyapunov exponents as a consequence a simple formula is found to compute the topological entropy of the attractor by means of a time series the results are used to compute the large deviations for positive lyapunov exponents

a spectral approach for quenched limit theorems for random expanding dynamical systems we prove quenched versions of i a large deviations principle ldp ii a central limit theorem clt and iii a local central limit theorem lclt for non autonomous dynamical systems a key advance is the extension of the spectral method commonly used in limit laws for deterministic maps to the general random setting we achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter while some versions of the ldp and clt have previously been proved with other techniques the local central limit theorem is to our knowledge a completely new result and one that demonstrates the strength of our method applications include non autonomous piecewise expanding maps defined by random compositions of the form t n 1 t t an important aspect of our results is that we only assume ergodicity and invertibility of the random driving in particular no expansivity or mixing properties are required

the gluing orbit property uniform hyperbolicity and large deviations principles for semiflows abstract we introduce a gluing orbit property weaker than specification for both continuous maps and flows we prove that flows with the c 1 robust gluing orbit property are uniformly hyperbolic and that every uniformly hyperbolic flow satisfies the gluing orbit property we also prove a level 1 large deviations principle and a level 2 large deviations lower bound for semiflows with the gluing orbit property as a consequence we establish a level 1 large deviations principle for hyperbolic flows and every continuous observable and also a level 2 large deviations lower bound finally since many non uniformly hyperbolic flows can be modeled as suspension flows we also provide criteria for such flows to satisfy uniform and non uniform versions of the gluing orbit property

averaging in dynamical systems and large deviations the paper treats ordinary differential equations of the form frac dz varepsilon left t right dt varepsilon b z varepsilon t f t y wheref t is a hyperbolic flow large deviations bounds for the averaging principle are obtained here in the form appeared previously in f1 f2 for the case when the flowf t is replaced by a markov process

free energy for brownian and geodesic homology existence and analyticity of the free energies associated with the asymptotic homology of brownian paths and geodesics are proved and a simple relation is found between them

relative entropy dimensions and large deviations for g measures we prove large deviation results for empirical measures with respect to an arbitrary g measure on a subshift of finite type by using an ergodic theorem for relative entropy we establish extending some results of cajar on billingsley dimensions to g measures we relate the relative entropy to the dimensions of saturated sets this allows a dimensional interpretation of the large deviation results and the study of the set of non generic points which are proved to be of full dimension

large deviations and a fluctuation symmetry for chaotic homeomorphisms we consider expansive homeomorphisms with the specification property we give a new simple proof of a large deviation principle for gibbs measures corresponding to a regular potential and we establish a general symmetry of the rate function for the large deviations of the antisymmetric part under time reversal of the potential this generalizes the gallavotti cohen fluctuation theorem to a larger class of chaotic systems

fourier transforms of gibbs measures for the gauss map we investigate under which conditions a given invariant measure mu for the dynamical system defined by the gauss map x mapsto 1 x mathrm mod 1 is a rajchman measure with polynomially decaying fourier transform begin aligned widehat mu xi o xi eta quad text as quad xi rightarrow infty end aligned we show that this property holds for any gibbs measure mu of hausdorff dimension greater than 1 2 with a natural large deviation assumption on the gibbs potential in particular we obtain the result for the hausdorff measure and all gibbs measures of dimension greater than 1 2 on badly approximable numbers which extends the constructions of kaufman and queffelec ramare our main result implies that the fourier stieltjes coefficients of the minkowski s question mark function decay to 0 polynomially answering a question of salem from 1943 as an application of the davenport erd s leveque criterion we obtain an equidistribution theorem for gibbs measures which extends in part a recent result by hochman shmerkin our proofs are based on exploiting the nonlinear and number theoretic nature of the gauss map and large deviation theory for hausdorff dimension and lyapunov exponents

large deviations for probabilistic cellular automata ii we obtain an upper large deviations bound which shows that for some models of probabilistic cellular automata which are far away from the product case the lower large deviation bound derived in eizenberg and kifer j stat phys 108 1255 1280 2002 is sharp and so the corresponding large deviations phenomena cannot be described via the traditional donsker varadhan form of the action functional for models which are close to the product case we derive approximate large deviations bounds using the donsker varadhan functional for the product case

large deviations for vector valued functionals of a markov chain lower bounds on obtient des bornes inferieures que l on exprime en termes des parametres de convergence de certains noyaux on donne une application aux mesures empiriques de chaines de markov

uniqueness of minimizer for countable markov shifts and equidistribution of periodic points for a finitely irreducible countable markov shift and a potential with summable variations we provide a condition on the associated pressure function which ensures that bowen s gibbs state the equilibrium state and the minimizer of the level 2 large deviations rate function are all unique and they coincide from this we deduce that the set of periodic points weighted with the potential equidistributes with respect to the gibbs equilibrium state as the periods tend to infinity applications are given to the gauss map and the bowen series map associated with a finitely generated free fuchsian group with parabolic elements

macroscopic determinism in interacting systems using large deviation theory we consider the quasi deterministic behavior of systems with a large number n of deterministically interacting constituents this work extends the results of a previous paper j statist phys 99 1225 1249 2000 to include vector valued observables on interacting systems the approach used here however differs markedly in that a level 1 large deviation principle ldp on joint observables rather than a level 2 ldp on empirical distributions is employed as before we seek a mapping t on the set of possibly vector valued macrostates such that when the macrostate is given to be a0 at time zero the macrostate at time t is t a0 with a probability approaching one as n tends to infinity we show that such a map exists and derives from a generalized dynamic free energy function provided the latter is everywhere well defined finite and differentiable we discuss some general properties of t relevant to issues of irreversibility and end with an example of a simple interacting lattice for which an exact macroscopic solution is obtained

large deviation principle in one dimensional dynamics we study the dynamics of smooth interval maps with non flat critical points for every such a map that is topologically exact we establish the full level 2 large deviation principle for empirical means in particular the large deviation principle holds for every non renormalizable quadratic map this includes the maps without physical measure found by hofbauer and keller and challenges the widely shared view of the large deviation principle as a refinement of laws of large numbers

large deviations for non uniformly expanding maps we obtain large deviation bounds for non uniformly expanding maps with non flat singularities or criticalities and for partially hyperbolic non uniformly expanding attracting sets that is given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map showing that the lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved as easy by products we deduce escape rates from subsets of the basins of physical measures for these types of maps the rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states

residuals for relative risk regression summary several possible definitions of residuals are given for relative risk regression with time varying covariates each such residual has a representation as an estimator of a stochastic integral with respect to the martingale arising from a subject s failure time counting process previously proposed residuals for individual study subjects and for specific time points are shown to be special cases of this definition as are previously derived regression diagnostics an illustration and various generalizations are also given able methods are required to detect various departures from modelling assumptions suitably defined residuals may play an important role in such identification however the nonparametric aspect of the model the possibility that modelled regression variables may be varying with follow up time and most importantly the usual presence of right censorship implies that specialized residual definitions are required the class of residuals considered here is most easily formulated using counting process notation for the failure time data let ni t yi t and zi t represent respectively for the ith subject the values of counting censoring and covariate processes at follow up time t i 1 n while ni u yi u zi u 0c u t specifies the corresponding histories for the ith subject prior to time t thus in a typical univariate failure time application ni with right continuous sample paths will take value zero prior to the time of failure on the ith subject and value one thereafter while yi with left continuous sample paths will take value one at times at which the ith subject is at risk for an observed failure and value zero otherwise the counting process ni can be uniquely decomposed so that for all t i

influential observations and outliers in regression statistics offered by cook 1977 and andrews and pregibon 1978 purport to reveal influential observations in a regression analysis detailed examination of these statistics shows that two different types of influence are being measured and this is illustrated with examples derived from a set of data given by mickey dunn and clark 1967 recommendations are given for obtaining the best use of the statistics available

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

two graphical displays for outlying and influential observations in regression summary the paper describes two procedures for detecting observations with outlying values either in the response variable or in the explanatory variables in multiple regression these procedures are presented as half normal plots with envelopes derived from simulation in order to avoid overinterpretation of the data analysis of a well known data set leads to the use of a data transformation a simple test for which is commended and to some comments on the relationship with robust regression the widespread availability of sophisticated computer software has made the fitting of multiple regression equations painless but the very ease of these procedures may cause insufficient care to be given to checking and scrutiny of the data transcription punching and data manipulation errors may lead to bad values both of the observations y and of the independent or explanatory variables x it is the purpose of the present paper to describe and exemplify two plots which provide checks against such bad values it is argued that these plots and a test for transformations should accompany any thorough regression analysis bad values of the response or outliers have long been detected by a variety of plots of residuals bad values of the explanatory variables are less easily detected even when they lead to an extreme point in the design space since the fitted equation will pass close to such an influential observation the residual may not be especially large even after allowance has been made for the small variance of the fitted value at such a point to detect such behaviour a quantity is needed which exhibits the dependence of the fitted model on each point or group of points discussions of such quantities are given by hoaglin welsch 1978 and more recently by belsley kuh welsch 1980 2 1 by cook weisberg 1980 and by pregibon 1981 the quantity used here is a modification of a statistic proposed by cook 1977 which is derived in 2 two half normal plots are presented in 3 as an aid to interpretation envelopes to the plots are generated by simulation examples of the plots are given in 4 both for simulated data and for the stack loss data given by brownlee 1965 p 454 transformations of this data set are discussed in 5 the paper concludes with some comments on the relationship with robust

large deviations for ℤ d actions we establish large deviations bounds for translation invariant gibbs measures of multidimensional subshifts of finite type this generalizes fo and partially c o and b where only full shifts were considered our framework includes in particular the hard core lattice gas models which are outside of the scope of fo c o and b

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

influence and a quadratic form in the andrews pregibon statistic a quadratic form found in the andrews pregibon ap statistic may be decomposed into two terms one of the terms has a simple geometric interpretation that relates naturally to the expression of influence found in cook s statistic the measure of influence derived from the quadratic form is easy to compute once the two factors of the ap statistic have been found this measure suggests a computing strategy for diagnostics and leads to an exact relation between the ap statistic and cook s statistic

random continued fractions levy constant and chernoff type estimate abstract given a stochastic process a n n 1 taking values in natural numbers the random continued fractions are defined as a 1 a 2 a n analogously to the continued fraction expansion of real numbers assume that a n n 1 is ergodic and the expectation e log a 1 we give a levy type metric theorem which covers that of real case presented by levy in 1929 moreover a corresponding chernoff type estimate is obtained under the conditions that a n n 1 is mixing and for each 0 t 1 e a 1 t

multiplicative ergodic theorem for a non irreducible random dynamical system we study the asymptotic properties of the trajectories of a discrete time random dynamical system in an infinite dimensional hilbert space under some natural assumptions on the model we establish a multiplica tive ergodic theorem with an exponential rate of convergence the assumptions are satisfied for a large class of parabolic pdes including the 2d navier stokes and complex ginzburg landau equations perturbed by a non degenerate bounded random kick force as a consequence of this er godic theorem we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system in particular we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible the proof relies on a refined version of the uniform feller property combined with some contraction and bootstrap arguments

deviations of trajectory averages and the defect in pesin s formula for anosov diffeomorphisms a large deviation theorem at the donsker varadhan level three is obtained for the convergence of trajectory averages of anosov diffeomorphisms it is possible to provide an explicit description of the rate function

unique equilibrium states for flows and homeomorphisms with non uniform structure abstract using an approach due to bowen franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the bowen property we show that this conclusion remains true using weaker non uniform versions of specification expansivity and the bowen property we also establish a corresponding result for homeomorphisms in the homeomorphism case we obtain the upper bound from the level 2 large deviations principle for the unique equilibrium state the theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic formalism in partially hyperbolic and non uniformly hyperbolic settings

some large deviation results for dynamical systems we prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics rate of escape problems and to shift spaces

predictive turbulence modeling by variational closure we show that a variational implementation of probability density function pdf closures has the potential to make predictions of general turbulence mean statistics for which a priori knowledge of the incorrectness is possible this possibility exists because of realizability conditions on effective potential functions for general turbulence statistics these potentials measure the cost for fluctuations to occur away from the ensemble mean value in empirical time averages of the given variable and their existence is a consequence of a refined ergodic hypothesis for the governing dynamical system navier stokes dynamics approximations of the effective potentials can be calculated within pdf closures by an efficient rayleigh ritz algorithm the failure of realizability within a closure for the approximate potential of any chosen statistic implies a priori that the closure prediction for that statistic is not converged the systematic use of these novel realizability conditions within pdf closures is shown in a simple 3 mode system of lorenz to result in a statistically improved predictive ability in certain cases the variational method allows an a priori optimum choice of free parameters in the closure to be made