nonequilibrium density profiles in lorentz tubes with thermostated boundaries we consider a long lorentz tube with absorbing boundaries particles are injected into the tube from the left end we compute the equilibrium density profiles in two cases the semi infinite tube in which case the density is constant and a long finite tube in which case the density is linear in the latter case we also show that convergence to equilibrium is well described by the heat equation in order to prove these results we obtain new results for the lorentz particle that are of independent interest first we show that a particle conditioned not to hit the boundary for a long time converges to the brownian meander second we prove several local limit theorems for particles having a prescribed behavior in the past 2014 wiley periodicals inc

diagonalization of the quadratic boson hamiltonian with zero modes i mathematical the necessary mathematical equipment is developed for an analysis of the zero modes connected with hamiltonians quadratic in boson creation and annihilation operators a cross matrix being a matrix of a certain partitioned form we consider the group g of conjuctivity transformations t where t is a 2m square cross matrix subjected to the restriction t it i i diag i i i is the m square unit matrix with respect to this group the concept of standard form is introduced for the set of positive semidefinite hermitian cross matrices d it is proved that any such matrix d can be transformed into a matrix e of the standard form according to t dt e t g an algorithm suitable as a basis for computer programs is given for finding standardized and standardizing matrices e and t for any given d

mutual information games in multiuser channels with correlated jamming we investigate the behavior of two users and one jammer in an additive white gaussian noise awgn channel with and without fading when they participate in a noncooperative zero sum game with the channel s input output mutual information as the objective function we assume that the jammer can eavesdrop on the channel and can use the information obtained to perform correlated jamming we also differentiate between the availability of perfect and noisy information about the user signals at the jammer under various assumptions on the channel characteristics and the extent of information available at the users and the jammer we show the existence or otherwise nonexistence of a simultaneously optimal set of strategies for the users and the jammer and characterize those strategies whenever they exist

stochastic processes originating in deterministic microscopic dynamics we investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics i e ifq t is the displacement of the test particle we letqa t q at a and consider the distribution of the trajectory qa t in the limit a the randomness of the motion is due entirely to the randomness of the initial state of the fluid test particle or both and the process is generally non markovian nevertheless it can be proven in some cases and we expect it to be true in many more that qa t looks like brownian motion in the limit a some results for simple model systems are presented

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

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

rates of mixing for non markov infinite measure semiflows we develop an abstract framework for obtaining optimal rates of mixing and higher order asymptotics for infinite measure semiflows previously such results were restricted to the situation where there is a first return poincare map that is uniformly expanding and markov as illustrations of the method we consider semiflows over non markov pomeau manneville intermittent maps with infinite measure and we also obtain mixing rates for semiflows over collet eckmann maps with nonintegrable roof function

steady state self diffusion at low density we prove that the motion of a test particle in a hard sphere fluid in thermal equilibrium converges in the boltzmann grad limit to the stochastic process governed by the linear boltzmann equation the convergence is in the sense of weak convergence of the path measures we use this result to study the steady state of a binary mixture of hard spheres of different colors but equal masses and diameters induced by color changing boundary conditions in the boltzmann grad limit the steady state is determined by the stationary solution of the linear boltzmann equation under appropriate boundary conditions

simplified bifurcation analysis of the sixteen vertex model based on the standard representation of equivalence classes of models the search for bifurcation points in the parameter space of sixteen vertex models as described in three papers by gaaff and hijmans1 and hijmans and schram2 3 canbe considerably simplified and clarified by representing the equivalence classes of models having the same partition function by their standard models instead of by their normal models the preference for the standard model arises from the fact that it is characterized by parameters belonging to the irreducible representations of the symmetry group of the partition function whereas the parameters defining the normal model belong to a mixed representation the standard model defined in terms of 10 characteristic parameters is unique apart from a finite stargroup of 96 operations arising from the freedom to permute the eigenvalues and eigenvectors of a characteristic 4 x 4 matrix qss bifurcations occur whenever the set of 10 characteristic parameters is invariant with respect to a non trivial little group of operations from the stargroup

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

on the bifurcations occurring in the parameter space of the sixteen vertex model ii bifurcations arising from degeneracies in the n matrix the analysis of the bifurcation structure in the 16 dimensional parameter space of homogeneous sixteen vertex models started in part i is completed in this paper as before equivalence classes of models having the same partition function are identified by means of a 4 4 diagonal matrix n and a pair of characteristic 2 2 matrices a and b the bifurcation classes of models studied in this part include also classes for which the n matrix shows degeneracies the four primary bifurcation classes uncovered in part i i e the general and complementary eight vertex models and the one sided and doubly cyclic models each give rise to a hierarchy of two kinds of subbifurcations on the one hand the models corresponding to the different patterns of degeneracies of the n matrix and on the other hand the models for which the rotation angles and characterizing the matrices a and b have values 2 na and 2 nb respectively with na and nb integers

on the time evolution of macroscopic systems autodiffusion a basse densite et theoreme de lanford correlations dans l etat initial auto diffusion a densite finie le gaz de lorentz et le theoreme de bunimovich et sinai

on the bifurcations occuring in the parameter space of the sixteen vertex model i discrete bifurcations abstract the 16 dimensional parameter space of homogeneous sixteen vertex models is scanned for bifurcation points i e points corresponding to models which possess extra symmetries not existing in nearby points equivalence classes of models having the same partition function are identified by means of a characteristic normal model represented by a 4 x 4 diagonal matrix n and a pair of 2 x 2 matrices a and b in this paper the matrix n is assumed to be non degenerate and the only bifurcations found are those associated with special types of matrices a and b i e matrices whose decomposition in terms of pauli matrices corresponds to a vector a a 1 a 2 a 3 or b b 1 b 2 b 3 that is invariant with respect to one or more elements of the cubic symmetry group the various bifurcation classes of models found include a the families of general and complementary eight vertex models b discrete sets of doubly and one sided cyclic models and c a number of secondary bifurcation classes within the eight vertex families among which is baxters symmetric eight vertex model

ergodic properties of the lozi mappings in this paper we construct the bowen ruelle measure for the lozi mapping an almost everywhere hyperbolic diffeomorphism of the plane we also derive some of its properties which are similar to those of an axiom a system

correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps we study a robust class of multidimensional non uniformly hyperbolic transformations considered by oliveira and viana ergod theory dyn syst 28 501 533 2008 for an open class of holder continuous potentials with small variation we show that the unique equilibrium state has exponential decay of correlations and that the distribution of hitting times is asymptotically exponential furthermore using that the equilibrium states satisfy a weak gibbs property we also prove log normal fluctuations of the return times around their average

on the boltzmann equation for the lorentz gas we consider the boltzmann grad limit for the lorentz or wind tree model we prove that if is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the poisson distribution with parameter 0 then the evolution of an initial a c particle density tends in the boltzmann grad limit to the solution of the boltzmann equation for the model as an intermediate step we prove that the process of the free path lengths and impact parameters induced by the lebesgue measure on a small region tends to a limiting independent process

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

the gaussian test channel with an intelligent jammer the problem of transmitting a sequence of identically distributed independent gaussian random variables through a gaussian memoryless channel with a given input power constraint in the presence of an intelligent jammer is considered the jammer taps the channel and feeds back a signal at a given energy level for the purpose of jamming the transmitting sequence under a square difference distortion measure which is to be maximized by the jammer and to be minimized by the transmitter and the receiver this correspondence obtains the complete set of optimal saddle point policies the solution is essentially unique and it is structurally different in three different regions in the parameter space which are determined by the signal to noise ratios and relative magnitudes of the noise variances the best maximin policy of the jammer is either to choose a linear function of the measurement he receives through channel tapping or to choose in addition and additively an independent gaussian noise sequence depending on the region where the parameters lie the optimal minimax policy of the transmitter is to amplify the input sequence to the given power level by a linear transformation and that of the receiver is to use a bayes estimator

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

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

intractable problems in control theory this paper is an attempt to understand the apparent intractability of problems in decentralized decision making using the concepts and methods of computational complexity we first establish that the discrete version of an important paradigm for this area proposed by witsenhausen is np complete thus explaining the failures reported in the literature to attack it computationally in the rest of the paper we show that the computational intractability of the discrete version of a control problem the team decision problem in our particular example can imply that there is no satisfactory continuous algorithm for the continuous version to this end we develop a theory of continuous algorithms and their complexity and a quite general proof technique which can prove interesting by themselves

limit theorems for dispersing billiards with cusps dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos exhibiting intermittent behavior that alternates between regular and chaotic patterns their statistical properties are therefore weak and delicate they are characterized by a slow power law decay of correlations and as a result the classical central limit theorem fails we prove that a non classical central limit theorem holds with a scaling factor of sqrt n log n replacing the standard sqrt n we also derive the respective weak invariance principle and we identify the class of observables for which the classical clt still holds

universal hitting time statistics for integrable flows the perceived randomness in the time evolution of chaotic dynamical systems can be characterized by universal probabilistic limit laws which do not depend on the fine features of the individual system one important example is the poisson law for the times at which a particle with random initial data hits a small set this was proved in various settings for dynamical systems with strong mixing properties the key result of the present study is that despite the absence of mixing the hitting times of integrable flows also satisfy universal limit laws which are however not poisson we describe the limit distributions for generic integrable flows and a natural class of target sets and illustrate our findings with two examples the dynamics in central force fields and ellipse billiards the convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices which is of independent interest its proof exploits ratner s measure classification theorem for unipotent flows and extends earlier work of elkies and mcmullen

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

the boltzmann grad limit of the periodic lorentz gas we study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density boltzmanngrad limit a generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a markov process with memory two

dynamics of horizontal like maps in higher dimension we study the regularity of the green currents and of the equilibrium measure associated to a horizontal like map in under a natural assumption on the dynamical degrees we estimate the speed of convergence towards the green currents the decay of correlations for the equilibrium measure and the lyapounov exponents we show in particular that the equilibrium measure is hyperbolic we also show that the green currents are the unique invariant vertical and horizontal positive closed currents the results apply in particular to henon like maps to regular polynomial automorphisms of and to their small perturbations

back to balls in billiards we consider a billiard in the plane with periodic configuration of convex scatterers this system is recurrent in the sense that almost every orbit comes back arbitrarily close to the initial point in this paper we study the time needed to get back in an e ball about the initial point in the phase space and also for the position in the limit when e 0 we establish the existence of an almost sure convergence rate and prove a convergence in distribution for the rescaled return times

recurrence properties of planar lorentz process first return and first hitting times local times and first intersection times are studied for planar finite horizon lorentz processes with a periodic configuration of scatterers their asymptotic behavior is analogous to the asymptotic behavior of the same quantities for the 2 d simple symmetric random walk cf classical results of darling kac 1957 and of erd s taylor 1960 moreover asymptotical distributions for phases in first hittings and in first intersections of lorentz processes are also described the results are also extended to the quasi one dimensional model of the linear lorentz process subject classification 37d50 billiards 60f05 weak theorems

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

the o 3 gauge transformation and 3 state vertex models the authors consider the o 3 gauge transformation for three state vertex models on lattices of coordination number three using an explicit mapping between o 3 and sl 2 they establish that there exist exactly six polynomials of the vertex weights which are fundamentally invariant under the o 3 transformation explicit expressions of these fundamental invariants are obtained in the case of symmetric vertex weights

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

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

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

advanced statistical properties of dispersing billiards a new approach to statistical properties of hyperbolic dynamical systems emerged recently it was introduced by l s young and modified by d dolgopyat it is based on coupling method borrowed from probability theory we apply it here to one of the most physically interesting models sinai billiards it allows us to derive a series of new results as well as make significant improvements in the existing results first we establish sharp bounds on correlations including multiple correlations then we use our correlation bounds to obtain the central limit theorem clt the almost sure invariance principle asip the law of iterated logarithms and integral tests

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

some information theoretic saddlepoints in order to study communication in the presence of jamming or vice versa one can formulate a two person zero sum game with mutual information as the payoff function since mutual information is convex concave in just the right way one often finds saddlepoint strategies which are simultaneously optimal for both players in this paper we shall discuss these saddlepoints for four distinct cases additive noise average power constraints and any combination of hard soft input output quantization one of these cases soft soft has been previously studied shannon dobrushin blachman but the other three appear to be new

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

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

a huygens principle for diffusion and anomalous diffusion in spatially extended systems we present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media for anisotropic systems strong chaos leads to diffusive behavior brownian motion with drift and weak chaos leads to superdiffusive behavior levy processes with drift for isotropic systems the drift term vanishes and strong chaos again leads to brownian motion we establish the existence of a nonlinear huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behavior in odd space dimensions

statistical properties of the universal limit map of grazing bifurcations in this paper the statistical properties of an interval map having a square root singular point which characterizes grazing bifurcations of impact oscillators are studied firstly we show that in some parameter regions the map admits an induced markov structure with an exponential decay tail of the return times then we prove that the map has a unique mixing absolutely continuous invariant probability measure finally by applying the markov tower method we prove that exponential decay of correlations and the central limit theorem hold for holder continuous observations

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

on the classification of the homogeneous 16 vertex models on a square lattice ii connection with a new type of phase transition abstract in the previous paper we have shown that the classification problem of the homogeneous real 16 vertex models on a square lattice as proposed by gaaff and hijmans is closely connected with the similar problem corresponding to the boson bogoliubov transformation in quantum mechanics this insight leads us in the present paper to use arguments and examples from papers dealing with the latter problem in a further study of the classification problem of the real 16 vertex models we also use a somewhat modified concept of standard or normal 16 vertex model which was introduced by hijmans and schram in a similar study concerning the complex 16 vertex models we show that rather than the single family found by hijmans and schram for the complex variant of the problem the set of equivalence classes of real 16 vertex models can be divided up into seven families this strongly suggests the existence of a new type of phase transitions which manifest themselves when the physical system in the 16 parameter space moves from one family to another

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

the galton board limit theorems and recurrence we study a particle moving in r2 under a constant external force and bouncing off a periodic array of convex domains scatterers the latter must satisfy a standard finite horizon condition to prevent ballistic collision free motion this model is known to physicists as galton board it is also identical to a periodic lorentz gas previous heuristic and experimental studies have suggested that the particle s speed v t should grow as t1 3 and its coordinate x t as t2 3 we prove these conjectures rigorously we also find limit distributions for the rescaled velocity t 1 3v t and position t 2 3x t in addition quite unexpectedly we discover that the particle s motion is recurrent that means that a ball dropped on an idealized galton board will roll down but from time to time it should bounce all the way back up with probability one

dissipative homoclinic loops of two dimensional maps and strange attractors with one direction of instability we prove that when subjected to periodic forcing of the form p t d h x y c sin t certain two dimensional vector fields with dissipative homoclinic loops generate strange attractors with sinai ruelle bowen measures for a set of forcing parameters of positive lebesgue measure the proof extends ideas of afraimovich and shilnikov and applies the recent theory of rank 1 maps developed by wang and young we prove a general theorem and then apply this theorem to an explicit model a forced duffing equation of the form d 2 q

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

fluctuation of the entropy production for the lorentz gas under small external forces in this paper we study the physical and statistical properties of the periodic lorentz gas with finite horizon driven to a non equilibrium steady state by the combination of non conservative external forces and deterministic thermostats a version of this model was introduced by chernov eyink lebowitz and sinai and subsequently generalized by chernov and the third author non equilibrium steady states for these models are srb measures and they are characterized by the positivity of the steady state entropy production rate our main result is to establish that the entropy production in this context equal to the phase space contraction satisfies the gallavotti cohen fluctuation relation the main tool needed in the proof is the family of anisotropic banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques

a complete characterization of minimax and maximin encoder decoder policies for communication channels with incomplete statistical description the problem is considered of transmitting a sequence of independent and identically distributed gaussian random variables over a channel whose statistical description is incomplete the channel is modeled as one that is conditionally gaussian with the unknown part being controlled by a so called jammer who may have access to the input to the encoder and operates under a given power constraint by adopting a game theoretic approach a complete set of solutions is obtained encoder and decoder mappings and least favorable distributions for the channel noise for this statistical decision problem under two different sets of conditions depending on whether the encoder mapping is deterministic or stochastic in the latter case existence of a mixed saddle point solution can be verified when a side channel of a specific nature is available between the transmitter and the receiver in the former case however only minimax and maximin solutions can be derived

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

bowen ruelle measures for certain piecewise hyperbolic maps we consider a class of piecewise c 2 lozi like maps and prove the existence of invariant measures with absolutely continuous conditional mea sures on unstable manifolds

linear programming based converses for finite blocklength lossy joint source channel coding a linear programming lp based framework is presented for obtaining converses for finite blocklength lossy joint source channel coding problems the framework applies for any loss criterion generalizes certain previously known converses and also extends to multi terminal settings the finite blocklength problem is posed equivalently as a nonconvex optimization problem and using a lift and project like method a close but tractable lp relaxation of this problem is derived lower bounds on the original problem are obtained by the construction of feasible points for the dual of the lp relaxation a particular application of this approach leads to new converses which recover and improve on the converses of kostina and verdu for finite blocklength lossy joint source channel coding and lossy source coding for finite blocklength channel coding the lp relaxation recovers the converse of polyanskiy poor and verdu and leads to a new improvement on the converse of wolfowitz showing thereby that our lp relaxation is asymptotically tight with increasing blocklengths for channel coding lossless source coding and joint source channel coding with the excess distortion probability as the loss criterion using a duality based argument a new converse is derived for finite blocklength joint source channel coding for a class of source channel pairs employing this converse the lp relaxation is also shown to be tight for all blocklengths for the minimization of the expected average symbolwise hamming distortion of a q ary uniform source over a q ary symmetric memoryless channel for any q in mathbb n the optimization formulation and the lift and project method are extended to networked settings and demonstrated by obtaining an improvement on a converse of zhou et al for the successive refinement problem for successively refinable source distortion measure triplets

the metric entropy of diffeomorphisms soit m une variete de riemann compacte c sans bord soit f un diffeomorphisme c 2 de m et soit m une mesure de probabilite de borel f invariante sur m soit h m f l entropie de f on donne des relations entre h m f et les proprietes geometriques de f et de m

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

limit laws and recurrence for the planar lorentz process with infinite horizon as bleher j stat phys 66 1 315 373 1992 observed the free flight vector of the planar infinite horizon periodic lorentz process s n n 0 1 2 belongs to the non standard domain of attraction of the gaussian law actually with the sqrt n log n scaling our first aim is to establish his conjecture that indeed frac s n sqrt n log n converges in distribution to the gaussian law a global limit theorem here the recent method of balint and gouezel commun math phys 263 461 512 2006 helped us to essentially simplify the ideas of our earlier sketchy proof szasz d varju t in modern dynamical systems and applications pp 433 445 2004 moreover we can also derive a the local version of the global limit theorem b the recurrence of the planar infinite horizon periodic lorentz process and finally c the ergodicity of its infinite invariant measure

on the classification of the homogeneous 16 vertex models on a square lattice i connection with the boson bogoliubov transformation abstract we consider the classification of all homogeneous 16 vertex models on a square lattice which was proposed by gaaff and hijmans we show that this classification can be translated into that is reformulated in terms of a classification of matrices which closely resembles one already investigated with some success in the literature viz the classification connected with the boson bogoliubov transformation in quantum mechanics the main difference between our approach and the one of gaaff and hijmans who also made a reformulation is that we can keep our considerations in conformity with the actual physical problem within the field of real rather than complex numbers in the accompanying paper this leads to results with features not present in the analysis over c which point to a possible connection with a new kind of phase transitions

worst case robust mimo transmission with imperfect channel knowledge in this paper we consider robust transmit strategies against the imperfectness of the channel state information at the transmitter csit for multi input multi output mimo communication systems following a worst case deterministic model the actual channel is assumed to be inside an ellipsoid centered at a nominal channel the objective is to maximize the worst case received signal to noise ratio snr or to minimize the worst case chernoff bound of the error probability thus leading to a maximin problem moreover we also consider the qos problem as a complement of the maximin design which minimizes the transmit power consumption and meanwhile keeps the received snr above a given threshold for any channel realization in the ellipsoid it is shown that for a general class of power constraints both the maximin and qos problems can be equivalently transformed into convex problems or even further into semidefinite programs sdps thus efficiently solvable by the numerical methods the most interesting result is that the optimal transmit directions i e the eigenvectors of the transmit covariance are just the right singular vectors of the nominal channel under some mild conditions this result leads to a channel diagonalizing structure as in the cases of perfect csit and statistical csit with mean or covariance feedback and reduces the complicated matrix valued problem to a scalar power allocation problem then we provide the closed form solution to the resulting power allocation problem

local limit theorem for randomly deforming billiards we study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure in the context of a planar periodic lorentz gas with finite horizon we consider random perturbations in the form of movements and deformations of scatterers we prove a central limit theorem for the cell index of planar motion as well as a mixing local limit theorem for piecewise holder continuous observables in the context of the infinite measure random system we prove limit theorems regarding visits to new obstacles and self intersections as well as decorrelation estimates the main tool we use is the adaptation of anisotropic banach spaces to the random setting

the problem of recurrence for lorentz processes we prove that for the planar lorentz process with a periodic configuration of scatterers the quasi local clt of the gaussian log n type holds for any 1 consequently for arbitrary 3 2 the probabilities that at the moment of thenth reflection this process lies in a square of size log n are asymptotically gaussian this implies that these events occur for infinitely many values ofn i e a weaker form of recurrence

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

local and networked mean square estimation with high dimensional log concave noise we consider the problem of mean square estimation of the state of a discrete time dynamical system having additive non gaussian noise we assume that the noise has the structure that at each time instant it is a projection of a fixed high dimensional noise vector with a log concave density we derive conditions which guarantee that as the dimension of this noise vector grows large the optimal estimator of the problem with gaussian noise becomes near optimal for the problem with non gaussian noise the results are derived by first showing an asymptotic gaussian lower bound on the minimum error which holds even for nonlinear systems the bound is asymptotically tight for linear systems for nonlinear systems this bound is tight provided and the noise has a strongly log concave density with some additional structure these results imply that the estimate obtained by employing the gaussian estimator on the non gaussian problem satisfies an approximate orthogonality principle and the difference between this estimate and the optimal estimate vanishes strongly in l 2 for linear systems with high dimensional log concave noise of this structure we get that the kalman filter serves as a near optimal estimator a key ingredient in the proofs is a recent central limit theorem of eldan and klartag the minimum mean square error is in general neither weakly upper semicontinuous nor weakly lower semicontinuous our results show that in the setting we consider it is weakly continuous we then consider mean square estimation over a noisy network where the problem is to jointly design strategies for sensing and estimation to minimize a composite quadratic cost these problems do not have a static information structure whereby the previously derived results do not apply we consider two different cases when the estimation part of the problem corresponds to estimating the original source sensed by the sensor and when this part corresponds to the estimation of the action of the sensor in the former case we show that results analogous to the above continue to hold the latter case corresponds to a variant of witsenhausen s problem wherein we show only partial results

energy transport and detailed verification of fourier heat law in a chain of colliding harmonic oscillators the authors study a simple nonlinear classical hamiltonian system with positive k entropy a model for heat conduction and they find that it obeys the fourier heat law numerical simulation of its dynamics can be performed very efficiently so they are able to explore it in detail they verify the fourier heat law and calculate the coefficient of thermal conductivity k by three independent methods the first is direct simulation i e simulating the dynamics of the chain between two heat reservoirs the second is the green kubo formalism which is derived in a self contained manner the third method the one sided heating of a semi infinite cold chain is new and gives the best results it yields the entire temperature dependence k t in a single numerical simulation and definitely demonstrates the validity of the fourier heat law at all temperatures for the given system they believe that this method can also be useful for other systems they derive analytically the asymptotic behaviour of the coefficient of thermal conductivity at low temperatures t to 0 and observe that it agrees with numerical results obtained by the green kubo formalism which gives by far the best results at very low temperatures

gaussian arbitrarily varying channels the em arbitrarily varying channel avc can be interpreted as a model of a channel jammed by an intelligent and unpredictable adversary we investigate the asymptotic reliability of optimal random block codes on gaussian arbitrarily varying channels gavc s a gavc is a discrete time memoryless gaussian channel with input power constraint p t and noise power n e which is further corrupted by an additive jamming signal the statistics of this signal are unknown and may be arbitrary except that they are subject to a power constraint p j we distinguish between two types of power constraints em peak and em average for peak constraints on the input power and the jamming power we show that the gavc has a random coding capacity for the remaining cases in which either the transmitter or the jammer or both are subject to average power constraints no capacities exist and only lambda capacities are found the asymptotic error probability suffered by optimal random codes in these cases is determined our results suggest that if the jammer is subject only to an average power constraint reliable communication is impossible at any positive code rate

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

shannon meets von neumann a minimax theorem for channel coding in the presence of a jammer we study the setting of channel coding over a family of channels whose state is controlled by an adversarial jammer by viewing it as a zero sum game between a finite blocklength encoder decoder team and the jammer the encoder decoder team choose stochastic encoding and decoding strategies to minimize the average probability of error in transmission while the jammer chooses a distribution on the state space to maximize this probability the min max value of the game is equivalent to channel coding for a compound channel we call this the shannon solution of the problem the max min value corresponds to finding a mixed channel with the largest value of the minimum achievable probability of error when the min max and max min values are equal the problem is said to admit a saddle point or von neumann solution while a shannon solution always exists the communicating team s problem is nonconvex for finite blocklengths whereby a von neumann solution may not exist despite this we show that the min max and max min values become equal asymptotically in the large blocklength limit for all but finitely many rates we explicitly characterize this limiting value as a function of the rate and obtain tight finite blocklength bounds on the min max and max min value as a corollary we get an explicit expression for the epsilon capacity of a compound channel under stochastic codes the first such result to the best of our knowledge our results demonstrate a deeper relation between the compound channel and mixed channel than was previously known they also show that the conventional information theoretic viewpoint articulated via the shannon solution coincides asymptotically with the game theoretic one articulated via the von neumann solution key to our results is the derivation of new finite blocklength upper bounds on the min max value of the game via a novel achievability scheme and lower bounds on the max min value obtained via the linear programming relaxation based approach we introduced in 2

recurrence properties of a special type of heavy tailed random walk in the proof of the invariance principle for locally perturbed periodic lorentz process with finite horizon a lot of delicate results were needed concerning the recurrence properties of its unperturbed version these were analogous to the similar properties of simple symmetric random walk however in the case of lorentz process with infinite horizon the analogous results for the corresponding random walk are not known either in this paper these properties are ascertained for the appropriate random walk this happens to be in the non normal domain of attraction of the normal law as a tool an estimation of the remainder term in the local limit theorem for the corresponding random walk is computed

locally perturbed random walks with unbounded jumps szasz and telcs j stat phys 26 3 1981 have shown that for the diffusively scaled simple symmetric random walk weak convergence to the brownian motion holds even in the case of local impurities if d 2 the extension of their result to finite range random walks is straightforward here however we are interested in the situation when the random walk has unbounded range concretely we generalize the statement of szasz and telcs j stat phys 26 3 1981 to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law we do this first for diffusively scaled random walks on zd d 2 having finite variance and second for random walks with distribution belonging to the non normal domain of attraction of the normal law this result can be applied to random walks with tail behavior analogous to that of the infinite horizon lorentz process these in particular have infinite variance and convergence to brownian motion holds with the superdiffusive sqrt n log n scaling

random walks with internal degrees of freedom between two absorbing barriers consider a random walk with a finite number of internal degrees of freedom and with zero drift by using a functional analytic approach based on the spectral theory of matrix polynomials the asymptotics of the first hitting probabilities is obtained when the distance of the barriers tends to infinity

one dimensional hard rod caricature of hydrodynamics we give here a rigorous deduction of the hydrodynamic equation which holds in the hydrodynamic limit for a model system of one dimensional identical hard rods interacting through elastic collisions the equation should be considered as the analog of the euler equation of real hydrodynamics owing to the degeneracy of the model it is written in terms of a functiong q v t expressing the density of particles with velocityv at the pointq at timet for this equation we prove an existence and uniqueness theorem in some natural class of functions our main result is the proof that if 0 is a class of initial states which are homogeneous on a scale much less than e 1 and if the corresponding particle densities tend ase 0 in the proper scale to the initial hydrodynamic densitygo q v then under some general assumptions on the states and ong0 the particle densities of the evolved states at timee 1t tend ase 0 to the unique solution of the hydrodynamic equation with initial conditiong0 the proof is completed by exhibiting a large class of initial families 0 which possess the required properties