the dynamics of pseudographs in convex hamiltonian systems we study the evolution under convex hamiltonian flows on cotangent bundles of compact manifolds of certain distinguished subsets of the phase space these subsets are generalizations of lagrangian graphs we call them pseudographs they emerge in a natural way from fathi s weak kam theory by this method we find various orbits which connect prescribed regions of the phase space our study is inspired by works of john mather as an application we obtain the existence of diffusion in a large class of a priori unstable systems and provide a solution to the large gap problem we hope that our method will have applications to more examples

a generic property of families of lagrangian systems we prove that a generic lagrangian has finitely many minimizing measures for every cohomology class

on asymptotic volume of finsler tori minimal surfaces in normed spaces and symplectic filling volume the main unconditional result of this paper theorem 3 states that every two dimensional affine disc in a normed space that is a disc contained in a two dimensional affine subspace is an area minimizing surface among all immersed discs with the same boundary with respect to the symplectic holmes thompson surface area to emphasize that this is not at all obvious it may be worth mentioning that a similar statement with rational chains in place of immersed discs is incorrect theorem 2 and that it is not known for surfaces that may not be topological discs the result still may not sound too exciting to the reader who never looked at the problem before even though it goes back to busemann s works in the 50 s see bes th and references there and the proof heavily relies on asymptotic geometry of tori we believe that it is more important that we embed this problem into a whole area of mostly open problems as well as give some partial results and suggest certain directions of how to attack them we begin with a trivial statement

d optimal 0 1 weighing designs for eight objects abstract let g m n max det w t w w m m n 0 1 a matrix w m m n 0 1 with det w t w g m n is called d optimal here we determine g m 8 for m large with the help of a variation of mitchell s detmax algorithm and describe the d optimal examples

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

notes on d optimal designs abstract the purpose of this paper is to exhibit new infinite families of d optimal 0 1 matrices we show that hadamard designs lead to d optimal matrices of size j mj and j 1 mj for certain integers j 3 mod 4 and all positive integers m for j a power of a prime and j 1 mod 4 supplementary difference sets lead to d optimal matrices of size j 2mj and j 1 2mj for all positive integers m we also show that for a given j and d sufficiently large about half of the entries in each column of a d optimal matrix are ones this leads to a new relationship between d optimality for 0 1 matrices and for 1 matrices known results about d optimal 1 matrices are then used to obtain new d optimal 0 1 matrices

euler poisson systems as action minimizing paths in the wasserstein space this paper uses a variational approach to establish existence of solutions t vt for the 1 d euler poisson system by minimizing an action we assume that the initial and terminal points 0 t are prescribed in mathcal p 2 mathbb r the set of borel probability measures on the real line of finite second order moments we show existence of a unique minimizer of the action when the time interval 0 t satisfies t these solutions conserve the hamiltonian and they yield a path t t in mathcal p 2 mathbb r when t y t is a dirac mass the euler poisson system reduces to ddot y y 0 the kinetic version of the euler poisson that is the vlasov poisson system was studied in ambrosio and gangbo comm pure appl math to appear as a hamiltonian system

aubry mather theory for conformally symplectic systems in this article we develop an analogue of aubry mather theory for a class of dissipative systems namely conformally symplectic systems and prove the existence of interesting invariant sets which in analogy to the conservative case will be called the aubry and the mather sets besides describing their structure and their dynamical significance we shall analyze their attracting repelling properties as well as their noteworthy role in driving the asymptotic dynamics of the system

a representation formula of viscosity solutions to weakly coupled systems of hamilton jacobi equations with applications to regularizing effect abstract based on a fixed point argument we give a dynamical representation of the viscosity solution to cauchy problem of certain weakly coupled systems of hamilton jacobi equations with continuous initial datum using this formula we obtain some regularity results related to the viscosity solution including a partial extension of lions regularizing effect 17 to the case of weakly coupled systems

an example of convex hamiltonian diffeomorphism where asymptotic distance from identity is strictly greater than the minimal action abstract we construct a convex hamiltonian diffeomorphism on the unit ball of cotangent bundle of t n n 2 where the asymptotic distance from identity is strictly greater than the minimal action

existence of c 1 critical subsolutions of the hamilton jacobi equation let m be a c second countable manifold without boundary we denote by tm the tangent bundle and by tm m the canonical projection a point in tm will be denoted by x v with x m and v tx m 1 x in the same way a point of the cotangent space t m will be denoted by x p with x m and p t x m a linear form on the vector space tx m we will suppose that g is a complete riemannian metric on m for v tx m the norm v is g v v 1 2 we will denote by the dual norm on t x m we will also use the notations v x for v tx m and p x for v t x m we will assume in the whole paper that h t m r is a function of class at least c2 which satisfies the following three conditions 1 uniform superlinearity for every k 0 there exists c k r such that

product integration methods for solving a system of nonlinear volterra integral equations in this paper the technique of subtracting out singularities is used to derive explicit and implicit product euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of volterra integral equations with a weakly singular kernel the convergence proofs of the numerical schemes are presented these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global lipschitz condition

towards a quantum analog of weak kam theory we discuss a quantum analogue of mather s minimization principle for lagrangian dynamics and provide some formal calculations suggesting the corresponding euler lagrange equation we then rigorously construct from the dual eigenfunctions of a certain non selfadjoint operator a candidate for a minimizer and recover aspects of weak kam theory in the limit as h 0 regarding our state as a quasimode we furthermore derive some error estimates although it remains an open problem to improve these bounds

on monotone abel volterra integral equations on the half line in this paper we reanalyze the trapezoidal method for the solution of nonlinear abel volterra integral equations on the half line we prove the convergence of the method in the uniform norm provided the nonlinearity is lipschitz continuous and strictly monotone

stability analysis of product t methods for abel integral equations of the second kind stability analysis of product methods for abel integral equations of the second kind based on the test equation y t 1 lambda int limits 0 1 frac y s t s alpha ds t geqq 0 0 0 and we investigate whether this property is inherited by numerical approximations toy

the monge problem for supercritical mane potentials on compact manifolds abstract we prove the existence of optimal transport maps for the monge problem when the cost is a finsler distance on a compact manifold our point of view consists in considering the distance as a mane potential and to rely on recent developments in the theory of viscosity solutions of the hamilton jacobi equation

almost periodically forced circle flows abstract we study general dynamical and topological behaviors of minimal sets in skew product circle flows in both continuous and discrete settings with particular attentions paying to almost periodically forced circle flows when a circle flow is either discrete in time and unforced i e a circle map or continuous in time but periodically forced behaviors of minimal sets are completely characterized by classical theory the general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies the topological complexity of the forcing space and the possible loss of mean motion property on one hand we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of denjoy sets of circle maps in the sense that they can be almost automorphic cantorian and everywhere non locally connected but on the other hand we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of denjoy theory for instance an almost periodically forced circle flow can be positively transitive and its minimal sets can be li yorke chaotic and non almost automorphic as an application of our results we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic sl 2 r valued continuous or discrete cocycle continuous almost periodically forced circle flows are among the simplest non monotone multi frequency dynamical systems they can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through mather theory in the damping free cases they also naturally arise in 2d almost periodic floquet theory as well as in climate models discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear schrodinger equations with almost periodic potentials they have been widely used as models for studying strange non chaotic attractors and intermittency phenomena during the transition from order to chaos hence the study of these flows is of fundamental importance to the understanding of multi frequency driven dynamical irregularities and complexities in non monotone dynamical systems

on the integrability of tonelli hamiltonians in this article we discuss a weaker version of liouville s theorem on the integrability of hamiltonian systems we show that in the case of tonelli hamiltonians the involution hypothesis on the integrals of motion can be completely dropped and still interesting information on the dynamics of the system can be deduced moreover we prove that on the n dimensional torus this weaker condition implies classical integrability in the sense of liouville the main idea of the proof consists in relating the existence of independent integrals of motion of a tonelli hamiltonian to the size of its mather and aubry sets as a byproduct we point out the existence of non trivial common invariant sets for all hamiltonians that poisson commute with a tonelli hamiltonian

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

hyperbolicity and exponential convergence of the lax oleinik semigroup abstract for a convex superlinear lagrangian l t m r on a compact manifold m it is known that there is a unique number c such that the lax oleinik semigroup l t c t c m r c m r has a fixed point moreover for any u c m r the uniform limit u lim t l t u c t exists in this paper we assume that the aubry set consists in a finite number of periodic orbits or critical points and study the relation of the hyperbolicity of the aubry set to the exponential rate of convergence of the lax oleinik semigroup

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

piecewise continuous collocation for integral equations the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous difficulties with the usual function space setting of l infty d are discussed and a satisfactory sense of point evaluation is given for elements of l infty d other approaches which are discussed include i reformulation as a degenerate kernel method ii the prolongation restriction framework of noble iii other function space settings and iv reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation

new rotation sets in a family of torus homeomorphisms we construct a family varphi t t in 0 1 of homeomorphisms of the two torus isotopic to the identity for which all of the rotation sets rho varphi t can be described explicitly we analyze the bifurcations and typical behavior of rotation sets in the family providing insight into the general questions of toral rotation set bifurcations and prevalence we show that there is a full measure subset of 0 1 consisting of infinitely many mutually disjoint non trivial closed intervals on each of which the rotation set mode locks to a constant polygon with rational vertices that the generic rotation set in the hausdorff topology has infinitely many extreme points accumulating on a single totally irrational extreme point at which there is a unique supporting line and that although rho varphi t varies continuously with t the set of extreme points of rho varphi t does not the family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set or by any directional ergodic measure

stability results for collocation methods for volterra integral equations abstract stability behavior is investigated for polynomial spline collocation applied to volterra integral equations with special emphasis on a weakly singular test equation the characterization of the corresponding stability domain is related to stability results for solutions of finite recursion relations as they occur for methods applied to ordinary differential equations estimates of the stability domains are established and as a consequence conditions for a 2 stability are obtained exploiting these estimates collocation approximations with constant linear and continuous splines are considered in more depth and several numerical illustrations are presented

a mathematical treatment of the fluorescence capillary fill device a mathematical model in the form of two coupled diffusion equations is provided for a competitive chemical reaction between an antigen and a labeled antigen for antibody sites on a cell wall boundary conditions are such that the problem is both nonlinear and nonlocal this is then recharacterized first as a pair of coupled singular integro differential equations and then as a system of four volterra integral equations the latter permits a proof of existence and uniqueness of the solution of the original problem small and large time asymptotic solutions are derived and from the first characterization a regular perturbation solution is obtained numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin

computing the effective hamiltonian using a variational approach a numerical method for homogenization of hamilton jacobi equations is presented and implemented as an l infty calculus of variations problem solutions are found by solving a nonlinear convex optimization problem the numerical method is shown to be convergent and error estimates are provided one and two dimensional examples are worked in detail comparing known results with the numerical ones and computing new examples the cases of nonstrictly convex hamiltonians and hamiltonians for which the cell problem has no solution are treated

robustness of optimal designs for correlated random variables abstract suppose that y y i is a normal random vector with mean xb and covariance 2 i n where b is a p dimensional vector b j x x ij is an n p matrix with x ij 1 1 this corresponds to a factorial design with 1 1 representing low or high level respectively or corresponds to a weighing design with 1 1 representing an object j with weight b j placed on the left and right of a chemical balance respectively e optimal designs z are chosen that are robust in the sense that they remain e optimal when the covariance of y i y i is 0 for i i within a smaller class of designs similar results are obtained with respect to a general class of optimality criteria which include the a and d criteria

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

largest j simplices in d cubes some relatives of the hadamard maximum determinant problem abstract this paper studies the computationally difficult problem of finding a largest j dimensional simplex in a given d dimensional cube the case in which j d is of special interest for it is equivalent to the hadamard maximum determinant problem it has been solved for infinitely many values of d but not for d 14 the subcase in which j d 3 mod 4 subsumes the famous problem on the existence of hadamard matrices the known results for the case j d are here summarized and used but the main focus is on fixed small values of j when j 1 the problem is trivial and when j 2 or j 3 it is here solved completely i e for all d beyond that the results are fragmentary but numerous and they lead to several attractive conjectures some other problems involving simplices in cubes are mentioned and the relationship of largest simplices to d optimal weighing designs is discussed

propagation in hamiltonian dynamics and relative symplectic homology the main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating hamiltonian is sufficiently large over the zero section the proof is based on floer homology and on the notion of a relative symplectic capacity applications include results about propagation properties of sequential hamiltonian systems periodic orbits on hypersurfaces hamiltonian circle actions and smooth lagrangian skeletons in stein manifolds

convergence of the solutions of the discounted hamilton jacobi equation we consider a continuous coercive hamiltonian h on the cotangent bundle of the compact connected manifold m which is convex in the momentum if u lambda m rightarrow mathbb r is the viscosity solution of the discounted equation begin aligned lambda u lambda x h x mathrm d x u lambda c h end aligned where c h is the critical value we prove that u lambda converges uniformly as lambda rightarrow 0 to a specific solution u 0 m rightarrow mathbb r of the critical equation begin aligned h x mathrm d x u c h end aligned we characterize u 0 in terms of peierls barrier and projected mather measures as a corollary we infer that the ergodic approximation as introduced by lions papanicolaou and varadhan in 1987 in their seminal paper on homogenization of hamilton jacobi equations selects a specific corrector in the limit

weak liouville arnol d theorems and their implications this paper studies the existence of invariant smooth lagrangian graphs for tonelli hamiltonian systems with symmetries in particular we consider tonelli hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the liouville arnol d theorem moreover we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable tonelli hamiltonians

aubry mather theory and hamilton jacobi equations we review and extend some results of jauslin kreiss moser and weinan e on the smooth approximation of mather sets

a variational principle associated to positive tilt maps in this paper we establish a variational principle for positive tilt maps this implies that the aubry mather set exists for positive tilt maps and most of the theory developed by mather for twist maps now applies for positive tilt maps

tonelli hamiltonians without conjugate points and c 0 integrability we prove that all the tonelli hamiltonians defined on the cotangent bundle t t n of the n dimensional torus that have no conjugate points are c 0 integrable i e t t n is c 0 foliated by a family fc of invariant c 0 lagrangian graphs assuming that the hamiltonian is c infty we prove that there exists a g delta subset gc of fc such that the dynamics restricted to every element of gc is strictly ergodic moreover we prove that the lyapunov exponents of every c 0 integrable tonelli hamiltonian are zero and deduce that the metric and topological entropies vanish

dynamics near an isolated m semi static homoclinic orbit abstract we show that there are infinitely many periodic orbits in any neighborhood of an isolated m semi static orbit homoclinic to an aubry set for time periodic positive lagrangian systems

mane s conjectures in codimension 1 we prove mane s conjectures 9 in the context of codimension 1 aubry mather theory 2011 wiley periodicals inc

optimal supersaturated designs we consider screening experiments where an investigator wishes to study many factors using fewer observations our focus is on experiments with two level factors and a main effects model with intercept since the number of parameters is larger than the number of observations traditional methods of inference and design are unavailable in 1959 box suggested the use of supersaturated designs and in 1962 booth and cox introduced measures for efficiency of these designs including e s2 which is the average of squares of the off diagonal entries of the information matrix ignoring the intercept for a design to be e s2 optimal the main effect of every factor must be orthogonal to the intercept factors are balanced and among all designs that satisfy this condition it should minimize e s2 this is a natural approach since it identifies the most nearly orthogonal design and orthogonal designs enjoy many desirable properties including efficient parameter estimation factor balance in an e s2 optimal d

robustness of a optimal designs abstract suppose that y y i is a normal random vector with mean xb and covariance 2 i n where b is a p dimensional vector b j x x ij is an n p matrix a optimal designs x are chosen from the traditional set d of a optimal designs for 0 such that x is still a optimal in d when the components y i are dependent i e for i i the covariance of y i y i is with 0 such designs depend on the sign of the general results are applied to x x ij where x ij 1 1 this corresponds to a factorial design with 1 1 representing low level or high level respectively or corresponds to a weighing design with 1 1 representing an object j with weight b j being weighed on the left and right of a chemical balance respectively

uniform error estimates of galerkin methods for monotone abel volterra integral equations on the half line we consider galerkin methods for monotone abel volterra integral equations of the second kind on the half line the l2 theory follows from kolodner s theory of monotone hammerstein equations we derive the loo theory from the l2 theory by relating the l2and loo spectra of operators of the form x b ax to one another here denotes convolution and b e l1 and a e loo as an extra condition we need b t o t a 1 with a 0 we also prove the discrete analogue in particular we verify that the galerkin matrix satisfies the discrete conditions

stability and robustness of collocation methods for abel type integral equations we study stability aspects of collocation methods for abel type integral equations of the first kind using piecewise polynomials these collocation methods may be formulated as projection methods stability is defined as the boundedness of the sequence of projectors in their natural setting robustness is essentially the optimal asymptotic insensitivity to perturbations in the data we show that stability and robustness are equivalent for the above collocation methods this allows us to obtain optimal error estimates for some methods that are well known to be robust we also present numerical results for some methods which appear to be robust

birkhoff periodic orbits for small perturbations of completely integrable hamiltonian systems with convex hamiltonians pour des systemes a n degres de liberte on montre l existence d au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l hypothese que le hamiltonien du systeme non perturbe a des surfaces d energie convexes pour les variables action angle

generic hyperbolicity of aubry sets on surfaces given a tonelli hamiltonian of class c 2 on the cotangent bundle of a compact surface we show that there is an open dense set of potentials in the c 2 topology for which the aubry set is hyperbolic in its energy level

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

a family of methods for abel integral equations of the second kind abstract a class of methods depending on some parameters are introduced for the numerical solution of the abel integral equations of the second kind some bounds on the parameters are determined so that the corresponding methods have infinite stability intervals

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

selection problems for a discount degenerate viscous hamilton jacobi equation abstract we prove that the solution of the discounted approximation of a degenerate viscous hamilton jacobi equation with convex hamiltonians converges to that of the associated ergodic problem we characterize the limit in terms of stochastic mather measures by using the nonlinear adjoint method and deriving a commutation lemma this convergence result was first proven by davini fathi iturriaga and zavidovique for first order hamilton jacobi equations

symplectic aspects of mather theory we prove that the so called aubry and mane sets introduced by john mather in lagrangian dynamics are symplectic invariants in order to do so we introduce a barrier in phase space and propose definitions of aubry and mane sets for non convex hamiltonian systems on montre que les ensembles dits d aubry et de mane introduits par john mather en dynamique lagrangienne sont des invariants symplectiques pour ceci on introduit une barriere sur l espace des phases et on definit des ensembles d aubry et de mather pour des systemes hamiltoniens non convexes

galerkin methods for two point boundary value problems for first order systems galerkin methods for two point boundary value problems for first order systems are studied for a model problem rigorous upper and lower estimates upon the error are derived these estimates show the previous work to be sharp in some cases and pessimistic in others a nonstandard least squares type of method is then analyzed this method gives the optimal accuracy of the usual least squares method when the system does not satisfy a regularity assumption the nonstandard method gives slightly more accurate answers measured in l 2 the linear system arising from the nonstandard method has condition number o h 1 contrasted with o h 2 for the usual least squares method

hyperbolicity of minimizers and regularity of viscosity solutions for a random hamilton jacobi equation we show that for a large class of randomly kicked hamilton jacobi equations the unique global minimizer is almost surely hyperbolic furthermore we prove that the unique forward and backward viscosity solutions though in general only lipshitz are smooth in a neighborhood of the global minimizer related results in the one dimensional case were obtained by e khanin et al ann math 2 151 3 877 960 2000 however the methods in the above paper are purely one dimensional and cannot be extended to the case of higher dimensions here we develop a completely different approach

global behaviors of weak kam solutions for exact symplectic twist maps abstract we investigated several global behaviors of the weak kam solutions u c x t parametrized by c h 1 t r for the suspended hamiltonian h x p t of the exact symplectic twist map we could find a family of weak kam solutions u c x t parametrized by c h 1 t r with c continuous and monotonic and t u c h x x u c c t c a e x t t 2 such that sequence of weak kam solutions u c c h 1 t r is 1 2 holder continuity of parameter r moreover for each generalized characteristic no matter regular or singular solving x s co p h x s c d u c x s s t s t x 0 x 0 x 0 t t 2 we evaluate it by a uniquely identified rotational number c h 1 t r this property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories besides we discussed this applies to high dimensional cases

propagation of singularities for weak kam solutions and barrier functions this paper studies the structure of the singular set points of nondifferentiability of viscosity solutions to hamilton jacobi equations associated with general mechanical systems on the n torus first using the level set method we characterize the propagation of singularities along generalized characteristics then we obtain a local propagation result for singularities of weak kam solutions in the supercritical case finally we apply such a result to study the propagation of singularities for barrier functions

on m semi static homoclinic orbits abstract we study m semi static homoclinic orbits to aubry sets in positive definite lagrangian systems