the scattering map in two coupled piecewise smooth systems with numerical application to rocking blocks abstract we consider a non autonomous dynamical system formed by coupling two piecewise smooth systems in r 2 through a non autonomous periodic perturbation we study the dynamics around one of the heteroclinic orbits of one of the piecewise smooth systems in the unperturbed case the system possesses two c 0 normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them these heteroclinic manifolds are foliated by heteroclinic connections between c 0 tori located at the same energy levels by means of the impact map we prove the persistence of these objects under perturbation in addition we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of melnikov like functions the heteroclinic manifolds allow us to define the scattering map which links asymptotic dynamics in the invariant manifolds through heteroclinic connections first order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds hence we have an essential tool for the construction of a heteroclinic skeleton which when followed can lead to the existence of arnold diffusion trajectories that on large time scales destabilize the system by further accumulating energy we validate all the theoretical results with detailed numerical computations of a mechanical system with impacts formed by the linkage of two rocking blocks with a spring

instability of high dimensional hamiltonian systems multiple resonances do not impede diffusion abstract we consider models given by hamiltonians of the form h i p q t e h i j 1 n 1 2 p j 2 v j q j e q i p q t e where i i r d t d p q r n t t 1 these are higher dimensional analogues both in the center and hyperbolic directions of the models studied in 28 29 43 and are usually called a priori unstable hamiltonian systems all these models present the large gap problem we show that for 0 e 1 under regularity and explicit non degeneracy conditions on the model there are orbits whose action variables i perform rather arbitrary excursions in a domain of size o 1 this domain includes resonance lines and hence large gaps among d dimensional kam tori this phenomenon is known as arnold diffusion the method of proof follows closely the strategy of 28 29 the main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions i i r d we show that since these multiple resonances happen in sets of codimension greater than one in the space of actions i they can be contoured this corresponds to the mechanism called diffusion across resonances in the physics literature the present paper however differs substantially from 28 29 on the technical details of the proofs we have taken advantage of the theory of the scattering map developed in 31 notably the symplectic properties which were not available when the above papers were written we have analyzed the conditions imposed on the resonances in more detail more precisely we have found that there is a simple condition on the melnikov potential which allows us to conclude that the resonances are crossed in particular this condition does not depend on the resonances so that the results are new even when applied to the models in 28 29

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

transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation we consider the cubic defocusing nonlinear schrodinger equa tion on the two dimensional torus we exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes this behavior is quantified by the growth of higher sobolev norms given any 1 k 1 s 1 we construct smooth initial data u0 withu0 h s k at some time t this growth occurs despite the hamiltonian s bound onu t h 1 and despite the conservation of the quantityu t l2

arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders we prove a form of arnold diffusion in the a priori stable case let h 0 p epsilon h 1 theta p t quad theta in mathbb t n p in b n t in mathbb t mathbb r mathbb t be a nearly integrable system of arbitrary degrees of freedom n geqslant 2 with a strictly convex h0 we show that for a generic epsilon h 1 there exists an orbit theta p satisfying p t p 0 l h 1 0 where l h 1 is independent of epsilon the diffusion orbit travels along a codimension 1 resonance and the only obstruction to our construction is a finite set of additional resonances

a group of non uniform exponential growth locally isomorphic to img z 2 i we prove that a sequence of marked three generated groups isomorphic to the iterated monodromy group of z 2 i converges to a group of non uniform exponential growth which is an extension of the infinite direct sum of cyclic groups of order 4 by a grigorchuk group

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

a randomized linear time algorithm to find minimum spanning trees we present a randomized linear time algorithm to find a minimum spanning tree in a connected graph with edge weights the algorithm uses random sampling in combination with a recently discovered linear time algorithm for verifying a minimum spanning tree our computational model is a unit cost random access machine with the restriction that the only operations allowed on edge weights are binary comparisons

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

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

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 the hausdorff dimension of the mather quotient under appropriate assumptions on the dimension of the ambient manifold and the regularity of the hamiltonian we show that the mather quotient is small in term of hausdorff dimension then we present applications in dynamics

not residually finite groups of intermediate growth commensurability and non geometricity abstract we construct a series of not residually finite groups of intermediate growth in particular we construct groups of intermediate growth that are not commensurable up to finite kernels with any residually finite group also we show that certain properties of groups among those solvability of the word problem and the property of having torsion of unbounded order are not weakly geometric the argument is based in particular on studying commensurability up to finite kernels

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

on commuting tonelli hamiltonians autonomous case we show that the aubry sets the mane sets and the barrier functions are the same for two commuting autonomous tonelli hamiltonians we also show the quasi linearity of functions from the viewpoint of dynamics and the existence of common c1 1 critical subsolution for their associated hamilton jacobi equations

aubry mather theory and periodic solutions of the forced burgers equation consider a hamiltonian system with hamiltonian of the form h x t p where h is convex in p and periodic in x a ndt and x2 r 1 it is well known that its smooth invariant curves correspond to smoothz 2 periodic solutions of the pde ut h x t u x 0 in this paper we establish a connection between the aubry mather theory of invariant sets of the hamiltonian system andz 2 periodic weak solutions of this pde by realizing the aubry mather sets as closed subsets of the graphs of these weak solutions we show that the complement of the aubry mather set on the graph can be viewed as a subset of the generalized unstable manifold of the aubry mather set defined in 2 24 the graph itself is a backward invariant set of the hamiltonian system the basic idea is to embed the globally minimizing orbits used in the aubry mather theory into the characteristic fields of the above pde this is done by making use of one and two sided minimizers a notion introduced in 12 and inspired by the work of morse on geodesics of type a 26 the asymptotic slope of the minimizers also known as the rotation number is given by the derivative of the homogenized hamiltonian defined in 21 as an application we prove that thez 2 periodic weak solution of the above pde with given irrational asymptotic slope is unique a similar connection also exists in multidimensional problems with the convex hamiltonian except that in higher dimensions two sided minimizers with a specified asymptotic slope may not exist c 1999 john wiley sons inc

a minimum spanning tree algorithm with inverse ackermann type complexity a deterministic algorithm for computing a minimum spanning tree of a connected graph is presented its running time is 0 m m n where is the classical functional inverse of ackermann s function and n respectively m is the number of vertices respectively edges the algorithm is comparison based it uses pointers not arrays and it makes no numeric assumptions on the edge costs

groups with infinite mod p schur multiplier abstract the non finiteness of the mod p schur multiplier of a finitely generated group g with trivial center implies the existence of uncountably many non residually finite non isomorphic central extension groups with kernel c p see thm a this phenomenon is related to the comparison of the cohomology of the profinite completion g of a group g and the cohomology of the group g itself see thm b according to j p serre 8 a group g is good if the cohomologies of g and g are naturally isomorphic on finite coefficients it is shown that non uniform arithmetic lattices of algebraic rank 1 groups over local fields of positive characteristic p are not good see thm c

a geometric mechanism of diffusion rigorous verification in a priori unstable hamiltonian systems abstract in this paper we consider a representative a priori unstable hamiltonian system with 2 1 2 degrees of freedom and we apply the geometric mechanism for diffusion introduced in a delshams r de la llave t m seara a geometric mechanism for diffusion in hamiltonian systems overcoming the large gap problem heuristics and rigorous verification on a model mem amer math soc 179 844 2006 viii 141 pp and generalized in a delshams g huguet geography of resonances and arnold diffusion in a priori unstable hamiltonian systems nonlinearity 22 8 2009 1997 2077 we provide explicit concrete and easily verifiable conditions for the existence of diffusing orbits the simplification of the hypotheses allows us to perform the straightforward computations along the proof and present the geometric mechanism of diffusion in an easily understandable way in particular we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold

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

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

random spanning trees and the prediction ofweighted graphs we investigate the problem of sequentially predicting the binary labels on the nodes of an arbitrary weighted graph we show that under a suitable parametrization of the problem the optimal number of prediction mistakes can be characterized up to logarithmic factors by the cutsize of a random spanning tree of the graph the cutsize is induced by the unknown adversarial labeling of the graph nodes in deriving our characterization we obtain a simple randomized algorithm achieving in expectation the optimal mistake bound on any polynomially connected weighted graph our algorithm draws a random spanning tree of the original graph and then predicts the nodes of this tree in constant expected amortized time and linear space experiments on real world data sets show that our method compares well to both global perceptron and local label propagation methods while being generally faster in practice

arnold diffusion in a priori chaotic symplectic maps we assume that a symplectic real analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder we prove that generically in the real analytic category the boundaries of the invariant cylinder are connected by trajectories of the map

geometry of arnold diffusion the goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of arnold diffusion we choose to speak of a specific model that of geometric rays in a periodic optical medium this model is equivalent to that of a particle in a periodic potential in mathbb r n with energy prescribed and to the geodesic flow in a riemannian metric on mathbb r n

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

a general mechanism of diffusion in hamiltonian systems qualitative results we present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable hamiltonian systems our approach relies on the scattering map outer dynamics and on the recurrence property of the inner dynamics restricted to a normally hyperbolic invariant manifold we apply topological methods to find trajectories that follow these two dynamics this method differs in several crucial aspects from earlier works there are virtually no assumptions on the inner dynamics as the method does not use at all the invariant objects for the inner dynamics e g primary and secondary tori lower dimensional hyperbolic tori and their stable unstable manifolds aubry mather sets the method applies when the unperturbed hamiltonian is not necessarily convex and of arbitrary degrees of freedom in addition this mechanism is easy to verify analytically or numerically in concrete examples as well as to establish diffusion in generic systems we include several applications such as bridging large gaps in a priori unstable models in any dimension and establishing diffusion in cases when the inner dynamics in a non twist map

the coexistence of quasi periodic and blow up solutions in a class of hamiltonian systems abstract the coexistence of quasi periodic solutions and blow up phenomena in a class of higher dimensional duffing type equations is proved in this paper moreover we show that the initial point sets for both kinds of solutions are of infinite lebesgue measure in the phase space for the part of quasi periodic solutions the tool we used is the small twist theorem for higher dimensional cases

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

growth of permutational extensions we study the geometry of a class of group extensions containing permutational wreath products which we call permutational extensions we construct for all k a torsion group kk with growth function v k k n sim exp n 1 1 alpha k qquad 2 3 3 alpha 2 2 2 alpha 2 1 1 alpha 2 and a torsion free group hk with growth function v h k n sim exp log n n 1 1 alpha k these are the first examples of groups of intermediate growth for which the asymptotics of their growth function is known

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

a particular minimization property implies c0 integrability abstract the manifold m being compact and connected and h being a tonelli hamiltonian such that t m is equal to the dual tiered mane set we prove that there is a partition of t m into invariant c 0 lagrangian graphs moreover among these graphs those that are c 1 cover a dense g subset of t m the dynamic restricted to each of these sets is non wandering

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

a new kind of lax oleinik type operator with parameters for time periodic positive definite lagrangian systems in this paper we introduce a new kind of lax oleinik type operator with parameters associated with positive definite lagrangian systems for both the time periodic case and the time independent case on one hand the family of new lax oleinik type operators with an arbitrary u in c m mathbb r 1 as initial condition converges to a backward weak kam solution in the time periodic case while it was shown by fathi and mather that there is no such convergence of the lax oleinik semigroup on the other hand the family of new lax oleinik type operators with an arbitrary u in c m mathbb r 1 as initial condition converges to a backward weak kam solution faster than the lax oleinik semigroup in the time independent case

the dfs fused lasso linear time denoising over general graphs the fused lasso also known as anisotropic total variation denoising is widely used for piecewise constant signal estimation with respect to a given undirected graph the fused lasso estimate is highly nontrivial to compute when the underlying graph is large and has an arbitrary structure but for a special graph structure namely the chain graph the fused lasso or simply 1d fused lasso can be computed in linear time in this paper we revisit a result recently established in the online classification literature herbster et al 2009 cesa bianchi et al 2013 and show that it has important implications for signal denoising on graphs the result can be translated to our setting as follows given a general graph if we run the standard depth first search dfs traversal algorithm then the total variation of any signal over the chain graph induced by dfs is no more than twice its total variation over the original graph this result leads to several interesting theoretical and computational conclusions letting m and n denote the number of edges and nodes respectively of the graph in consideration it implies that for an underlying signal with total variation t over the graph the fused lasso properly tuned achieves a mean squared error rate of t2 3n 2 3 moreover precisely the same mean squared error rate is achieved by running the 1d fused lasso on the dfs induced chain graph importantly the latter estimator is simple and computationally cheap requiring o m operations to construct the dfs induced chain and o n operations to compute the 1d fused lasso solution over this chain further for trees that have bounded maximum degree the error rate of t2 3n 2 3 cannot be improved in the sense that it is the minimax rate for signals that have total variation t over the tree finally several related results also hold for example the analogous result holds for a roughness measure defined by the l0 norm of differences across edges in place of the total variation metric

on the number of mather measures of lagrangian systems in 1996 ricardo ricardo mane discovered that mather measures are in fact the minimizers of a universal infinite dimensional linear programming problem this fundamental result has many applications of which one of the most important is to the estimates of the generic number of mather measures mane obtained the first estimation of that sort by using finite dimensional approximations recently we were able with gonzalo contreras to use this method of finite dimensional approximation in order to solve a conjecture of john mather concerning the generic number of mather measures for families of lagrangian systems in the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem we use a notion of countably rectifiable sets of finite codimension in banach and frechet spaces which may deserve independent interest

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

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

geometric properties of the scattering map of a normally hyperbolic invariant manifold given a normally hyperbolic invariant manifold for a map f whose sta ble and unstable invariant manifolds intersect transversally we consider its associated scattering map that is the map that given an asymptotic orbit in the past gives the asymptotic orbit in the future we show that when f and are symplectic resp exact symplectic then the scattering map is symplectic resp exact symplectic furthermore we show that in the exact symplectic case there are extremely easy formulas for the primitive function which have a variational interpretation as dierence of actions we use this geometric information to obtain ecien t perturbative calculations of the scattering map using deformation theory this perturbation theory generalizes and extends several results already obtained using the melnikov method analogous results are true for hamiltonian o ws the proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems hence the results can be used to obtain intersection properties of objects of any type we also reexamine the calculation of the scattering map in a geodesic o w perturbed by a quasi periodic potential we show that the geometric theory reproduces the results obtained in dls06b using methods of fast slow systems moreover the geometric theory allows to compute perturbatively the dependence on the slow variables which does not seem to be accessible to the previous methods

orbits of unbounded energy in quasi periodic perturbations of geodesic flows abstract we show that certain mechanical systems including a geodesic flow in any dimension plus a quasi periodic perturbation by a potential have orbits of unbounded energy the assumptions we make in the case of geodesic flows are a the metric and the external perturbation are smooth enough b the geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection c the frequency of the external perturbation is diophantine d the external potential satisfies a generic condition depending on the periodic orbit considered in b the assumptions on the metric are c 2 open and are known to be dense on many manifolds the assumptions on the potential fail only in infinite codimension spaces of potentials the proof is based on geometric considerations of invariant manifolds and their intersections the main tools include the scattering map of normally hyperbolic invariant manifolds as well as standard perturbation theories averaging kam and melnikov techniques we do not need to assume that the metric is riemannian and we obtain results for finsler or lorentz metrics indeed there is a formulation for hamiltonian systems satisfying scaling hypotheses we do not need to make assumptions on the global topology of the manifold nor on its dimension

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

on the isolated points in the space of groups we investigate the isolated points in the space of finitely generated groups we give a workable characterization of isolated groups and study their hereditary properties various examples of groups are shown to yield isolated groups we also discuss a connection between isolated groups and solvability of the word problem

lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions the globally positive diffeomorphisms of the 2n dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite for these globally positive diffeomorphisms an aubry mather theory was developed by garibaldi and thieullen that provides the existence of some minimizing measures using the two green bundles g and g that can be defined along the support of these minimizing measures we will prove that there is a deep link between the angle between g and g along the support of the considered measure mu the size of the smallest positive lyapunov exponent of mu the tangent cone to the support of mu

destruction of invariant curves in the restricted circular planar three body problem by using comparison of action the classical principle of least action says that orbits of mechanical systems extremize action an important subclass are those orbits that minimize action in this paper we utilize this principle along with aubry mather theory to construct birkhoff regions of instability for a certain three body problem given by a hamiltonian system of 2 degrees of freedom we believe that these methods can be applied to construct instability regions for a variety of hamiltonian systems with 2 degrees of freedom the hamiltonian model we consider describes dynamics of a sun jupiter comet system and under some simplifying assumptions we show the existence of instabilities for the orbit of the comet in particular we show that a comet which starts close to an orbit in the shape of an ellipse of eccentricity e 0 66 can increase in eccentricity up to e 0 96 in the sequels to this paper we extend the result to beyond e 1 and show the existence of ejection orbits such orbits are initially well within the range of our solar system this might give an indication of why most objects rotating around the sun in our solar system have relatively low eccentricity

transitivity of generic semigroups of area preserving surface diffeomorphisms we prove that the action of the semigroup generated by a cr generic pair of area preserving diffeomorphisms of a compact orientable surface is transitive

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

lasry lions approximations for discounted hamilton jacobi equations abstract we study the lasry lions approximation using the kernel determined by the fundamental solution with respect to a time dependent tonelli lagrangian this approximation process is also applied to the viscosity solutions of the discounted hamilton jacobi equations

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