stability of statistical properties in two dimensional piecewise hyperbolic maps we investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions we accomplish such a program in the case of two dimensional maps with uniformly bounded second derivative but we are confident that the present approach can be successful in much greater generality we hope including higher dimensional billiards for the class of systems at hand we obtain a complete description of the srb measures their statistical properties and their stability with respect to many types of perturbations including deterministic and random perturbations and holes

closed form expressions for the ber ser of ofdm systems with an integer time offset this paper provides a detailed mathematical analysis of the impact of integer time offsets on the performance of ofdm systems although a number of approximate expressions for the probability of error in ofdm systems with integer time offset can be found in the literature to the best of our knowledge this work is the first that provides exact expressions in particular we derive exact closed form expressions for the bit error rate ber and the symbol error rate ser of bpsk qpsk and 16 qam modulation for transmission over both awgn and rayleigh fading channels the effect of the fractional carrier frequency offset cfo is taken into consideration in the derivations this makes the derived expressions more useful in evaluating the probability of error for ofdm systems with cfo only especially in cases like 16 qam modulation for transmission over rayleigh fading channels where exact closed form expressions are not available for ofdm systems with a large number of subcarriers an approximate method for evaluating the ber ser is given finally numerical results are included to demonstrate the exactness of the derived expressions and the accuracy of the approximate method

etching a two dimensional mathematical approach a mathematical model for the etching of a semi infinite active surface is presented the model assumes that the transport of the active species occurs solely by diffusion it is shown that when the diffusion field propagates much faster than the etched surface the problem can be solved by a singular perturbation technique which distinguishes a near field in the area where the moving surface and the non etchable mask meet and a far field where edge effects may be disregarded to first order the leading terms of a composite expansion are given from which the shape of the moving boundary can be determined at all times

survival probability for the stadium billiard abstract we consider the open stadium billiard consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides due to the hyperbolic nature of the stadium billiard the initial decay of trajectories due to loss through the hole appears exponential however some trajectories bouncing ball orbits persist and survive for long times and therefore form the main contribution to the survival probability function at long times using both numerical and analytical methods we concur with previous studies that the long time survival probability for a reasonably small hole drops like c o n s t a n t t i m e 1 here we obtain an explicit expression for the constant

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

gibbs markov young structures with stretched exponential tail for partially hyperbolic attractors abstract in this work we extend the results obtained by gouezel in 12 to partially hyperbolic attractors we study a forward invariant set k on a riemannian manifold m whose tangent space splits as dominated decomposition t k m e c u e s for which the center unstable direction e c u is non uniformly expanding on some local unstable disk we prove that the stretched exponential decay of recurrence times for an induced scheme can be deduced under the assumption of stretched exponential decay of the time that typical points need to achieve some uniform expanding in the center unstable direction as an application of our results we obtain exponential decay of correlations and exponential large deviations for a class of partially hyperbolic diffeomorphisms considered in 1

computation of the bit error rate of coherent m ary psk with gray code bit mapping efficient computation of the bit error rate ber for the coherent m ary psk signals with gray code bit mapping is considered a closed form expression for the exact ber of 8 ary psk is presented tight upper and lower bounds on ber are also obtained for m ary psk with larger m

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

statistical properties for flows with unbounded roof function including the lorenz attractor for geometric lorenz attractors including the classical lorenz attractor we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors we also obtain the functional central limit theorem and moment estimates as well as iterated versions of these results a consequence is deterministic homogenisation convergence to a stochastic differential equation for fast slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two

entry times distribution for mixing systems we consider the return times dynamics to bowen balls for continuous maps on metric spaces which have invariant probability measures with certain mixing properties these mixing properties are satisfied for instance by systems that allow young tower constructions we show that the higher order return times to bowen balls are in the limit poisson distributed we also provide a general result for the asymptotic behavior of the recurrence time for bowen balls for ergodic systems and those with specification

quenched decay of correlations for slowly mixing systems we study random towers that are suitable to analyse the statistics of slowly mixing random systems we obtain upper bounds on the rate of quenched correlation decay in a general setting we apply our results to the random family of liveranisaussol vaienti maps with parameters in 0 1 0 1 chosen independently with respect to a distribution on 0 1 and show that the quenched decay of correlation is governed by the fastest mixing map in the family in particular we prove that for every 0 for almost every 0 1 z the upper bound n 1 1 0 holds on the rate of decay of correlation for holder observables on the fibre over for three different distributions on 0 1 discrete uniform quadratic we also derive sharp asymptotics on the measure of return time intervals for the quenched dynamics ranging from n 1 0 to log n 1 0 n 1 0 to log n 2 0 n 1 0 respectively

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

equidistribution for nonuniformly expanding dynamical systems and application to the almost sure invariance principle let t m mt m m be a nonuniformly expanding dynamical system such as logistic or intermittent map let v m rdv m rd be an observable and vn n 1k 0v tkvn k 0n 1v tk denote the birkhoff sums given a probability measure on m we consider vn as a discrete time random process on the probability space m m in smooth ergodic theory there are various natural choices of such as the lebesgue measure or the absolutely continuous t invariant measure they give rise to different random processes we investigate relation between such processes we show that in a large class of measures it is possible to couple redefine on a new probability space every two processes so that they are almost surely close to each other with explicit estimates of closeness the purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by melbourne and nicol

phase transitions for countable markov shifts we study the analyticity of the topological pressure for some one parameter families of potentials on countable markov shifts we relate the non analyticity of the pressure to changes in the recurrence properties of the system we give sufficient conditions for such changes to exist and not to exist we apply these results to the manneville pomeau map and use them to construct examples with different critical behavior

asymptotic and numerical solutions for the two dimensional solidification of a liquid half space this paper concerns the twodimensional inward solidification of a halfspace filled with molten material at its fusion temperature the temperature along half of the boundary surface is instantaneou

etching through a slit the problem of etching through a slit shaped orifice is formulated in mathematical terms although a full solution to this problem can be obtained only by the application of numerical methods it is shown that certain aspects of it can be dealt with by analytical means asymptotic solutions that are valid in certain parts of the time domain are derived early intermediate and late asymptotes are considered these analytical results are compared with numerical ones and a satisfactory agreement is found emphasizing the validity of the two approaches

decay of correlations for slowly mixing flows we show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of a class of nonuniformly hyperbolic dieomorphisms for which young proved polynomial decay of correlations roughly speaking in situations where the decay rate o 1 n has previously been proved for dieomorphisms we establish the decay rate o 1 t for flows applications include certain classes of semidispersing billiards as well as dispersing billiards with vanishing curvature in addition we obtain results for suspension flows with unbounded roof functions in particular the classical planar lorentz flow with a doubly periodic array of circular scatterers has decay rate 1 t as anticipated by physicists

dispersing billiards with cusps slow decay of correlations dispersing billiards introduced by sinai are uniformly hyperbolic and have strong statistical properties exponential decay of correlations and various limit theorems however if the billiard table has cusps corner points with zero interior angles then its hyperbolicity is nonuniform and statistical properties deteriorate until now only heuristic and experimental results existed predicting the decay of correlations as mathcal o 1 n we present a first rigorous analysis of correlations for dispersing billiards with cusps

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

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

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

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 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 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

asymptotic solutions to the stefan problem with a constant heat source at the moving boundary in 1989 huppert formulated mathematical models of the melting and or freezing that occurs when a hot fluid begins to flow turbulently over a cold substrate he solved the resulting stefan problems numerically and presented certain asymptotic results this note revisits one of his problems and adds to the mathematical analysis of its asymptotic structure generalizations and relationships with other results are also noted

on differentiability of srb states for partially hyperbolic systems consider a one parameter family of diffeomorphisms f e such that f 0 is an anosov element in a standard abelian anosov action having sufficiently strong mixing properties let e be any u gibbs state for f e we prove theorem 1 that for any c function a the map e e a is differentiable at e 0 this implies corollary 2 2 that the difference of birkhoff averages of the perturbed and unperturbed systems is proportional to e we apply this result corollary 3 3 to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique srb measure with good statistical properties

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

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

rapid mixing for the lorenz attractor and statistical limit laws for their time 1 maps we prove that every geometric lorenz attractor satisfying a strong dissipativity condition has superpolynomial decay of correlations with respect to the unique sinai ruelle bowen measure moreover we prove the central limit theorem and almost sure invariance principle for the time 1 map of the flow of such attractors in particular our results apply to the classical lorenz attractor

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

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

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

the spectrum of the billiard laplacian of a family of random billiards random billiards are billiard dynamical systems for which the reflection law giving the post collision direction of a billiard particle as a function of the pre collision direction is specified by a markov scattering operator p billiards with microstructure are random billiards whose markov operator is derived from a microscopic surface structure on the boundary of the billiard table the microstructure in turn is defined in terms of what we call a billiard cellq the shape of which completely determines the operator p this operator defined on an appropriate hilbert space is bounded self adjoint and for the examples considered here a hilbert schmidt operator a central problem in the statistical theory of such random billiards is to relate the geometric characteristics of q and the spectrum of p we show for a particular family of billiard cell shapes parametrized by a scale invariant curvature k fig 2 that the billiard laplacian p i is closely related to the ordinary spherical laplacian and indicate by partly analytical and partly numerical means how this provides asymptotic information about the spectrum of p for small values of k it is shown in particular that the second moment of scattering about the incidence angle closely approximates the spectral gap of p

hitting time statistics and extreme value theory we consider discrete time dynamical systems and show the link between hitting time statistics the distribution of the first time points land in asymptotically small sets and extreme value theory distribution properties of the partial maximum of stochastic processes this relation allows to study hitting time statistics with tools from extreme value theory and vice versa we apply these results to non uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws with no extra condition on the speed of mixing for example we also give applications of our theory to higher dimensional examples for which we also obtain classical extreme value laws and exponential hitting time statistics for balls we extend these ideas to the subsequent returns to asymptotically small sets linking the poisson statistics of both processes

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

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

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

a recursive algorithm for the exact ber computation of generalized hierarchical qam constellations hierarchical constellations offer a different degree of protection to the transmitted messages according to their relative importance as such they find interesting application in digital video broadcasting systems as well as wireless multimedia services although a great deal of attention has been devoted in the literature to the study of the bit error rate ber performance of uniform quadrature amplitude modulation qam constellations very few results were published on the ber performance of hierarchical qam constellations we obtain exact and generic expressions in m for the ber of the generalized hierarchical m pam pulse amplitude modulation constellations over additive white gaussian noise awgn and fading channels we also show how these expressions can be extended to generalized hierarchical m qam constellations square and rectangular for the awgn case these expressions are in the form of a weighted sum of complementary error functions and are solely dependent on the constellation size m the carrier to noise ratio and a constellation parameter which controls the relative message importance because of their generic nature these new expressions readily allow numerical evaluation for various cases of practical interest

rate of chaotic mixing and boundary behavior we discuss rigorous results on the rate of mixing for an idealized model of a class of fluid mixing device these show that the decay of correlations of a scalar field is governed by the presence of boundaries in the domain and in particular by the behavior of the modeled fluid at such boundaries

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

on the continuity of the srb entropy for endomorphisms we consider classes of dynamical systems admitting markov induced maps under general assumptions which in particular guarantee the existence of srb measures we prove that the entropy of the srb measure varies continuously with the dynamics we apply our result to a vast class of non uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the srb measure in particular we show that the srb entropy of viana maps varies continuously with the map

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

escape rates and physically relevant measures for billiards with small holes we study the billiard map corresponding to a periodic lorentz gas in 2 dimensions in the presence of small holes in the table we allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers for a large class of smooth initial distributions we establish the existence of a common escape rate and normalized limiting distribution this limiting distribution is conditionally invariant and is the natural analogue of the srb measure of a closed system finally we prove that as the size of the hole tends to zero the limiting distribution converges to the smooth invariant measure of the billiard map

on the asymptotic moments and edgeworth expansions for some processes in random dynamical environment we prove that certain asymptotic moments exist for some random distance expanding dynamical systems and markov chains in random dynamical environment and compute them in terms of the derivatives at 0 of an appropriate pressure function it will follow that these moments satisfy the relations that the asymptotic moments gamma k lim n rightarrow infty n frac k 2 mathbb e sum i 1 n x i k of sums of independent and identically distributed centered random variables satisfy under certain mixing conditions we will also estimate the convergence rate towards these limits the arguments in the proof of these results yield that the partial sums generated by the random ruelle perron frobenius triplets and all of their parametric derivatives considered as functions on the base corresponding to appropriate random transfer or markov operators satisfy several probabilistic limit theorems such as the central limit theorem we will also obtain certain edgeworth asymptotic expansions related to the central limit theorem for such processes our proofs rely on a parametric random complex ruelle perron frobenius theorem which replaces some of the spectral techniques used in literature in order to obtain limit theorems for deterministic dynamical systems and markov chains

analytical and numerical solutions to two dimensional moving interface problems with applications to the solidification of killed steel ingots this paper is concerned with the development of methods for investigating moving interface problems that involve more than one space variable the work was motivated by problems arising from the british steel corporation scunthorpe concerning the solidification of killed steel ingots first we investigate certain two dimensional analytical solutions which simulate particular aspects of ingot solidification these relate to the inward solidification of a liquid half space either when the wall temperature varies with position along the wall or when newton cooling takes place at the wall with the local heat transfer coefficient a function of wall position other approximate analytical approaches with use of the heat balance integral method and formulation in terms of an integro differential equation are discussed briefly and their limited application to two dimensional problems noted a computer procedure is then developed for solving these half space problems and this yields results in exact agreement with the analytical solutions in the work special attention is given to obtaining reliable information for small stefan number beta l c p t f t 0 since this relates to values found in the steel making process in the major part of the paper a two dimensional mathematical model of ingot solidification is developed numerical solutions are given for an ingot when the mould surface varies in geometry and in composition good agreement is obtained with radio isotope tracer tests performed at the british steel corporation scunthorpe

average of product of two gaussian q functions and its application to performance analysis in nakagami fading the average of the product of two gaussian q functions having arguments as different scaled versions of the same nakagami distributed fading gain magnitude is derived in closed form both the cases when 1 the fading parameter is an integer and 2 the fading parameter is an odd multiple of one half are considered the results are applied to evaluating in closed form a the symbol error probability sep of rectangular quadrature amplitude modulation with maximal ratio combining in independent nakagami fading and b the outage probability of a two user synchronous code division multiple access system having signal to noise ratio snr imbalance among users in nakagami fading it is found in application a that with increase in the degree of fading the robustness to decision distance imbalance between in phase and quadrature signals increases in addition there exists an optimum decision distance ratio at which the sep has a minimum in application b we find that the outage probability increases sharply with snr imbalance factor at low snr imbalance and tends to reach saturation as the imbalance increases

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

a useful integral for wireless communication theory and its application to rectangular signaling constellation error rates a useful integral representing the average over rayleigh fading of the product of two gaussian q functions is solved in closed form a closed form solution for the symbol error probability of general rectangular quadrature amplitude modulation in rayleigh fading is derived

rates of convergence in the clt for two dimensional dispersive billiards we consider billiards in the two dimensional torus with convex obstacles central limit theorems have been established for regular functions for the billiard transformation in 2 1 and 14 we are interested here in the problem of the rate of convergence in this paper we establish a rate in for any 0 for the billiard transformation by adapting the proof of 7 6 8 in our proof we use a strong decorrelation result obtained by the method developped in 14 for the study of general hyperbolic systems moreover we establish a rate of convergence in in the central limit theorem for the billiard flow

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

subexponential decay of correlations we describe a method for proving subexponential lower bounds for correlations functions and apply it to study decay of correlations for maps with countable markov partitions one result is that ls young s upper estimates y2 are optimal in many situations our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on banach spaces which we apply to the iterates of the transfer operator

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

spectral gap and transience for ruelle operators on countable markov shifts we find a necessary and sufficient condition for the ruelle operator of a weakly holder continuous potential on a topologically mixing countable markov shift to act with spectral gap on some rich banach space we show that the set of potentials satisfying this condition is open and dense for a variety of topologies we then analyze the complement of this set in a finer topology and show that among the three known obstructions to spectral gap weak positive recurrence null recurrence transience transience is open and dense and null recurrence and weak positive recurrence have empty interior

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

exponential decay of correlations for piecewise cone hyperbolic contact flows we prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form in dimension three this is the first time exponential decay of correlations is proved for continuous time dynamics with singularities on a manifold our proof combines the second author s version liverani in ann math 159 1275 1312 2004 of dolgopyat s estimates for contact flows and the first author s work with gouezel j mod dyn 4 91 137 2010 on piecewise hyperbolic discrete time dynamics

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

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

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

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

structure size and statistical properties of chaotic components in a mixed type hamiltonian system we perform a detailed study of the chaotic component in mixed type hamiltonian systems on the example of a family of billiards introduced by robnik in j phys a math gen 16 3971 1983 jphac50305 447010 1088 0305 4470 16 17 014 the phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times the structure of the chaotic component is discerned from the cells visited by the chaotic orbit the fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box counting method the cell filling dynamics is compared to a model of uncorrelated motion the so called random model robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 and deviations attributed to sticky objects in the phase space are found the statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run the stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times the recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed the standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter lastly the decay of correlations and the diffusion of momenta is analyzed

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

large deviation probabilities for correlated gaussian processes and intermittent dynamical systems in its classical version the theory of large deviations makes quantitative statements about the probability of outliers when estimating time averages if time series data are identically independently distributed we study large deviation probabilities ldps for time averages in short and long range correlated gaussian processes and show that long range correlations lead to subexponential decay of ldps a particular deterministic intermittent map can depending on a control parameter also generate long range correlated time series we illustrate numerically in agreement with the mathematical literature that this type of intermittency leads to a power law decay of ldps the power law decay holds irrespective of whether the correlation time is finite or infinite and hence irrespective of whether the central limit theorem applies or not

smooth approximation of stochastic differential equations consider an ito process x satisfying the stochastic differential equation dx a x dt b x dw where a b are smooth and w is a multidimensional brownian motion suppose that wn has smooth sample paths and that wn converges weakly to w a central question in stochastic analysis is to understand the limiting behavior of solutions xn to the ordinary differential equation dxn a xn dt b xn dwn the classical wong zakai theorem gives sufficient conditions under which xn converges weakly to x provided that the stochastic integral b x dw is given the stratonovich interpretation the sufficient conditions are automatic in one dimension but in higher dimensions the correct interpretation of b x dw depends sensitively on how the smooth approximation wnis chosen in applications a natural class of smooth approximations arise by setting wn t n 1 2 nt0v sds where t is a flow generated e g by an ordinary differential equation and v is a mean zero observable under mild conditions on t we give a definitive answer to the interpretation question for the stochastic integral b x dw our theory applies to anosov or axiom a flows t as well as to a large class of nonuniformly hyperbolic flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on t the methods used in this paper are a combination of rough path theory and smooth ergodic theory

the lorenz attractor is mixing we study a class of geometric lorenz flows introduced independently by afraimovic bykov sil nikov and by guckenheimer williams and give a verifiable condition for such flows to be mixing as a consequence we show that the classical lorenz attractor is mixing

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

statistical stability and continuity of srb entropy for systems with gibbs markov structures we present conditions on families of diffeomorphisms that guarantee statistical stability and srb entropy continuity they rely on the existence of horseshoe like sets with infinitely many branches and variable return times as an application we consider the family of henon maps within the set of benedicks carleson parameters

computing and bounding the first order marcum q function a geometric approach a geometric interpretation of the first order marcum q function q a b is introduced as the probability that a complex gaussian random variable with real mean a takes on values outside of a disk co b of radius b centered at the origin o this interpretation engenders a fruitful approach for deriving new representations and tight upper and lower bounds on q a b the new representations obtained involve finite range integrals with pure exponential integrands they are shown to be simpler and more robust than their counterparts in the literature the new bounds obtained include the generic exponential bounds which involve an arbitrarily large number of exponential functions and the simple erfc bounds which involve just a few erfc functions together with exponential functions in some cases the new generic exponential bounds approach the exact value of q a b as the number of exponential terms involved increases these generic exponential bounds evaluated with only two terms and the new simple erfc bounds are much tighter than the existing exponential bounds in most cases especially when the arguments a and b are large thus in many applications requiring further analytical manipulations of q a b these new bounds can lead to some closed form results which are better than the results available so far

exact error probability analysis of rectangular qam for single and multichannel reception in nakagami m fading channels in this contribution we derive exact closed form expressions for the average symbol error probability sep of arbitrary rectangular quadrature amplitude modulation qam for single and multichannel diversity reception over independent but not necessarily identically distributed nakagami m fading channels the diversity branches may hence exhibit identical or distinctive power levels and their associated nakagami indexes need not be the same our work extends previous results pertaining to nondiversity reception of m ary rectangular qam over rayleigh fading channels and multichannel reception of m ary square qam over nakagami m fading channels for a given number l of diversity branches and a corresponding set of arbitrary real valued nakagami indexes not less than 1 2 our sep results are expressed in terms of gauss s hypergeometric function 2f1 and lauricella s multivariate hypergeometric function fd l of l variables both of which can be efficiently evaluated using standard numerical softwares

the geometric approach for constructing sinai ruelle bowen measures an important class of physically relevant measures for dynamical systems with hyperbolic behavior is given by sinai ruelle bowen srb measures we survey various techniques for constructing srb measures and studying their properties paying special attention to the geometric push forward approach after describing this approach in the uniformly hyperbolic setting we review recent work that extends it to non uniformly hyperbolic systems