an integral equation approach for the valuation of american style down and out calls with rebates in this paper an integral equation approach is adopted to price american style down and out calls instead of using the probability theory as used in the literature we use the continuous fourier sine transform to solve the partial differential equation system governing the option prices as a way of validating our approach we show that the early exercise premium representation for american style down and out calls without rebate can be re derived by using our approach we then examine the case that time dependent rebates are included in the contract of american style down and out calls as a result a more general integral representation for the price of an american style down and out call with the presence of an extra term associated with the rebate is obtained our numerical method based on the newly derived integral representation appears to be efficient in computing the price and the hedging parameters for american style down and out calls with rebates in addition significant effects of rebates on the option prices and the optimal exercise boundaries are illustrated through selected numerical results

exponential decay of correlations for strongly coupled toom probabilistic cellular automata we investigate the low noise regime of a large class of probabilistic cellular automata including the north east center model of toom they are defined as stochastic perturbations of cellular automata belonging to the category of monotonic binary tessellations and possessing a property of erosion we prove for a set of initial conditions exponential convergence of the induced processes toward an extremal invariant measure with a highly predominant spin value we also show that this invariant measure presents exponential decay of correlations in space and in time and is therefore strongly mixing

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

breaking of ergodicity in expanding systems of globally coupled piecewise affine circle maps to identify and to explain coupling induced phase transitions in coupled map lattices cml has been a lingering enigma for about two decades in numerical simulations this phenomenon has always been observed preceded by a lowering of the lyapunov dimension suggesting that the transition might require changes of linear stability yet recent proofs of co existence of several phases in specially designed models work in the expanding regime where all lyapunov exponents remain positive in this paper we consider a family of cml composed by piecewise expanding individual map global interaction and finite number n of sites in the weak coupling regime where the cml is uniformly expanding we show mathematically for n 3 and numerically for n ge 3 that a transition in the asymptotic dynamics occurs as the coupling strength increases the transition breaks the milnor attractor into several chaotic pieces of positive lebesgue measure with distinct empiric averages it goes along with various symmetry breaking quantified by means of magnetization type characteristics despite that it only addresses finite dimensional systems to some extend this result reconciles the previous ones as it shows that loss of ergodicity symmetry breaking can occur in basic cml independently of any decay in the lyapunov dimension

diffusion in energy conserving coupled maps we consider a dynamical system consisting of subsystems indexed by a lattice each subsystem has one conserved degree of freedom energy the rest being uniformly hyperbolic the subsystems are weakly coupled together so that the sum of the subsystem energies remains conserved we prove that the subsystem energies satisfy the diffusion equation in a suitable scaling limit

an open system approach for the characterization of spatio temporal chaos we investigate the structure of the invariant measure of space time chaos by adopting an open system point of view we consider large but finite windows of formally infinite one dimensional lattices and quantify the effect of the interaction with the outer region by mapping the problem on the dynamical characterization of localized perturbations this latter task is performed by suitably generalizing the concept of lyapunov spectrum to cope with perturbations that propagate outside the region under investigation as a result we are able to introduce a volume propagation velocity i e the velocity with which ensembles of localized perturbations tend to fill volumes in the neighbouring regions

equilibrium phase transitions in coupled map lattices a pedestrian approach a class of piecewise linear coupled map lattices with simple symbolic dynamics is constructed it can be solved analytically in terms of the statistical mechanics of spin lattices the corresponding hamiltonian is written down explicitly in terms of the parameters of the map the approach follows the line of recent mathematical investigations but the presentation is kept elementary so that phase transitions in the dynamical model can be studied in detail although the method works only for map lattices with repelling invariant sets some of the conclusions i e the role of local curvature of the single site map and properties of the nearest neighbour coupling might play an important role for phase transitions in general dynamical systems

coupled map lattices one step forward and two steps back abstract we discuss the general notion of lattice dynamical systems and some recent progress in their studies we suggest also the approach that allows to extract some information on the dynamics of finitely extended real systems from the results obtained on the infinitely extended artificial ones

spatial correlations and synchronization in coupled map lattices with long range interactions abstract we used numerical diagnostics to quantify spatial disorder and its relation with temporal chaos for a one dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice distance in a power law fashion the main tool is spatial return plots whose properties are used to obtain information about the chaotic synchronized states of the system a spatial correlation integral is introduced to characterize the clustering of points in the spatial return plots

srb measures for a class of partially hyperbolic attractors in hilbert spaces abstract in this paper we study the existence of srb measures and their properties for infinite dimensional dynamical systems in a hilbert space we show several results including i if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions then it has at least one srb measure ii if the attractor is uniformly hyperbolic and the system is topological mixing and the splitting is holder continuous then there exists a unique srb measure which is mixing iii if the attractor is uniformly hyperbolic and the system is non wondering and the splitting is holder continuous then there exist at most finitely many srb measures iv for a given hyperbolic measure there exist at most countably many ergodic components whose basin contains an observable set

limit theorems for coupled analytic maps in 23 h h rugh gave a new and simpler method to prove existence of an invariant measure with nice mixing properties for some weakly coupled analytic maps he proved in fact a spectral gap property for a well defined transfer operator we modify this method to construct generalized transfer operators associated to potentials and preserve the spectral gap for small potentials this allows us to prove new limit results for these systems central limit theorem moderate deviations principle and a partial large deviations result these results are available under wide classes of observables and initial measures

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

smooth dependence of thermodynamic limits of srb measures the thermodynamic limits of srb measures for uniformly hyperbolic sets are smoothly dependent on the map in an appropriate functional space

absolute continuity of stable foliations for systems on banach spaces abstract we prove the absolute continuity of stable foliations for c 1 maps of banach spaces satisfying a globally defined infinitesimal invariant cones condition proofs of regularity for center and stable manifolds needed for the main theorem are included our results are applicable to dynamical systems generated by ordinary partial or functional differential equations including non autonomous differential equations that are periodic in time

analyticity of the srb measure of a lattice of coupled anosov diffeomorphisms of the torus we consider the thermodynamic limit of a d dimensional lattice of hyperbolic dynamical systems on the 2 torus interacting via weak and nearest neighbor coupling we prove that the srb measure is analytic in the strength of the coupling the proof is based on symbolic dynamics techniques that allow us to map the srb measure into a gibbs measure for a spin system on a d 1 dimensional lattice this gibbs measure can be studied by an extension decimation of the usual cluster expansion techniques

dimension formula for random transformations we consider compositions of random diffeomorphisms and show that the dimension of sample measures equals lyapunov dimension as conjectured in the nonrandom case by yorke et al

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

a decomposition approach via fourier sine transform for valuing american knock out options with rebates we present an innovative decomposition approach for computing the price and the hedging parameters of american knock out options with a time dependent rebate our approach is built upon i the fourier sine transform applied to the partial differential equation with a finite time dependent spatial domain that governs the option price and ii the decomposition technique that partitions the price of the option into that of the european counterpart and an early exercise premium our analytic representations can generalize a number of existing decomposition formulas for some european style and american style options a complexity analysis of the method together with numerical results show that the proposed approach is significantly more efficient than the state of the art adaptive finite difference methods especially in dealing with spot prices near the barrier numerical results are also examined in order to provide new insight in the significant effects of the rebate on the option price the hedging parameters and the optimal exercise boundary

phase transition and correlation decay in coupled map lattices for a coupled map lattice with a specific strong coupling emulating stavskaya s probabilistic cellular automata we prove the existence of a phase transition using a peierls argument and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoupling originally developed for weak coupling this implies the exponential decay in space and in time of the correlation functions of the invariant measures

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

random walks in space time mixing environments we prove that random walks in random environments that are exponentially mixing in space and time are almost surely diffusive in the sense that their scaling limit is given by the wiener measure

uniqueness of the srb measure for piecewise expanding weakly coupled map lattices in any dimension we prove the existence of a unique srb measure for a wide range of multidimensional weakly coupled map lattices these include piecewise expanding maps with diffusive coupling

strange attractors in periodically kicked limit cycles and hopf bifurcations we prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with srb measures when certain simple dynamical systems are kicked at periodic time intervals the settings considered include limit cycles and stationary points undergoing hopf bifurcations

uniqueness and exponential decay of correlations for some two dimensional spin lattice systems we consider a two dimensional lattice spin system which naturally arises in dynamical systems called coupled map lattice the configuration space of the spin system is a direct product of mixing subshifts of finite type the potential is defined on the set of all squares in z2 and decays exponentially with the linear size of the square via the polymer expansion technique we prove that for sufficiently high temperatures the limit gibbs distribution is unique and has an exponential decay of correlations

random walk in markovian environment we prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on z d we assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is markovian with strong spatial and temporal mixing properties

central extensions of current groups in two dimensions the paper is devoted to generalization of the theory of loop groups to the two dimensional case to every complex riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex lie group g by the jacobian of this surface this extension is topologically nontrivial as in the loop group case orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalg bundles over the surface when the surface is the torus elliptic curve classification of coadjoint orbits is related to linear difference equations in one variable and to classification of conjugacy classes in the loop group we study integral orbits for which the kirillov kostant form is a curvature form for some principal torus bundle the number of such orbits for a given level is finite as in the loop group case conjecturedly they correspond to analogues of integrable modules occurring in conformal field theory

strange attractors with one direction of instability we give simple conditions that guarantee for strongly dissipative maps the existence of strange attractors with a single direction of instability and certain controlled behaviors only the d 2 case is treated in this paper although our approach is by no means limited to two phase dimensions we develop a dynamical picture for the attractors in this class proving they have many of the statistical properties associated with chaos positive lyapunov exponents existence of srb measures and exponential decay of correlations other results include the geometry of fractal critical sets nonuniform hyperbolic behavior symbolic coding of orbits and formulas for topological entropy

spatial recurrence plots we propose an extension of the recurrence plot concept to perform quantitative analyzes of roughness and disorder of spatial patterns at a fixed time we introduce spatial recurrence plots srps as a graphical representation of the pointwise correlation matrix in terms of a two dimensional spatial return plot this technique is applied to the study of complex patterns generated by coupled map lattices which are characterized by measures of complexity based on srps we show that the complexity measures we propose for srps provide a systematic way of investigating the distribution of spatially coherent structures such as synchronization domains in lattice profiles this approach has potential for many more applications e g in surface roughness analyzes

almost every real quadratic map is either regular or stochastic in this paper we complete a program to study measurable dynamics in the real quadratic family our goal was to prove that almost any real quadratic map pc z t x2 c c c 2 1 4 has either an attracting cycle or an absolutely continuous invariant measure the final step completed here is to prove that the set of infinitely renormalizable parametric values c c 2 1 4 has zero lebesgue measure we derive this from a renormalization theorem which asserts uniform hyperbolicity of the full renormalization operator this theorem gives the most general real version of the feigenbaum coullet tresser universality simultanuously for all combinatorial types

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

a dynamical proof of the multiplicative ergodic theorem we shall give a proof of the following result of oseledec in which gl d denotes the space of invertible d d real matrices denotes any norm on the space of d d matrices and log t max 0 log t for t 0

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

new conditions for non uniform behaviour of linear cocycles over flows abstract we give a characterization of tempered exponential dichotomies for linear cocycles over flows in terms of the spectral properties of certain linear operators we consider noninvertible linear cocycles acting on infinite dimensional spaces and our approach avoids the use of lyapunov norms finally we apply obtained results to give new conditions for uniform exponential stability of linear cocycles

characterization of dominated splittings for operator cocycles acting on banach spaces abstract versions of the oseledets multiplicative ergodic theorem for cocycles acting on infinite dimensional banach spaces have been investigated since the pioneering work of ruelle in 1982 and are a topic of continuing research interest for a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions a necessary and sufficient condition for the existence of such a dominated splitting also known as exponential separation was recently given by j bochi and n gourmelon for invertible finite dimensional cocycles in discrete time we extend this result to cocycles of injective bounded and possibly noncompact linear maps acting on banach spaces in both discrete and continuous time using an essentially geometric approach based on a notion of approximate singular value decomposition in banach spaces our method is constructive and in the finite dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

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

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

sternberg theorems for random dynamical systems in this paper we prove the smooth conjugacy theorems of sternberg type for random dynamical systems based on their lyapunov exponents we also present a stable and unstable manifold theorem with tempered estimates that are used to construct conjugacy 2005 wiley periodicals inc

ising type and other transitions in one dimensional coupled map lattices with sign symmetry we consider a one dimensional lattice of expanding antisymmetric maps 1 1 1 1 with nearest neighbor diffusive coupling for such systems it is known that if the coupling parameter e is small there is unique stationary in time state which is chaotic in space time a disputed question is whether such systems can exhibit ising type phase transitions as e grows beyond some critical value ec we present results from computer experiments which give definite indication that such a transition takes place the mean square magnetization appears to diverge as e approaches some critical value with a critical exponent around 0 9 we also study other properties of the coupled map system

entropy limit theorems and variational principles for disordered lattice systems we study infinite volume limits and gibbs states of disordered lattice systems with bounded and continuous potentials our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes we find that many familiar results of invariant potentials such as large deviation theorems variational principles and equivalence of ensembles continue to hold for disordered models with suitably modified statements

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

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

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

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

equilibrium states for random lattice models of hyperbolic type abstract we study the structural stability of random coupled map lattice models of hyperbolic type under certain metrics furthermore by describing the thermodynamic formalism of the underlying random spin lattice system we prove the existence of equilibrium states for equi holder continuous random functions on lattice models under the conditions of random weak interaction and translation invariance and present some partial results on the uniqueness of equilibrium states

four dimensional realization of two dimensional current groups the quotient realization of the central extensions of the current groups over riemann surfaces is achieved by means of the leray residue theory this approach replaces de rham cohomology in the classical wznw construction for affine lie groups

linear response function for coupled hyperbolic attractors the srb measures of a hyperbolic system are widely accepted as the measures that are physically relevant it has been shown by ruelle that they depend smoothly on the system furthermore ruelle showed by a separate argument that the first derivative i e the linear response function admits a geometric interpretation in this paper we consider thermodynamic limits of srb measures in lattices of coupled hyperbolic attractors in a previous paper using markov partitions and thermodynamic formalism we had established the smooth dependence of thermodynamic limits of srb measures here we establish that the linear response function admits a geometric interpretation the formula is analogous to the one found by ruelle for finite dimensional systems if one term is reinterpreted appropriately we show that the limiting derivative is the thermodynamic limit of the derivatives in finite volume we also obtain similar results for the derivatives of the entropy

absolute continuity of stable foliations for mappings of banach spaces we prove the absolute continuity of stable foliations for mappings of banach spaces satisfying conditions consistent with time t maps of certain classes of dissipative pdes this property is crucial for passing information from submanifolds transversal to the stable foliation to the rest of the phase space it is also used in proofs of ergodicity absolute continuity of stable foliations is well known in finite dimensional hyperbolic theory on banach spaces the absence of nice geometric properties poses some additional difficulties

dynamical systems on lattices with decaying interaction ii hyperbolic sets and their invariant manifolds abstract this is the second part of the work devoted to the study of maps with decay in lattices here we apply the general theory developed in fontich et al 2011 3 to the study of hyperbolic sets in particular we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map we also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds

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

on the definition of srb measures for coupled map lattices we consider srb measures of coupled map lattices the emphasis is given to a definition according to which a srb measure is an invariant probability measure whose projections onto finite dimensional subsystems are absolutely continuous with respect to the lebesgue measure we show that coupled map lattices which are close to an uncoupled expanding map have typically an infinite number of srb measures in particular we give a counterexample to the bricmont kupiainen conjecture

lyapunov exponents periodic orbits and horseshoes for semiflows on hilbert spaces two settings are considered flows on finite dimensional riemann ian manifolds and semiflows on hilbert spaces with conditions consistent with those in systems defined by dissipative parabolic pdes under certain assump tions on lyapunov exponents and entropy we prove the existence of geometric structures called horseshoes this implies in particular the presence of infinitely many periodic solutions for diffeomorphisms of compact manifolds analogous results are due to a katok here we extend katok s results to i continuous time and ii infinite dimensions courant institute of mathematical sciences new york university 251 mercer street new york new york 10012 e mail address lian cims nyu edu courant institute of mathematical sciences new york university 251 mercer street new york new york 10012 e mail address lsy cims nyu edu

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

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

dynamical systems on lattices with decaying interaction i a functional analysis framework abstract we consider weakly coupled map lattices with a decaying interaction that is we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites we develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds this framework is very well suited to study systematically invariant objects one obtains that the invariant objects are essentially local we use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties i e the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets in the companion paper by fontich et al 2011 10

entropy volume growth and srb measures for banach space mappings we consider c 2 frechet differentiable mappings of banach spaces leaving invariant compactly supported borel probability measures and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces srb measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds equivalently the sum of positive lyapunov exponents of the map in addition to numerous difficulties incurred by our infinite dimensional setting a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions the results here generalize previously known results for diffeomorphisms of finite dimensional riemannian manifolds and are applicable to dynamical systems defined by large classes of dissipative parabolic pdes

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

mathematical theory of lyapunov exponents this paper reviews some basic mathematical results on lyapunov exponents one of the most fundamental concepts in dynamical systems the first few sectionscontainsomeverygeneralresultsinnonuniformhyperbolictheory we consider f where f is an arbitrary dynamical system and is an arbitrary invariant measure and discuss relations between lyapunov exponents and several dynamical quantities of interest including entropy fractal dimension and rates of escape the second half of this review focuses on observable chaos characterized by positive lyapunov exponents on positive lebesgue measure sets much attention is given to srb measures a very special kind of invariantmeasuresthatofferawaytounderstandobservablechaosindissipative systems paradoxical as it may seem given a concrete system it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied case studies will be discussed the final section is on noisy or stochastically perturbed systems for which we present a dynamical picture simpler than that for purely deterministic systems in this short review we have elected to limit ourselves to finite dimensional systems and to discrete time the phase space which is assumed to be r d or a riemannian manifold is denoted by m throughout the lebesgue or the riemannian measure on m is denoted by m and the dynamics are generated by iterating a self map of m written f m for flows the reviewed results

infinite dimensional srb measures we review the basic steps leading to the construction of a sinai ruelle bowen srb measure for an infinite lattice of weakly coupled expanding circle maps and we show that this measure has exponential decay of space time correlations first using the perron frobenius operator one connects the dynamical system of coupled maps on a d dimensional lattice to an equilibrium statistical mechanical model on a lattice of dimension d 1 this lattice model is for weakly coupled maps in a high temperature phase and we use a general but very elementary method to prove exponential decay of correlations at high temperatures

random perturbations of axiom a basic sets in this paper we study small random diffeomorphism type perturbations of an axiom a basic set by means of the structural stability of such a basic set with respect to time dependent perturbations and by means of the markov partition of the basic set we apply the thermodynamic formalism of random subshifts of finite type to this situation obtaining some ergodic theoretic results concerning equilibrium states

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

equilibrium measures for coupled map lattices existence uniqueness and finite dimensional approximations we consider coupled map lattices of hyperbolic type i e chains of weakly interacting hyperbolic sets attractors over multi dimensional lattices we describe the thermodynamic formalism of the underlying spin lattice system and then prove existence uniqueness mixing properties and exponential decay of correlations of equilibrium measures for a class of holder continuous potential functions with a sufficiently small holder constant we also study finite dimensional approximations of equilibrium measures in terms of lattice systems approximations and lattice spin systems d approximations we apply our results to establish existence uniqueness and mixing property of srb measures as well as obtain the entropy formula

kac moody groups topology of the dirac determinant bundle and fermionization the relation between kac moody groups and algebras and the determinant line bundle of the massless dirac operator in two dimensions is clarified analogous objects are studied in four space time dimensions and a generalization of witten s fermionization mechanism is presented in terms of the topology of the dirac determinant bundle

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

dimension of hyperbolic measures of random diffeomorphisms we consider dynamics of compositions of stationary random c 2 diffeomorphisms we will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional this is an extension to random diffeomorphisms of the main result of barreira pesin and schmeling 1999 which proves the eckmann ruelle dimension conjecture for a deterministic diffeomorphism

sinai ruelle bowen measures for lattice dynamical systems for weakly coupled expanding maps on the unit circle bricmont and kupiainen showed that the sinai ruelle bowen srb measure exists as a gibbs state via thermodynamic formalism we prove that this srb measure is indeed the unique equilibrium state for a holder continuous potential function on the infinite dimensional phase space for a more general class of lattice systems that are small perturbations of the uncoupled map lattice we present the variational principle the entropy formula and the formula for the potential function for the srb measures for coupled map lattices with nearest neighbor interactions we give an explicit formula of the potential function for the srb measure and consequently obtain the entropy in terms of coupling parameters

spatio temporal chaos a solvable model abstract a solvable coupled map lattice model exhibiting spatio temporal chaos is studied exact expressions are obtained for the spectra of lyapunov exponents as a function of the model parameters although the model has spatio temporal structure the time series measured at a single lattice site are shown to consist of independent identically distributed samples for several values of the model parameters for these parameter values the spatial series measured at a fixed time also consist of independent identically distributed samples in these cases the information dimension density is 1 but the information entropy density depends on the model parameters thus the model is an example where the information entropy density can be obtained neither from a time series measured at a single lattice site nor from a spatial series measured at a fixed time we conclude that in studying only a time series or a spatial series without any knowledge of the system one could be easily led into thinking that there is no spatio temporal structure for a full characterization of the system structure in time and space will have to be considered simultaneously

the entropy formula for srb measures of lattice dynamical systems we give a detailed proof of the entropy formula for srb measures of coupled hyperbolic attractors over integer lattices we show that the topological pressure for the potential function of the srb measure is zero

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

pricing american style parisian down and out call options we propose an integral equation approach for pricing american style parisian down and out call options under the blackscholes framework for this type of options the knock out feature is activated only if the underlying asset price continuously remains below a pre determined barrier for a sufficiently long period of time as such the corresponding pricing problem becomes a three dimensional 3 d free boundary problem instead of a two dimensional 2 d one as is the case of one touch barrier options and this poses a computational challenge in our approach we first reduce the 3 d problem to a 2 d one and then by applying the fourier sine transform to the resulting 2 d problem we can derive a pair of coupled integral equations governing the option price at any given time in terms of i the option price at the barrier and ii the optimal exercise boundary at that time this pair of coupled integral equations can be solved using the newtonraphson iterative procedure after which the option price the optimal exercise boundary and the hedging parameters can be obtained in a straightforward manner a complexity analysis of the method together with numerical results show that the proposed approach is robust and significantly more efficient than existing uniform finite difference methods with cranknicolson timestepping especially in dealing with spot prices near the barrier numerical results are also examined in order to provide new insight into several interesting properties of the option price and the optimal exercise boundary